Unitary Cycles on Shimura Curves

and the Shimura Lift I

Siddarth Sankaran

Received: April 4, 2013

Communicated by Peter Schneider

Abstract. This paper concerns two families of divisors, which we call the ‘orthogonal’ and ‘unitary’ special cycles, defined on integral models of Shimura curves. The orthogonal family was studied extensively by Kudla-Rapoport-Yang, who showed that they are closely related to the Fourier coefficients of modular forms of weight 3/2, while the unitary divisors are analogues of cycles appearing in more recent work of Kudla-Rapoport on unitary Shimura varieties. Our main result relates these two families by (a formal version of) the Shimura lift.

2010 Mathematics Subject Classification: 14G35, 11G18, 11F30.

Contents

1 Introduction  1404
2 Shimura curves and global special cycles  1409
3 Local cycles on the Drinfeld upper half-plane  1415
3.1 Drinfeld space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1416
3.2 The structure of a local unitary cycle . . . . . . . . . . . . . . . 1420
3.3 Relation to local orthogonal special cycles . . . . . . . . . . . . 1436
3.4 p-adic uniformizations . . . . . . . . . . . . . . . . . . . . . . . 1440
4 Proof of the main theorem  1444
4.1 The (formal) Shimura lift . . . . . . . . . . . . . . . . . . . . . 1444
4.2 The Shimura lift formula in fibres of bad reduction . . . . . . . 1445
4.3 The main theorem and applications . . . . . . . . . . . . . . . . 1459

1 Introduction

In a series of works leading up to the monograph [KRY], Kudla, Rapoport and Yang study a family of arithmetic divisors that lie in the first arithmetic Chow groups of integral models of Shimura curves. One of their main results states that if one assembles these divisors into a formal generating series, the result is modular of weight 3/2: in effect, pairing this generating series with a suitable linear functional yields the q-expansion of a weight 3/2 modular form. In this paper, together with its forthcoming sequel, we take up the study of the Shimura lift of this generating series. Our main result relates the Shimura lift to a generating series comprised of ‘unitary’ divisors, that are analogues of the cycles constructed by Kudla-Rapoport in their more recent work [KR3] on unitary Shimura varieties. In the present work, we focus on the geometry of the two families of divisors, and prove that their members satisfy relations that mirror the relations between the Fourier coefficients of a holomorphic modular form and those of its Shimura lift; as we discuss in more detail below, the key step is determining the local structure of the cycles in a formal neighbourhood of a prime of bad reduction. Let B be a rational indefinite quaternion algebra with discriminant D_{B}, and fix a maximal order O_{B}. The Shimura curve C_{B} is the moduli stack over Spec(ℤ) that parametrizes pairs \underline{A} = (A, ι); here A is an abelian scheme over some base scheme S, equipped with an O_{B}-action

ι: O_{B} → End(A)

that satisfies a determinant condition, cf. Definition 2.1. This stack is then an integral model for the classical Shimura curve associated to B. For a positive integer n, Kudla, Rapoport and Yang define (more or less, see Definition 2.3) an ‘orthogonal’ special cycle Z^{o}(n) as the moduli space of diagrams

ξ: A → A,

where A is a O_{B}-abelian surface (i.e. a point of C_{B}), and ξ is a traceless O_{B}- linear endomorphism of A such that ξ^{2} = −n. We may view Z^{o}(n) as a cycle on C_{B} via the natural forgetful map, and form the orthogonal generating series (which, to emphasize the connection to modular forms, we have suggestively written as a q-expansion in terms of the variable τ ∈ ℌ):

Φ^{o}(τ) := Z^{o}(0) + ∑_{n>0} Z^{o}(n) q_{τ}^{n} ∈ Div(C_{B})⟦q_{τ} ⟧.

Here Div(C_{B}) is the group of divisors on C_{B}, i.e., the free abelian group generated by the closed irreducible substacks of C_{B} that are étale-locally Cartier, and Z^{o}(0) is an appropriate constant term, cf. (4.23). The ‘unitary’ cycles are also defined by a moduli problem. Let k = ℚ(\sqrt{Δ}) be an imaginary quadratic field, where Δ < 0 is squarefree, and denote its ring of integers by o_{k}. Throughout this paper, we assume that (i) Δ is even and (ii) every prime p|D_{B} is inert in k. The second condition implies the existence of an embedding ϕ: o_{k} → O_{B}, and we fix one such embedding for the moment. Consider the moduli stack E^{+} over Spec(o_{k}) that parametrizes tuples

\underline{E} = (E, i_{E}, λ_{E}).

Here E is an elliptic curve over some base scheme S over Spec(o_{k}), endowed with an o_{k}-action

i_{E}: o_{k} → End(E)

and a compatible principal polarization λ_{E}. We also assume that on the Lie algebra Lie(E), the action induced by i_{E} agrees with the structural morphism o_{k} → O_{S}. Let E^{−} denote the moduli space of tuples \underline{E} = (E, i_{E}, λ_{E}) as before, except now we insist that on Lie(E), the action of i_{E} is given by the conjugate of the structural morphism. Finally we take

E = E^{+} ∐ E^{−}

to be the disjoint union of these two stacks over Spec(o_{k}). Fix a base scheme S over o_{k}, and suppose we are given a pair of S-points \underline{E} ∈ E(S) and \underline{A} = (A, ι) ∈ C_{B}(S). Following [KR3], we form the space of special homomorphisms

Hom_{o_{k},ϕ}(E, A) := {y ∈ Hom_{S}(E, A) | y ∘ i_{E}(a) = ι (ϕ(a)) ∘ y for all a ∈ o_{k}},

which is equipped with an o_{k}-hermitian form h_{E,A}^{ϕ}, cf. (2.2). For an integer m > 0, the unitary special cycle Z(m, ϕ) is the moduli stack over Spec(o_{k}) that parametrizes tuples (\underline{E}, \underline{A}, y), where \underline{E} and \underline{A} are as above and

y ∈ Hom_{o_{k},ϕ}(E, A)  such that h_{E,A}^{ϕ}(y, y) = m.  (1.1)

Again, we may view Z(m, ϕ) as a cycle on C_{B/o_{k}} := C_{B} × Spec(o_{k}) via the natural forgetful map, and in fact it turns out that Z(m, ϕ) is a divisor. We also define the following rescaled version:

Z^{∗}(m, ϕ) := Z (\frac{m}{gcd(m, D_{B})}, ϕ)

Finally, we form the unitary generating series:

Φ^{u}(τ) := Z(0) + \frac{1}{2h(k)} ∑_{m>0} ( ∑_{[ϕ]∈Opt/ O_{B}^{×}} Z(m, ϕ) + Z^{∗}(m, ϕ)) q_{τ}^{m}

∈ Div(C_{B/o_{k}}) ⟦q_{τ} ⟧ ⊗_{ℤ} ℚ

for an appropriate constant term Z(0), cf. (4.24), and the sum on [ϕ] runs over an equivalence class of optimal embeddings (cf. the notation section below).

Our main theorem describes the relationship between Φ^{o} and Φ^{u} in terms of the Shimura lift, which is a classical operation on modular forms. It takes as its input a modular form F of half-integral weight together with a squarefree integer parameter t, and yields a modular form Sh(F) of even integral weight. Moreover, when F = ∑ a(n)q^{n} is holomorphic, there are explicit formulas (depending on t together with the weight, level, and character) for the Fourier coefficients of Sh(F) in terms of those Fourier coefficients of F that are of the form a(tm^{2}). We define the formal Shimura lift to be the operator on formal power series determined by these formulas, cf. §4.1.

Main Theorem (see Theorem 4.10). Suppose k = ℚ(\sqrt{Δ}), where Δ < 0 is a squarefree even integer and assume further that every prime dividing D_{B} is inert in k. Let Φ_{/o_{k}}^{o} denote the generating series

Φ_{/o_{k}}^{o}(τ) = Z^{o}(0)_{/o_{k}} + ∑_{n>0} Z^{o}(n)_{/o_{k}} q_{τ}^{n} ∈ Div(C_{B/o_{k}})⟦q_{τ} ⟧

obtained by taking the base change to o_{k} of each coefficient. Then we have an equality of formal generating series

Sh(Φ_{/o_{k}}^{o})(τ) = Φ^{u}(τ),  (1.2)

in Div(C_{B/o_{k}})⟦q_{τ} ⟧ ⊗ ℚ.

We now give an outline of the proof. It turns out that both the orthogonal and unitary divisors have vertical components only at primes p|D_{B}. This leads us to study the p-adic uniformization of the Shimura curve C_{B}, which relates the formal completion along its fibre at such a prime p to (a formal model of) the Drinfeld p-adic upper half-plane D. We also have p-adic uniformizations for the orthogonal and unitary special cycles, which are expressed in terms of linear combinations of analogous ‘local’ cycles defined on D. Let 𝔽 = 𝔽_{p}^{alg} be an algebraic closure of 𝔽_{p}, and let W = W (𝔽) the ring of Witt vectors. Denote by Nilp the category of W -schemes such that p is locally nilpotent, and for a scheme S ∈ Nilp, let \overline{S} := S ×_{W} 𝔽. Then the Drinfeld upper half-plane D parametrizes tuples \underline{X} = (X, ι_{X}, ρ_{X}), where X is a p-divisible group of height 4 and dimension 2, over a base scheme S ∈ Nilp, together with a ‘special’ action ι_{X}: O_{B,p} → End(X) of the maximal order O_{B,p} (cf. §3.1); finally ρ_{X} is an O_{B,p}-linear quasi-isogeny

ρ_{X}: X ×_{S} \overline{S} → 𝕏 ×_{𝔽}\overline{S}

of height 0, where 𝕏 is some fixed p-divisible group over 𝔽 endowed with a special O_{B,p}-action. In this picture, the local analogues of the orthogonal and unitary cycles are described as the deformation loci of homomorphisms of pdivisible groups. A complete description of the orthogonal cycles can be found in [KR1], and so our aim in Section 3 is provide the same for the unitary cycles. Recent work of Kudla and Rapoport [KR4] describes the special fibre of D in terms of the Bruhat-Tits tree for SU(C), where C is the split 2-dimensional hermitian space over k_{p} (recall that by assumption, k_{p} is an unramified quadratic extension of ℚ_{p}). Combining this description with a healthy dose of Grothendieck-Messing theory, we are able to write down explicit equations for a local unitary cycle as a closed formal subscheme of D, and we consequently obtain a precise description of its irreducible components. This description parallels the one found in [KR1] for the orthogonal cycles, and by comparing the two formulas, we obtain the following key result (Theorems 3.17 and 3.19): any local orthogonal cycle that appears in the p-adic uniformization of an orthogonal cycle Z^{o}(|Δ|n^{2}) can be expressed as sum of two (explicitly determined) local unitary cycles. We now describe the p-adic uniformizations. If we fix an 𝔽-valued point \underline{A} = (A, ι_{A}) ∈ C_{B}(𝔽), then the space of O_{B}-linear quasi-endomorphisms

B′ := End_{O_{B}}(A)_{ℚ}

is a definite quaternion algebra over ℚ with discriminant D_{B}/p. The p-adic uniformization of the Shimura curve C_{B} can be expressed in the following way: there is a finite subgroup Γ′ ⊂ (B′)^{×} acting on D, such that if we let C̃_{B} denote the base change to W of the formal completion of C_{B} along its fibre at p, then there is an isomorphism

C̃_{B} ≃ [Γ′\ D]

of formal stacks^{1} over W. Similarly, let Z̃^{o}(n) and Z̃(m, ϕ) denote the base change to W of the formal completions of the special cycles Z^{o}(n) and Z(m, ϕ) along their fibres at p. Viewed as a cycle on C̃_{B}, we may express an orthogonal cycle as a sum

Z̃^{o}(n) = ∑_{\begin{matrix}ξ∈Ω^{o}(n) \\ mod Γ′ \end{matrix}}[Z^{o}(ξ[p^{∞}])]

where [Z^{o}(ξ[p^{∞}])] is the projection to [Γ′\ D] of a local orthogonal cycle on D, and

Ω^{o}(n) ⊂ {b′ ∈ B′ | T r(b′) = 0}

is a Γ′-invariant set of vectors of reduced norm n satisfying a certain integrality property, cf. Theorem 3.21. For the unitary cycles, we fix a triple \underline{E} ∈ E(𝔽). A unitary cycle then decomposes as

Z̃(m, ϕ) = \frac{1}{|o_{k}^{×}|} ∑_{[𝔞]∈Cl(k)} ( ∑_{\begin{matrix}β∈Ω^{+}(m,𝔞,ϕ) \\ mod Γ′ \end{matrix}}[Z(β[p^{∞}])] + ∑_{\begin{matrix}β′∈Ω^{−}(m,𝔞,ϕ) \\ mod Γ′ \end{matrix}}[Z(β′[p^{∞}])]) ,

1Here and throughout this paper, we use the term ‘isomorphism of formal stacks’ in a rather cavalier fashion. What we mean in all cases is the following: upon fixing sufficiently deep level structure away from p, one obtains a formal scheme on each side, together with a covering map from the corresponding stack. Our assertion is that there is an isomorphism of formal schemes compatible with the automorphisms of the covering maps, and the morphisms induced by varying the level structure, in the natural way.

Documenta Mathematica 18 (2013) 1403–1464

where Cl(k) is the class group of k, and as before [Z(β[p^{∞}])] and [Z(β′[p^{∞}])] are local unitary cycles. The sets Ω^{±}(m, 𝔞, ϕ) appearing above are subsets

Ω^{±}(m, 𝔞, ϕ) ⊂ Hom(E, A) ⊗_{ℤ} ℚ,

consisting of quasi-morphisms of a specified norm, a linearity (or anti-linearity, in the case of Ω^{−}) condition with respect to the action of o_{k}, and again satisfying an integrality condition, cf. Theorem 3.22. Thus, in order to compare the unitary and orthogonal cycles, we need to compare the indexing sets Ω^{o}(n) and Ω^{±}(m, 𝔞, ϕ), at least in the case that the squarefree part of n is equal to |Δ|, and as ϕ varies among classes of optimal embeddings. This task, which amounts to a study in the arithmetic of quaternion algebras, is carried out in §4.2. Together with the description of the local cycles discussed previously, we arrive at a relationship between the two families of cycles, in a formal neighbourhood at p|D_{B}, that matches exactly the formula for the Fourier coefficients of the Shimura lift. Since the vertical components of the cycles only occur at such primes, the main theorem follows immediately, cf. Theorem 4.10. I would like to conclude the introduction by placing this result in the context of the sequel [San2] to the present work, whose aim is to establish the same Shimura lift formula in the first arithmetic Chow group of C_{B} (in the sense of Gillet-Soulé). Recall that in [KRY], the authors prove that the generating series

Φ̂^{o}(τ) := ∑_{n∈ℤ} Ẑ^{o}(n, v) q_{τ}^{n}, ℑ(τ) = v

is a non-holomorphic modular form of weight 3/2; here the coefficients are arithmetic classes

Ẑ^{o}(n, v) = (Z^{o}(n), Gr^{o}(n, v)) ∈ ĈH^{1}(C_{B})

for appropriate choices of Green functions Gr^{o}(n, v), and constant term Ẑ^{o}(0, v). The modularity of Φ̂^{o}(τ) means that in particular, applying a linear functional to its coefficients yields the q-expansion of a modular form in the usual sense (in fact, their proof of modularity amounts to showing that this is true for a well-chosen set of linear functionals). In [San2], we augment the unitary cycles with Green functions Gr(m, v, ϕ) and obtain classes

Ẑ(m, v, ϕ) = (Z(m, ϕ), Gr(m, v, ϕ)) ∈ ĈH^{1}(C_{B})

and, together with an appropriate constant term Ẑ(0, v), define

Φ̂^{u}(τ) := Ẑ(0, v) + \frac{1}{2h(k)} ∑_{m∈ℤ} ∑_{[ϕ]} [Ẑ(m, v, ϕ) + Ẑ^{∗} (m, v, ϕ)] q^{m}.

The main result of [San2] identifies this generating series with the Shimura lift of Φ̂^{o}, by combining the geometric results of the present work with calculations that relate the Green functions. In particular, applying a linear functional to both series yields an identity involving the classical Shimura lift of modular forms. Some linear functionals are ‘geometric’ in nature, i.e., they only depend on the cycles and not the Green functions. When this is the case, the main result of the present work already implies that the Shimura lift relation holds for the corresponding modular forms, and we give a few examples at the end of this paper that will play a role in the sequel.

Acknowledgments: This paper is an extension of the work carried out in my Ph. D. thesis; I am grateful to my advisor S. Kudla for introducing me to the subject, and his consistent encouragement. I would also like to thank B. Howard, M. Rapoport, and the anonymous referee for helpful comments. This work was supported by the SFB/TR 45 ‘Periods, Moduli Spaces, and Arithmetic of Algebraic Varieties’ of the DFG (German Research Foundation).

Notation:

• B is an indefinite quaternion algebra over ℚ, with discriminant D_{B}, and with reduced trace and norm denoted by T rd and Nrd respectively. We fix a maximal order O_{B} ⊂ B.

• k = ℚ(\sqrt{Δ}) is an imaginary quadratic field, with ring of integers o_{k}. We denote the non-trivial Galois automorphism by a ↦ a′. We also assume throughout this paper that

(i) Δ < 0 is a squarefree even integer; (ii) and every prime p|D_{B} is inert in k. In particular, D_{B} is odd.

• By definition, an embedding ϕ: o_{k} ↪ O_{B} is optimal if

ϕ(o_{k}) = ϕ(k) ∩ O_{B} .  (1.3)

Let Opt denote the set of optimal embeddings. Note that O_{B}^{×} acts on Opt by conjugation: for ξ ∈ O_{B}^{×}, set

(ξ · ϕ)(a) := (Ad_{ξ} ∘ ϕ)(a) = ξ · ϕ(a) · ξ^{−1}, a ∈ o_{k}.

Denote by Opt/ O_{B}^{×} the set of equivalence classes of optimal embeddings under this action.

• χ_{k} is the quadratic character associated to k, so that for a prime p,

χ_{k}(p) = \begin{cases}1, & if p splits, \\ 0, & if p is ramified, \\ −1, & if p is inert. \end{cases}  (1.4)

2 Shimura curves and global special cycles

In this section, we recall the construction of Shimura curves, and the global orthogonal and unitary special cycles on them.

Definition 2.1 (Shimura curve). Let C_{B} denote the moduli problem which associates to a scheme S over Spec(ℤ) the category whose objects are pairs

C_{B}(S) = {(A, ι)},

where (i) A is an abelian surface over S, and (ii) ι: O_{B} → End_{S}(A) is an action of O_{B} on A. We also require that for every b ∈ O_{B},

det(T − ι(b)|_{Lie(A)}) = T^{2} − T rd(b)T + Nrd(b) ∈ O_{S}[T].

Proposition 2.2 (cf. [KRY], Proposition 3.1.1). The moduli problem C_{B} is representable by a Deligne-Mumford (DM) stack, which we also denote by C_{B}. It is regular, proper and flat over Spec(ℤ) of relative dimension 1, and smooth over Spec ℤ[D_{B}^{−1}].

The orthogonal special cycles, as constructed in e.g. [KRY], are also defined by a moduli problem:

Definition 2.3. For n ∈ ℤ_{>0}, let Z^{o}(n)^{♯} denote the DM stack which represents the following moduli problem over Spec ℤ: to a scheme S/ ℤ, we let Z^{o}(n)^{♯}(S) be the category of tuples (A, ι, ξ), where

(i) (A, ι) ∈ C_{B}(S), and (ii) ξ ∈ End_{O_{B}}(A) is an O_{B}-linear endomorphism of A with T r(ξ) = 0 and

ξ^{2} = −n. We view Z^{o}(n)^{♯} as a cycle on C_{B} via the natural forgetful map Z^{o}(n)^{♯} → C_{B}, which is finite and unramified by [KRY, §3.4]. Let Z^{o}(n) denote the Cohen-Macauleyfication of Z^{o}(n)^{♯}, as in p. 55 of loc. cit., so that Z^{o}(n) is of pure codimension 1 in C_{B}.

We now turn to the unitary special cycles, following [KR3]. Fix once and for all an element θ ∈ O_{B} such that θ^{2} = −D_{B}. Given a point (A, ι) ∈ C_{B}(S), there exists a unique principal polarization λ_{A}^{0} on A such that the Rosati involution φ ↦ φ^{†} satisfies

ι(b)^{†} = (λ_{A}^{0})^{−1} ∘ ι(b)^{∨} ∘ λ_{A}^{0} = ι(θ^{−1} b^{ι} θ), for all b ∈ O_{B},

cf. [How, §3.1].

Lemma 2.4. Let ϕ: o_{k} → O_{B} be an embedding. Then

(i) T rd(θϕ(\sqrt{Δ})) ≠ 0; (ii) Assume, without loss of generality, that T rd(θϕ(\sqrt{Δ})) > 0, by replacing

θ by −θ if necessary. Then the isogeny

λ_{A,ϕ} := λ_{A}^{0} ∘ ι(θ ϕ(\sqrt{Δ})).  (2.1)

is a (non-principal) polarization.

Proof. (i) Since θ^{2} and Δ are both negative, the statement follows immediately from the assumption that B is indefinite. (ii) We need to check that on geometric points, the map λ_{A,ϕ} is induced by an ample line bundle; therefore we assume A is an abelian variety over an algebraically closed field. The endomorphism ι(θϕ(\sqrt{Δ})) is symmetric with respect to †, and so lies in the image of the map

NS(A) → End(A)

where NS(A) is the Néron-Severi group. An element in this image comes from an ample bundle if and only if it is totally positive, i.e. its characteristic polynomial has positive real roots, cf. [Mum, §21]. The roots of the characteristic polynomial are the same as those of its minimal polynomial, cf. loc. cit. p. 203, and ι(θϕ(\sqrt{Δ})) has minimal polynomial

P (x) = x^{2} − T rd(θϕ(\sqrt{Δ})) · x + Nrd(θϕ(\sqrt{Δ})).

We may fix an isomorphism B ⊗ ℝ ≃ M_{2}(ℝ) such that T rd and Nrd are identified with the trace and determinant respectively, and the positive involution

b ↦ b^{†} = θ^{−1} · b^{ι} · θ

is identified with the transpose operator. Then θϕ(\sqrt{Δ}) is identified with a symmetric matrix A = A^{t}; the conditions tr(A) > 0 and det(A) = |Δ|D_{B} > 0 then imply that the roots of P (x) are real and positive.

Note that the Rosati involution ∗ associated to the polarization λ_{A,ϕ} satisfies

ι_{A} (ϕ(a))^{∗} = ι_{A} (ϕ(a′)) , for all a ∈ o_{k}.

Let ℰ^{+} denote the moduli stack over Spec(o_{k}) such that for a base scheme S/o_{k} , the S-points parametrize tuples

ℰ^{+}(S) = \{\underline{E} = (E, i_{E}, λ_{E})\};

here E is an elliptic curve over S endowed with an action i_{E}: o_{k} → End(E), and a principal polarization λ_{E} such that the induced Rosati involution ^{∗} satisfies

i_{E}(a)^{∗} = i_{E}(a′).

We further impose the condition that the action of o_{k} on Lie(E) induced by i_{E} agrees with the action given via the structural morphism o_{k} → O_{S}. Similarly, we define ℰ^{−} to be the moduli space of tuples \underline{E} = (E, i_{E}, λ_{E}) as above, except we require that the action on Lie(E) induced by i_{E} is equal to the conjugate of the structural morphism. Finally, we take

ℰ = ℰ^{+} ∐ ℰ^{−}

to be the disjoint union of these stacks. Suppose S is a scheme over o_{k}, and we are given two points \underline{A} ∈ C_{B}(S) and \underline{E} ∈ ℰ (S). We form the space of special homomorphisms:

Hom_{ϕ}(\underline{E}, \underline{A}) := \{y ∈ Hom(E, A) | y ∘ i_{E}(a) = ι_{A}(ϕ(a)) ∘ y, for all a ∈ o_{k}\}

This space comes equipped with an o_{k}-hermitian form h_{E,A}^{ϕ} defined by

h_{E,A}^{ϕ}(s, t) := (λ_{E})^{−1} ∘ t^{∨} ∘ λ_{A,ϕ} ∘ s ∈ End(E, i_{E}) ≃ o_{k}.  (2.2)

Definition 2.5 (Unitary special cycles.). Suppose m ∈ ℤ_{>0} and ϕ: o_{k} → O_{B} is an optimal embedding. Let Z(m, ϕ) denote the DM stack over Spec(o_{k}) representing the following moduli problem: for a scheme S/o_{k}, we define Z(m, ϕ)(S) to be the category of tuples

Z(m, ϕ)(S) = \{(\underline{E}, \underline{A}, y)\}

where (i) \underline{E} ∈ ℰ (S), (ii) \underline{A} ∈ C_{B}(S), and (iii) y ∈ Hom_{o_{k},ϕ}(\underline{E}, \underline{A}) such that h_{E,A}^{ϕ}(y, y) = m.

By the proof of [KR3, Proposition 2.10], which applies verbatim to the present setting, the forgetful map

Z(m, ϕ) → C_{B/o_{k}} := C_{B} ×_{ℤ} Spec(o_{k})

is finite and unramified, and so we may view Z(m, ϕ) as a cycle on C_{B/o_{k}}; abusing notation, we shall refer to the cycle by the same symbol, and hope that context will suffice to clarify which instance of the notation is intended.

Proposition 2.6. The cycle Z(m, ϕ) is a divisor on C_{B/o_{k}}, i.e., each irreducible component is of codimension 1. Moreover, it has vertical components in characteristic p if and only if (i) p divides D_{B} and (ii) ord_{p}m > 0.

Proof. By the complex uniformization [San, Theorem 3.3.3], the irreducible components of the generic fibre Z(m, ϕ)_{k} are 0-dimensional, and so their closures in C_{B/o_{k}} (i.e. the horizontal components) are 1-dimensional. Suppose p|D_{B}. In Section 3.2 below, we show that Z(m, ϕ) is a divisor in a formal neighbourhood of the fibre at p by writing down explicit equations. In particular, Theorem 3.14 asserts that Z(m, ϕ) contains vertical irreducible components in this fibre if and only if ord_{p}(m) > 0. It remains to consider the fibre at primes 𝔭 ⊂ o_{k} with (𝔭, D_{B}) = 1. Suppose z ∈ Z(m, ϕ)(𝔽) is a geometric point corresponding to a triple (\underline{E}, \underline{A}, y) over 𝔽, for an algebraic closure 𝔽 of o_{k}/𝔭. Let x ∈ C_{B/o_{k}}(𝔽) denote the point below z. We need to prove that (i) the cycle Z(m, ϕ) is a divisor at x and (ii) there are no vertical components, i.e., that the data (\underline{E}, \underline{A}, y) can be lifted to characteristic zero and Z(m, ϕ) does not contain the entire fibre at 𝔭.

Let W be the completion of the maximal unramified extension of the local ring o_{k,𝔭}, and denote by ART the category of local Artinian W-algebras with residue field 𝔽. The natural morphism

o_{k,𝔭} → W

endows any W-algebra with an o_{k,𝔭} structure, and so in particular every object of ART can be viewed as an o_{k}-algebra in this way. Let R_{z} and R_{x} denote the étale local rings of z and x respectively. Then R_{x} pro-represents the functor of deformations of the data \underline{A} = (A, ι_{A}) to objects of ART. By the Serre-Tate theorem, giving a deformation of \underline{A} is equivalent to giving a deformation of the underlying p-divisible group (with induced O_{B} ⊗ ℤ_{p}- action):

\underline{X} = (X, ι_{X}) := (A[p^{∞}], ι_{A} ⊗ ℤ_{p}).

Similarly, the local ring R_{z} pro-represents deformations of the data (\underline{E}, \underline{A}, y), which in turn are equivalent to deformations of

(\underline{Y} , \underline{X}, b) := (\underline{E}[p^{∞}], \underline{A}[p^{∞}], y[p^{∞}]),

where \underline{Y} = (Y, i_{Y} , λ_{Y}) is the p-divisible group attached to E, together with its induced o_{k} ⊗ ℤ_{p} action, and principal polarization. More precisely, the O_{B}-action on A and the o_{k}-action on E induce actions

ι_{X}: O_{B} ⊗ℤ_{p} → End_{ℤ_{p}}(X),  and  i_{Y} : o_{k} ⊗ ℤ_{p} → End_{ℤ_{p}}(Y)

respectively, such that

b ∘ i_{Y} (a) = ι_{X}(ϕ(a)) ∘ b,  for all a ∈ o_{k,p} := o_{k} ⊗ ℤ_{p} .  (2.3)

Recalling that p ∤ D_{B}, we may fix an isomorphism O_{B} ⊗ ℤ_{p} ≃ M_{2}(ℤ_{p}) such that ϕ(\sqrt{Δ}) is identified with the matrix ( \begin{matrix}& 1 \\ Δ \end{matrix} ); note that when p = 2, this is possible on account of the assumption that |Δ| is even. The two idempotents

e_{1} = (\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix})  and  e_{2} = (\begin{matrix}0 & 0 \\ 0 & 1 \end{matrix})

determine a splitting X = X^{o} × X^{o}, where X^{o} is a p-divisible group of dimension 1 and height 2; keep in mind that X^{o} itself does not inherit any additional endomorphism structure. In a similar manner, any deformation of \underline{X} = (X, ι_{X}) decomposes into a product of two copies of a deformation of X^{o}. Hence, Def(X, ι_{X}) = Def(X^{o}), and it is a well-known fact, cf. e.g. [MZ, Theorem 3.8], that the latter deformation problem is pro-represented by Spf(W⟦t⟧). Our aim is to analyze the deformation locus Def(\underline{Y} , \underline{X}, b) as a closed formal subscheme of Def(\underline{X}) = Def(X^{o}) = Spf(W⟦t⟧). Under the splitting X = X^{o} × X^{o}, we have a decomposition b = (b_{1}, b_{2}), where b_{i} ∈ Hom(Y, X^{o}) are non-zero morphisms. Morever, because of (2.3),

b_{2} = b_{1} ∘ i_{Y} (\sqrt{Δ}).

Thus, the problem of deforming (\underline{Y} , \underline{X}, b) is equivalent to deforming (\underline{Y} , X^{o}, b_{1}). The key tool is the theory of canonical lifts: given a p-divisible group 𝔤 over 𝔽 of height 2 and dimension 1 together with an action

i_{𝔤}: o_{k,p} → End(𝔤),

there exists a canonical lift \underline{𝔊} = (𝔊, i_{𝔊}) over W. This fact is a consequence of the existence of Serre-Tate canonical coordinates in the case when 𝔤 is ordinary, cf. [Mes, Appendix], and of Gross’ theory of (quasi-)canonical liftings when 𝔤 is supersingular, cf. [Gro]. Moreover, for any object R ∈ ART, there exists a unique lift of \underline{𝔤} to R: namely, the base change \underline{𝔊}_{/R} of \underline{𝔊} to R. In particular, if we let \underline{𝔜} be the canonical lift of \underline{Y} as above, then the problem of deforming (\underline{Y} , X^{o}, b_{1}) to R ∈ ART is equivalent to finding deformations 𝔛^{o} of X^{o} such that b_{1} lifts to a morphism 𝔜_{/R} → 𝔛^{o}. It is not hard to show that the locus in Def(X^{o}) where b_{1} lifts is cut out by a single equation, cf. the proof of [Wew, Prop. 5.1]. Note that in our case, this equation cannot be 0. If it were, then R_{x} = R_{z}, and so Z(m, ϕ) would contain a positive-dimensional component in the generic fibre; this would contradict the description of the complex points above. Moreover, we claim Z(m, ϕ) cannot contain the entire fibre (C_{B})_{𝔭}. Note that if X^{o} and Y are isogenous, they are either both super-singular or both ordinary. But Y is necessarily ordinary when 𝔭 is a split prime, and supersingular otherwise, as it admits an action of o_{k} ⊗ ℤ_{p}; thus taking X^{o} to be supersingular in the first case, or ordinary in the second, yields a point in the fibre (C_{B})_{𝔭} that cannot lie on Z(m, ϕ). It remains to show that there always exists a lift to characteristic 0. Suppose first that X^{o} and Y are ordinary. Then p splits in k and End(X^{o}) = End(Y) = o_{k,p}. Fixing an isomorphism X^{o} ≃ Y, we have

b_{1} ∈ Hom(Y, X^{o}) ≃ End(X^{o}) ≃ o_{k,p},

which lifts to an endomorphism of the canonical lift 𝔛^{o} of X^{o}. Next, we consider the case where X^{o} and Y are supersingular; this necessarily implies that p is non-split in k. Let D be the division quaternion algebra over ℚ_{p} with maximal order denoted by O_{D}, and fix a uniformizer Π ∈ O_{D}. Suppose first that p = 𝔭^{2} is ramified in k, and let ϖ ∈ o_{k,p} be a uniformizer. Then we may identify

End(Y) ≃ O_{D}

such that i_{Y} (ϖ) is identified with Π. We may also fix an isomorphism

X^{o} ≃ Y,  (2.4)

which induces isomorphisms

End(X^{o}) ≃ Hom(Y, X^{o}) ≃ End(Y) ≃ O_{D} in the natural way. Without loss of generality, we may normalize (2.4) such that b_{1} is identified with Π^{t} ∈ O_{D} for some t. We let i_{X^{o}}: o_{k,p} → End(X^{o}) denote the action given by composing i_{Y} with the isomorphism (2.4). Appealing again to Gross’ results, this action determines a canonical lifting \underline{X}^{o} = (X^{o}, i_{X^{o}}) of (X, i_{X}) to W, and it is clear by construction that b_{1} also lifts. Finally, we suppose that p = 𝔭 is inert in k. We may fix an identification End(Y) ≃ O_{D} such that

Π · i_{Y} (a) = i_{Y} (a′) · Π, for all a ∈ o_{k,p}.

As before, we may also fix an isomorphism α: X^{o} ⟶^{∼} Y such that b_{1} is identified with Π^{t} for some t. We then define an action i_{X^{o}}: o_{k,p} → End(X^{o}) by the formula:

i_{X^{o}}(a) := \begin{cases}α^{−1} ∘ i_{Y} (a) ∘ α,  if t is even, \\ α^{−1} ∘ i_{Y} (a′) ∘ α,  if t is odd. \end{cases}

By construction, the morphism b_{1} will lift to a morphism 𝔜 → 𝔛^{o}, where 𝔜 and 𝔛^{o} are the canonical lifts of Y and X^{o} to W determined by their respective o_{k,p}-actions as above.

3 Local cycles on the Drinfeld upper half-plane

In this section, we discuss the local analogues of the unitary and orthogonal special cycles. Suppose p|D_{B}, so that in particular p is inert in k and p ≠ 2. Let k_{p} denote the completion at p, and let δ ∈ k_{p} denote the image of \sqrt{Δ}. Throughout this section, we fix an embedding

ϕ: o_{k,p} ↪ O_{B,p},  (3.1)

where o_{k,p} and O_{B,p} are the maximal orders in k_{p} and B_{p} respectively. We also fix a uniformizer Π ∈ O_{B,p} such that Πϕ(a) = ϕ(a′)Π, for all a ∈ o_{k,p}. Let 𝔽 = 𝔽_{p}^{alg} denote a fixed algebraic closure of 𝔽_{p}, and W = W (𝔽) the ring of Witt vectors, with Frobenius endomorphism σ: W → W. We fix an embedding τ_{0} : o_{k,p}/(p) → 𝔽, which lifts uniquely to an embedding τ_{0} : o_{k,p} → W. In addition, we let τ_{1} denote the conjugate embedding (or its lift to characteristic 0). Note that these maps allow us to view 𝔽 and W as o_{k}-algebras. Let Nilp denote the category of W -schemes for which the ideal sheaf generated by p is locally nilpotent, and for a scheme S ∈ Nilp, we set

\overline{S} := S ×_{W} Spec(𝔽).

Additionally, we fix a trivialization of the prime-to-p roots of unity in 𝔽:

𝔸_{f}^{p}(1) := (∏_{ℓ≠p} \underleftarrow{lim}_{n} μ_{ℓ^{n}}(𝔽)) ⊗_{ℤ} ℚ ≃ 𝔸_{f}^{p}

Finally, we set ℤ̂^{p} := ∏_{ℓ≠p} ℤ_{ℓ}, and if M is any ℤ-module, we define M̂^{p} := M ⊗_{ℤ} ℤ̂^{p}.

3.1 Drinfeld space

In this section, we recall the definition of Drinfeld space as a moduli space of p-divisible groups, as well as an ‘alternative’ description of its special fibre as a union of projective lines indexed by hermitian lattices, as in [KR4]. Following [BC], we define a special formal O_{B,p}-module over a scheme S to be a pair (X, ι_{X}) consisting of a p-divisible group X of height 4 and dimension 2 over S, and a map

ι_{X}: O_{B,p} → End(X),

which satisfies the following special condition: the Lie algebra Lie(X) is (locally on S) a free O_{S} ⊗_{W,τ_{0}} o_{k,p} module of rank 1. Here Lie(X) is viewed as an O_{S} ⊗ o_{k,p}-module via the action of o_{k,p} on X induced by the composition ι_{X} ∘ ϕ. Fix once and for all a special formal O_{B,p}-module (𝕏, ι_{𝕏}) over 𝔽; note such a pair is unique up to isogeny, by the classification of Dieudonné isocrystals over 𝔽 [BC, Proposition II.5.2]. The pair (𝕏, ι_{𝕏}) serves as a “base point” for the following moduli problem:

Definition 3.1. Let D denote the moduli problem on Nilp that associates to a scheme S the category of isomorphism classes of tuples

D(S) = {(X, ι_{X}, ρ_{X})} / ≃,

consisting of • a special formal O_{B}-module (X, ι_{X}) over S,

• and an O_{B,p}-linear quasi-isogeny

ρ_{X}: X ×_{S} \overline{S} → 𝕏 ×_{Spec(𝔽)}\overline{S}

of height 0.

An isomorphism between two tuples (X, ι, ρ) and (X′, ι′, ρ′) is an isomorphism α: X → X′ which is O_{B,p}-equivariant, and such that ρ = ρ′ ∘ (α ×_{S} S_{0}). The functor D is then represented by (a formal model of) the Drinfeld upperhalf plane, and in particular is a formal scheme over Spf(W); see [BC] for a discussion of this result. Next, we recall the ‘alternative’ description of the reduced locus of D as given in [KR4]. Crucial to this description is the following theorem:

Theorem 3.2 (Drinfeld, cf. §III.4 of [BC]). Suppose (X, ι_{X}) is a special formal O_{B,p}-module, over any base S ∈ Nilp. Then there exists a principal polarization λ_{X}^{0} on X such that

(λ_{X}^{0})^{−1} ∘ ι_{X}(b)^{∨} ∘ λ_{X}^{0} = ι_{X} (Π b^{ι} Π^{−1})  for all b ∈ O_{B,p};  (3.2)

here the map b ↦ b^{ι} is the involution of B. Moreover, λ_{X}^{0} is unique up to multiplication by ℤ_{p}^{×}.

For the base point 𝕏, we shall fix once and for all a polarization λ_{𝕏}^{0} as in Drinfeld’s theorem. Then, for any point (X, ι_{X}, ρ_{X}) ∈ D(S), there is a unique principal polarization λ_{X}^{0} satisfying (3.2), and such that the diagram

X ×_{S} \overline{S} ⟶^{λ_{X}^{0}×\overline{S}} X^{∨} ×_{S} \overline{S}
ρ_{X} ↓  ↓^{ρ_{X}^{∨}}
𝕏 ×_{𝔽}\overline{S} ⟶_{λ_{𝕏}^{0}×\overline{S}} 𝕏^{∨} ×_{𝔽} \overline{S}

commutes. Thus, for any point (X, ι_{X}, ρ) ∈ D(S), we may associate another polarization λ_{X,ϕ} by the formula

λ_{X,ϕ} := λ_{X}^{0} ∘ ι_{X} (Π ϕ(δ)) .  (3.3)

Note that the Rosati involution ∗ induced by λ_{X,ϕ} has the property

ι_{X} (ϕ(a))^{∗} = ι_{X} (ϕ(a′)) , for all a ∈ o_{k,p}.

Let M(𝕏) be the Dieudonné module of 𝕏 over W = W (𝔽), with Frobenius and Verschiebung operators denoted by F and V respectively. We have a decomposition

M(𝕏) = M(𝕏)_{0} ⊕ M(𝕏)_{1},

where

M(𝕏)_{i} := {m ∈ M(𝕏) | (ι_{𝕏} ∘ ϕ)(a) · m = τ_{i}(a)m, for all a ∈ o_{k,p}} . (3.4)

Let N(𝕏) := M(𝕏) ⊗_{ℤ} ℚ be the rational Dieudonné module, with induced decomposition

N(𝕏) = N(𝕏)_{0} ⊕ N(𝕏)_{1}.

Note that the operator pV^{−2} = V^{−1}F: N(𝕏)_{0} → N(𝕏)_{0} is a σ^{2}-linear operator, and hence the space of invariants

C := (N(𝕏)_{0})^{V^{−1}F =1}  (3.5)

is a two-dimensional vector space over k_{p}, where k_{p} acts on C via the embedding τ_{0} : k_{p} → W. The polarization λ_{𝕏,ϕ} as defined in (3.3) induces an alternating pairing

{·, ·}_{𝕏} : N(𝕏) × N(𝕏) → W_{ℚ}

such that for all x, y ∈ N(𝕏), we have

{F x, y}_{𝕏} = σ ({x, V y}_{𝕏}) .

Thus, if we define

h(x, y) := \frac{1}{pδ} \{x, F y\}_{𝕏} ,  (3.6)

it is straightforward to verify that the restriction of h to C defines a hermitian form; we denote this restricted form again by h. We shall shortly see that (C, h) is in fact split, cf. Remark 3.4. Let q(x) = h(x, x) denote the corresponding quadratic form.

Definition 3.3. (i) If L is a W -lattice in N(𝕏)_{0}, we let

L^{♯} := {n ∈ N(𝕏)_{0} | h(n, L) ⊂ W }.

Note that (L^{♯})^{♯} = pV^{−2}L. Similarly, if Λ ⊂ C is an o_{k,p}-lattice, we set

Λ^{♯} := {v ∈ C | h(v, Λ) ⊂ o_{k,p}}.

(ii) Suppose Λ ⊂ C. We say Λ is a vertex lattice of type 0 (resp. type 2) if Λ^{♯} = Λ (resp. Λ^{♯} = pΛ). We shall use the term “vertex lattice” to mean a vertex lattice of type 0 or 2. (iii) Let B denote the Bruhat-Tits tree for SU(C), which is a graph with the following description. The vertices are vertex lattices, and edges can only occur between vertex lattices of differing type. Two vertex lattices Λ and Λ′ of type 0 and 2 respectively are joined by an edge if and only if

pΛ′ ⊂ Λ ⊂ Λ′,

where the successive quotients are 𝔽_{p^{2}} vector spaces of dimension 1. In particular, this graph is a p + 1-regular tree.

Suppose x = (X, ι_{X}, ρ_{X}) ∈ D(𝔽). We may use the quasi-isogeny ρ_{X} to identify the Dieudonné module M(X) as a W -lattice inside of N(𝕏). Furthermore, as ρ_{X} is o_{k,p}-linear, we have M(X)_{i} = M(X) ∩ N(𝕏)_{i} for i = 0, 1, where M(X)_{i} is defined in the same way as (3.4). Hence, to any point x, we may associate a chain of W -lattices B ⊂ A, where

B = M(X)_{0}, A = (V M(X)_{1})^{♯}.

By [KR4, Corollary 2.3], we have either B^{♯} = B or A^{♯} = pA, or both. If both conditions are satisfied, then we say the point x is superspecial; otherwise, x is ordinary. We say a point is special if both A and B are pV^{−2}-invariant, so in particular superspecial points are special. This construction yields a bijection between D(𝔽) and pairs of W -lattices B ⊂ A such that either B^{♯} = B or A^{♯} = pA. Moreover, if B^{♯} = B, then B = Λ ⊗_{o_{k,p}} W for some vertex lattice Λ of type 0; on the other hand, if A^{♯} = pA, then A = Λ′ ⊗ W for a vertex lattice Λ′ of type 2 , cf. [KR4, Corollary 2.3]. Suppose Λ is a vertex lattice of type 0. We may define a map

ℙ_{Λ}(𝔽) := ℙ(p^{−1}Λ/Λ)(𝔽) → D(𝔽),

by sending a line ℓ ⊂ (p^{−1}Λ/Λ) ⊗_{𝔽_{p^{2}}} 𝔽 to the pair of lattices B ⊂ A, where B = Λ_{W} = Λ ⊗ W, and A is the inverse image of ℓ in p^{−1}Λ_{W}. If Λ′ is a vertex lattice of type 2, we obtain a map

ℙ_{Λ′}(𝔽) := ℙ(Λ′/pΛ′)(𝔽) → D(𝔽),

defined by sending a line ℓ′ ⊂ (Λ′/pΛ′) ⊗_{𝔽_{p^{2}}} 𝔽 to the pair of lattices B ⊂ A, where A = Λ′_{W}, and B is the inverse image of ℓ′ in Λ′_{W}. Note that if Λ and Λ′ are neighbours in B, i.e if pΛ′ ⊂ Λ ⊂ Λ′, then the lines

ℓ = Λ′ ⊗_{o_{k,p},τ_{0}} 𝔽 ∈ ℙ(p^{−1}Λ/Λ)(𝔽), ℓ′ = Λ ⊗ 𝔽 ∈ ℙ(Λ′/pΛ′)(𝔽)

define the same point of D(𝔽); this point is superspecial, and all superspecial points arise in this way. By [KR4, Proposition 2.4], the above maps are induced by embeddings of schemes over 𝔽:

ℙ_{Λ} → D_{red}, Λ a vertex lattice,

where D_{red} is the underlying reduced subscheme of the formal scheme D, and the collection of such maps, as Λ varies among the vertex lattices, yield a cover of D_{red} by projective lines.

Remark 3.4. (i) In [KR1, §1], there is a similar description of the special fibre of D, but it is given in terms of homothety classes of ℤ_{p}-lattices. These two descriptions are essentially the same. To start, note that the operator

ϵ := Π_{𝕏}^{−1} ∘ V : N(𝕏) → N(𝕏), Π_{𝕏} := ι_{𝕏}(Π)

is 0-graded and commutes with F^{−1}V, and hence restricts to a Galoissemilinear operator ϵ: C → C. Without loss of generality, we may assume 𝕏 = 𝕐 × 𝕐, where 𝕐 is a supersingular p-divisible group of height 2 and dimension 1 over 𝔽. Then the Dieudonné module M(𝕏) has a basis {e_{0}, e_{1}, f_{0}, f_{1}} consisting of vectors that are both F^{−1}V - and ϵ-invariant, and such that

M(𝕏)_{0} = W · e_{0} ⊕ W · f_{1}, M(𝕏)_{1} = W · e_{1} ⊕ W · f_{0},

cf. [BC, §III.4.5]. The set {e_{0}, f_{1}} is an ϵ-invariant k_{p}-basis for C such that h(e_{0}, f_{1}) = δ and h(e_{0}, e_{0}) = h(f_{1}, f_{1}) = 0, and in particular, we see that C is split. Let Λ_{0} := span_{o_{k,p}}(e_{0}, f_{1}), which is a vertex lattice of type 0, i.e. Λ_{0}^{♯} = Λ_{0}. Now suppose γ ∈ End_{O_{B,p}}(𝕏) ⊗_{ℤ_{p}} ℚ_{p}. Then γ also acts on C, and is described by a matrix of the form

[γ] = (\begin{matrix}a & b \\ c & d \end{matrix}) , a, b, c, d ∈ ℚ_{p} .

with respect to the basis {e_{0}, f_{1}}. In particular (at least, when γ is invertible), the matrix [γ] lies in GU(C) with [γ]^{∗} = det(γ) · [γ]^{−1}. One can then verify directly that the map γ ↦ [γ] induces an isomorphism

{γ ∈ End_{O_{B,p}}(𝕏) ⊗ ℚ_{p}, det(γ) = 1} ⟶^{∼} SU(C).  (3.7)

Suppose [L] is a homothety class of ℤ_{p}-lattices in M(𝕏)_{0}^{ϵ=1} = C^{ϵ=1} ≃ (ℚ_{p})^{2}, and L ∈ [L]. Then there exists γ ∈ End_{O_{B,p}}(𝕏)_{ℚ_{p}} such that

γ(L) = L_{0} := span_{ℤ_{p}}{e_{0}, f_{1}}.

Set Λ := L ⊗_{ℤ_{p}} o_{k,p}, so that

Λ^{♯} = ([γ]^{∗})^{−1} · Λ_{0}^{♯} = det(γ)^{−1}[γ] · Λ_{0} = det(γ)^{−1}Λ.

Hence, by scaling by a power of p, there is a unique representative L ∈ [L] such that L ⊗ o_{k,p} is a vertex lattice, whose type depends on the parity of ord_{p} det(γ). Conversely, suppose Λ ⊂ C is a vertex lattice. Since SU(C) acts transitively on the set of lattices of a given type, there exists an element [γ] ∈ SU(C) such that [γ] · Λ is equal to one of Λ_{0} or Λ′_{0} := span_{o_{k,p}}(p^{−1}e_{0}, f_{1}), depending on the type of Λ. In either case, by (3.7), the transformation [γ] commutes with ϵ, and so Λ admits an ϵ-invariant basis. Thus the ℤ_{p}-lattice Λ^{ϵ=1} determines a homothety class of ℤ_{p}-lattices [L] := [Λ^{ϵ=1}]. (ii) Suppose x = (X, ι_{X}, ρ_{X}) ∈ D(𝔽) , and let M = M_{0} ⊕ M_{1} denote its Diedonné module, where each M_{i} is viewed as a W -lattice in N(𝕏)_{i}. Each lattice M_{i} is self-dual with respect to the pairing induced by the polarization λ_{X}^{0} described in Theorem 3.2, and it is then a straightforward calculation to show

M_{0}^{♯} = Π^{−1}V M_{0},  and  (V M_{1})^{♯} = Π^{−1}M_{1},

where the ♯ denotes the dual with respect to the hermitian form h. Recall that x is said to be “0-critical” in the sense of [BC] if and only if ΠM_{0} = V M_{0}, which is equivalent to the relation M_{0}^{♯} = M_{0}, i.e. x ∈ ℙ_{Λ}(𝔽) for a vertex lattice Λ of type 0. Similarly x is “1-critical” if and only if ΠM_{1} = V M_{1}, which is equivalent to the relation ((V M_{1})^{♯})^{♯} = p(V M_{1})^{♯}. This last condition is then equivalent to the condition x ∈ ℙ_{Λ′}(𝔽) for a vertex lattice Λ′ of type 2. In particular, this discussion implies that our use of the terms “ordinary”, “special” and “superspecial” coincides with their use in [KR1, §1]. ⋄

3.2 The structure of a local unitary cycle

We begin this section by defining the (local) unitary special cycles, whose construction is due to Kudla and Rapoport [KR2]. To start, fix a triple \underline{𝕐} = (𝕐, i_{𝕐}, λ_{𝕐}) over 𝔽 consisting of

(i) a (supersingular) p-divisible group 𝕐 of dimension 1 and height 2 over 𝔽; (ii) an action i_{𝕐}: o_{k,p} → End(𝕐) such that on Lie(𝕐), this action coincides

with the action of o_{k,p} via the embedding τ_{0} : o_{k,p}/(p) → 𝔽; (iii) and finally, a principal polarization λ_{𝕐} such that the induced Rosati in-

volution * satisfies

i_{𝕐}(a)^{∗} = i_{𝕐}(a′), for all a ∈ o_{k,p}.

We define two spaces of special homomorphisms:

𝕍_{ϕ}^{+} := {b ∈ Hom(𝕐, 𝕏) ⊗_{ℤ_{p}} ℚ_{p} | b ∘ i_{𝕐} (a) = ι_{𝕏} (ϕ(a)) ∘ b for all a ∈ o_{k,p}}

(3.8)

and

𝕍_{ϕ}^{−} := {b ∈ Hom(𝕐, 𝕏) ⊗_{ℤ_{p}} ℚ_{p} | b ∘ i_{𝕐} (a) = ι_{𝕏} (ϕ(a′)) ∘ b for all a ∈ o_{k,p}} .

(3.9)

Using the polarization λ_{𝕐}, and the polarization λ_{𝕏,ϕ} as defined in (3.3), we may construct natural hermitian forms h^{+} and h^{−} on 𝕍_{ϕ}^{+} and 𝕍_{ϕ}^{−} respectively; these are defined by the formulas

h^{+}(b_{1}, b_{2}) := λ_{𝕐}^{−1} ∘ b_{2}^{∨} ∘ λ_{𝕏,ϕ} ∘ b_{1} ∈ End_{o_{k}}(\underline{𝕐}^{+}) ⊗_{ℤ_{p}} ℚ_{p} ≃ k_{p}  (3.10)
h^{−}(b_{1}, b_{2}) := λ_{𝕐}^{−1} ∘ b_{1}^{∨} ∘ λ_{𝕏,ϕ} ∘ b_{2} ∈ End_{o_{k}}(\underline{𝕐}^{+}) ⊗_{ℤ_{p}} ℚ_{p} ≃ k_{p}.  (3.11)

Let q^{±}(b) := h^{±}(b, b) denote the corresponding quadratic forms. Our next step is to relate the spaces 𝕍_{ϕ}^{±} to the hermitian space (C, h). Let M(𝕐) denote the Dieudonné module over W = W (𝔽) attached to 𝕐; this is a free W -module of rank 2. As before, we have a grading M(𝕐) = M(𝕐)_{0} ⊕ M(𝕐)_{1}, where

M(𝕐)_{i} := {m ∈ M(𝕐) | i_{𝕐}(a) · m = τ_{i}(a)m, for all a ∈ o_{k,p}}.

Moreover, we may choose generators f_{0} and f_{1} for M(𝕐)_{0} and M(𝕐)_{1} respectively, such that V f_{0} = f_{1}, V f_{1} = pf_{0}, and that the alternating form {·, ·}_{𝕐} defined by the polarization λ_{𝕐} satisfies

{f_{0}, f_{1}}_{𝕐} = δ;  (3.12)

see [KR2, Remark 2.5]. Suppose that b ∈ 𝕍_{ϕ}^{+}. Abusing notation, we denote the corresponding map on (rational) Dieudonné modules again by

b: N(𝕐) = M(𝕐) ⊗ ℚ → N(𝕏).

Let b := b(f_{0}). Then since b is o_{k,p}-linear, we have that b ∈ N(𝕏)_{0}, and furthermore,

V^{−1}F b = pV^{−2}b = p b(V^{−2}f_{0}) = b,

which implies b ∈ C. Finally, we note that

b(f_{1}) = b(V f_{0}) = V b(f_{0}) = V b,

and so b is determined by b. We therefore obtain an isomorphism

φ^{+} : 𝕍_{ϕ}^{+} → C,  b ↦ b := b(f_{0}).  (3.13)

One readily checks that q(φ^{+}b) = p^{−1}q^{+}(b), where q and q^{+} are the quadratic forms on C and 𝕍_{ϕ}^{+} defined by (3.6) and (3.10) respectively. In a similar manner, if b ∈ 𝕍_{ϕ}^{−}, then b := b(f_{1}) ∈ C, and b is again determined by b. Hence we obtain an isomorphism

φ^{−}: 𝕍_{ϕ}^{−} → C,  b ↦ b := b(f_{1})  (3.14)

such that q(φ^{−}b) = q^{−}(b). Finally, we come to the definition of the unitary special cycles, as in [KR2]. Note that the p-divisible group \underline{𝕐} = (𝕐, i_{𝕐}), together with its o_{k,p}-action, admits a canonical lift \underline{𝕐}_{W} = (𝕐_{W} , i_{𝕐_{W}}) to W. For any S ∈ Nilp, we let \underline{𝕐}_{S} = (𝕐_{S}, i_{𝕐_{S}}) denote the base change 𝕐_{S} = 𝕐_{W} ×_{W} S, with induced o_{k,p}-action.

Definition 3.5. For b ∈ 𝕍_{ϕ}^{±}, we define the local unitary special cycle Z(b) by the following moduli problem: for S ∈ Nilp, let Z(b)(S) denote the set of points (X, ι_{X}, ρ) ∈ D(S) such that quasi-isogeny

ρ^{−1} ∘ b: 𝕐 ×_{𝔽} \overline{S} → X ×_{S} \overline{S}

lifts to a morphism of p-divisible groups 𝕐_{S} → X.

Remark 3.6. (i) If such a lift exists, then it is unique, by rigidity for pdivisible groups. In particular, if the lift exists, then it is o_{k,p}-linear (resp. o_{k,p}-antilinear) whenever b ∈ 𝕍_{ϕ}^{+} (resp. b ∈ 𝕍_{ϕ}^{−}). (ii) By [RZ, Proposition 2.9], the moduli problem Z(b) is represented by a closed formal subscheme of D. ⋄ Our first result is the following description of the 𝔽-points of the special cycles.

Proposition 3.7. (i) Suppose b ∈ 𝕍_{ϕ}^{−}, and b = φ^{−}b ∈ C is the corresponding vector, cf. (3.14). Let Λ be any vertex lattice. Then

Z(b)(𝔽) ∩ ℙ_{Λ}(𝔽) = \begin{cases}∅, & if b ∉ Λ, \\ a single point \{x\}, & if b ∈ Λ − pΛ, \\ ℙ_{Λ}(𝔽), & if b ∈ pΛ. \end{cases}

If the second case above occurs, then Λ is necessarily of type 0, and the point x is special. Moreover x is superspecial if and only if ord_{p}q^{−}(b) > 0. (ii) Similarly, for b ∈ 𝕍_{ϕ}^{+} with b = φ^{+}b, cf. (3.13), we have

Z(b)(𝔽) ∩ ℙ_{Λ}(𝔽) = \begin{cases}∅, & if b ∉ Λ, \\ a single point \{x\}, & if b ∈ Λ − pΛ, with Λ^{♯} = pΛ \\ ℙ_{Λ}(𝔽), & if b ∈ Λ with Λ^{♯} = Λ, \\ ℙ_{Λ}(𝔽), & if b ∈ pΛ, Λ arbitrary. \end{cases}

In the second case, the unique point x is special, and is superspecial if and only if ord_{p}q^{+}(b) > 0.

Proof. (i) Suppose b ∈ 𝕍_{ϕ}^{−}. We observe that a point x = (\underline{X}, ρ_{X}) ∈ D(𝔽) is in Z(b)(𝔽) if and only if, upon identifying M(X) with a W -lattice in N(𝕏), we have

b(M(𝕐)) ⊂ M(X) ⟺ b(f_{0}) ∈ M(X)_{1}, and b(f_{1}) ∈ M(X)_{0}
⟺ b = b(f_{1}) ∈ V M(X)_{1};

the last equivalence follows from the relation V b(f_{0}) = b(f_{1}). Recall also that b(f_{0}) ∈ C = N(𝕏)_{0}^{F V^{−1}}, so

x = (\underline{X}, ρ_{X}) ∈ Z(b)(𝔽) ⟺ b ∈ V M(X)_{1} ∩ C.

Now suppose x ∈ ℙ_{Λ}(𝔽) ∩ Z(b)(𝔽), with Λ^{♯} = pΛ. By construction, x corresponds to a lattice pair B ⊂ A, where

A = V M(X)_{1}^{♯} = Λ ⊗_{τ_{0}} W.

On the other hand, note

pA = A^{♯} = (V M_{1}(X)^{♯})^{♯} = F M_{1}(X).

Thus, as x ∈ Z(b)(𝔽) as well, then

b ∈ V M_{1}(X) ∩ C = F M_{1}(X) ∩ C = pA ∩ C.

However, as pA = pΛ_{W}, the above line is true for all x ∈ ℙ_{Λ}(𝔽), as soon as it is true for a single point. Hence we have ℙ_{Λ}(𝔽) ⊂ Z(b)(𝔽). Now suppose x ∈ ℙ_{Λ}(𝔽) with Λ^{♯} = Λ. By construction, this means M(X)_{0} is identified with Λ ⊗_{τ_{0}} W, and so if ℙ_{Λ}(𝔽) ∩ Z(b) ≠ ∅, then we must have b ∈ Λ. Furthermore, any x ∈ ℙ_{Λ}(𝔽) is determined by the sequence of inclusions of 𝔽-codimension 1

pM(X)_{0} ⊂ V M(X)_{1} ⊂ M(X)_{0}
||  ||
pΛ ⊗ W  Λ ⊗ W.

Hence, if b ∈ pΛ, then b ∈ V M(X)_{1} for all (\underline{X}, ρ_{X}) ∈ ℙ_{Λ}(𝔽), and so

Z(b)(𝔽) ∩ ℙ_{Λ}(𝔽) = ℙ_{Λ}(𝔽).

If on the other hand b ∈ Λ\pΛ, and ℙ_{Λ}(𝔽)∩Z(b)(𝔽) ≠ ∅, then this intersection necessarily consists of a single point x = (X, ρ_{X}): namely, the unique point with

V M(X)_{1} = W · b + pΛ_{W} ⊂ Λ_{W} .

Note in this case, (pV^{−2})V M(X)_{1} = V M(X)_{1} as both Λ_{W} and b are pV^{−2}- invariant.

Now by construction, we have

Λ_{W} ⊂ V M(X)_{1}^{♯} ⊂ p^{−1}Λ_{W} .

If ord_{p}q(b) = 0, then p^{−1}b ∉ V M(X)_{1}^{♯}, and so the point is ordinary. On the other hand, if ord_{p}q(b) > 0, then

V M(X)_{1}^{♯} = W · p^{−1}b + Λ_{W} = p^{−1}V M(X)_{1} = p^{−1} (V M(X)_{1}^{♯})^{♯} ,

and so this point is superspecial. (ii) Suppose b ∈ 𝕍_{ϕ}^{+}, and let b = b(f_{0}) = φ^{+}(b). A point x = (\underline{X}, ρ_{X}) lies in Z(b)(𝔽) if and only if b ∈ M(X)_{0}. The lemma follows via a similar argument to the previous case.

Lemma 3.8. Fix an ϵ-invariant basis {v_{0}, v_{1}} of C (cf. Remark 3.4) such that

h(v_{0}, v_{0}) = h(v_{1}, v_{1}) = 0, h(v_{0}, v_{1}) = −h(v_{1}, v_{0}) = δ.

Suppose that b = a_{0}v_{0} + a_{1}v_{1}, with ord_{p}q(b) either 0 or −1. Set b′ := ϵ(b) = a′_{0}v_{0} + a′_{1}v_{1} and

Λ_{b} := span_{o_{k,p}} {b, b′} .

If ord_{p}q(b) = 0, then Λ_{b} is of type 0 (i.e Λ_{b}^{♯} = Λ_{b}), and is the unique lattice of type 0 such that b ∈ Λ_{b} − pΛ_{b}. Similarly, if ord_{p}q(b) = −1, then Λ_{b} is of type 2, and is the unique lattice of type 2 such that b ∈ Λ_{b} − pΛ_{b}.

Proof. Suppose ord_{p}q(b) = 0 and Λ is a type 0 lattice with b ∈ Λ − pΛ. Then there exists an element γ ∈ SU(C) such that

γ · Λ = Λ_{b},

as SU(C) acts transitively on the set of type 0 lattices. We may assume without loss of generality that q(b) = 1. Note the vectors b and b′ form an orthogonal basis for C, with h(b, b) = −h(b′, b′) = 1. Let

[γ] = (\begin{matrix}x & y \\ w & z \end{matrix})

denote the matrix representation of γ with respect to the basis {b, b′}. Since b = ( \begin{matrix}1 \\ 0 \end{matrix} ) ∈ Λ, we have x, w ∈ o_{k,p}. On the other hand, the equation γ · γ^{∗} = 1 implies

xx′ − yy′ = −ww′ + zz′ = 1, yw′ = xz′,

which implies that y, w ∈ o_{k,p} as well. Hence γ and γ^{∗} stabilize Λ_{b}, which yields the result Λ = Λ_{b}. The proof in the case ord_{p}q(b) = −1 is similar.

Remark 3.9. Suppose b ∈ 𝕍_{ϕ}^{±}, with ord_{p}q^{±}b = 0. Let b = φ^{±}b ∈ C denote the corresponding vector. Then by Lemma 3.8, there is a unique lattice Λ_{⊙} such that b ∈ Λ_{⊙} − pΛ_{⊙}, and Λ_{⊙} is of type 0 (resp. of type 2) if b ∈ 𝕍_{ϕ}^{−} (resp. b ∈ 𝕍_{ϕ}^{+}). Combining this observation with Proposition 3.7, we find that

Z(b)(𝔽) = {x}

is a single ordinary point lying in the component ℙ_{Λ_{⊙}}(𝔽). ⋄

Our goal for the remainder of this section is to give a complete description of the special cycles Z(b) as cycles on D, as in Theorem 3.14 below. We do this by writing down equations using the (formal) affine open cover described in [KR1, §1], which consist of affine schemes of two types:

1. Ω̂_{Λ}^{ord} ≃ Spf W [T, (T^{p} − T)^{−1}]^{∨}, for each vertex lattice Λ, and

2. Ω̂_{[Λ,Λ′]} ≃ Spf W [T_{1}, T_{2}, (T_{1}^{p−1}−1)^{−1}(T_{2}^{p−1}−1)^{−1}]^{∨}, for each pair of neighbouring vertex lattices Λ and Λ′.

In both cases above, the superscript ^{∨} denotes completion along the ideal generated by p, and the isomorphisms are determined by a choice of basis for Λ. The underlying set of Ω̂_{Λ}^{ord} is the set of ordinary points ℙ_{Λ}^{ord} in ℙ_{Λ} (that is, the complement of the superspecial points), while the underlying set of Ω̂_{[Λ,Λ′]} is the union

ℙ_{Λ}^{ord} ∪ ℙ_{Λ′}^{ord} ∪ {x}

where x is the superspecial point at the intersection of ℙ_{Λ} ∩ ℙ_{Λ′}. For Λ a type 0 lattice, with neighbour Λ′ of type 2, we have open immersions (cf. [KR1, §1])

Ω̂_{Λ}^{ord} → Ω̂_{[Λ,Λ′]}, induced by T_{0} ↦ T, T_{1} ↦ pT^{−1}

and

Ω̂_{Λ′}^{ord} → Ω̂_{[Λ,Λ′]}, induced by T_{0} ↦ pT^{−1}, T_{1} ↦ T.

We first consider a type 0 lattice Λ = Λ^{♯}. Let b ∈ 𝕍_{ϕ}^{±}, with ord_{p}q^{±}(b) ≥ 0 and such that Z(b) ∩ Ω̂_{Λ}^{ord} ≠ ∅. Let b = φ^{±}b ∈ C the corresponding vector; by Proposition 3.7, we must have b ∈ Λ. Fix an ϵ-invariant basis {v_{0}, v_{1}} of Λ with respect to which the hermitian form h has matrix

h ∼ (\begin{matrix}& δ \\ −δ \end{matrix}) ,  (3.15)

and write

b = a_{0}v_{0} + a_{1}v_{1},  where a_{0}, a_{1} ∈ o_{k,p}.  (3.16)

Proposition 3.10. Let b ∈ 𝕍_{ϕ}^{±}, and suppose Z(b) ∩ Ω̂_{Λ}^{ord} ≠ ∅. Then, with notation as in the previous paragraph,

Z(b) ∩ Ω̂_{Λ}^{ord} ≃ Spf W [T, (T^{p} − T)^{−1}]^{∨}/(f), where

f := \begin{cases}a_{0}T + a_{1}, & if b ∈ 𝕍_{ϕ}^{−} \\ p(a′_{0}T + a′_{1}), & if b ∈ 𝕍_{ϕ}^{+} . \end{cases}  (3.17)

Proof. The proof of this proposition is modelled on arguments by Terstiege found in the proofs of [Ter, Propositions 2.8 and 4.5]. We consider the following cases:

Case 1: b ∈ 𝕍_{ϕ}^{−}, with b = φ^{−}b ∈ Λ.

Fix a point x ∈ Z(b)(𝔽) ∩ ℙ_{Λ}(𝔽) corresponding to (\overline{X}, ι\overline{X}, ρ) ∈ D(𝔽). At present, we do not require that x be ordinary (i.e. \overline{X} may be superspecial); in fact the argument we are about to discuss applies equally well in the superspecial case, and so it will be expedient to consider this slightly more general setting here. Consider the (complete) local ring

R = O_{D,x}.

Let m_{R} denote the maximal ideal of R, and I denote the ideal corresponding to Z(b). For the purposes of the proposition, we need to prove that I is generated by the image of f in R. Note R is Noetherian, which implies that I is complete for the p-adic topology. Hence if we set

R_{n} := R/p^{n}R, I_{n} = I + p^{n}R ⊂ R_{n},

it will suffice to prove that I_{n} is generated by the image of f in R_{n}, for each n. Let m_{n} denote the maximal ideal of R_{n}. We set

A := R_{n}/m_{n}I_{n}, A′ := R_{n}/I_{n}.

Then the kernel J := I_{n}/m_{n}I_{n} of the projection A → A′ satisfies J^{2} = 0, and hence is endowed with a PD structure. Moreover, to prove the proposition, it suffices (by Nakayama) to show that J is generated by the image of f in A (which, abusing notation, we shall henceforth denote as f). Finally, we note that both A and A′ can be viewed as o_{k,p}-algebras via the fixed embedding τ_{0} : o_{k,p} → W composed with the respective structural morphisms. Now associated to the rings A and A′ are two points in the moduli space D, which in turn correspond to formal O_{B}-modules X and X′, both of whose special fibres are equal to \overline{X}. Moreover, by assumption, X′ is in Z(b)(A′). In other words, the map

β\overline{X} := ρ^{−1} ∘ b: 𝕐 → \overline{X}

is a morphism of p-divisible groups, which lifts to a morphism

β_{X′}: 𝕐_{A′} → X′.  (3.18)

Here 𝕐_{A′} is the base change of the canonical lift 𝕐_{W} to A′.

Let 𝔻(𝕐_{A′} /·) denote the Grothendieck-Messing crystal of 𝕐_{A′}, which carries a ℤ /2 ℤ grading induced by the action of o_{k,p}. In particular, for a P D extension B → A′, the canonical lift of 𝕐_{A′} (together with its o_{k,p}-action) to B is determined by the Hodge filtration:

𝔻(𝕐_{A′} /B)_{1} = F_{𝕐_{B}} ⊂ 𝔻(𝕐_{A′} /B).  (3.19)

This is a consequence of the requirement that o_{k,p} act on Lie(𝕐_{B}) with signature (1, 0). Turning to the p-divisible groups X and X′, we let 𝔻(X/·) and 𝔻(X′/·) denote their respective Grothendieck-Messing crystals. We then have a diagram

\begin{aligned}𝔻(X/A) & ≃ & 𝔻(X′/A) \\ ↓ & mod J \\ & & 𝔻(X′/A′) \end{aligned}  (3.20)

Let B → A′ be a PD-extension as above. Since X′ ∈ Z(b)(A′), there is a map of B-modules

𝔻(β_{X′}): 𝔻(𝕐_{A′} /B) → 𝔻(X′/B)

induced by (3.18). By Grothendieck-Messing theory, if X̃ is a lift of X′ to B which corresponds to an O_{B,p}-stable direct summand F ⊂ 𝔻(X′/B), then

X̃ ∈ Z(b)(B)  ⟺  𝔻(β_{X′})(F_{𝕐_{B}}) ⊂ F.

Fixing any basis vector f_{B} ∈ F_{𝕐_{B}} = 𝔻(𝕐_{A′} /B)_{1}, and recalling that b is o_{k,p}- antilinear, the above condition can be rewritten as

X̃ ∈ Z(b)(B)  ⟺  𝔻(β_{X′})(f_{B}) ∈ F_{0}.  (3.21)

For convenience, we fix a basis vector f′_{1} ∈ 𝔻(𝕐_{A′} /A′)_{1}, and a lift to a basis vector f_{1} ∈ 𝔻(𝕐_{A} /A)_{1}. We want to express the condition (3.21) in terms of the affine chart

Ω̂_{[Λ,Λ′]} ≃ Spf W [T_{0}, T_{1}, (T_{0}^{p−1} − 1)^{−1}, (T_{1}^{p−1} − 1)^{−1}]/(T_{0}T_{1} − p)^{∨}. (3.22)

As p is nilpotent in A, the point X ∈ Ω̂_{[Λ,Λ′]}(A) is simply determined by a map of W -algebras

W [T_{0}, T_{1}, (T_{0}^{p−1} − 1)^{−1}, (T_{1}^{p−1} − 1)^{−1}]/(T_{0}T_{1} − p) → A.  (3.23)

Let t ∈ A denote the image of T_{0}, so that t^{p−1} − 1 ∈ A^{×}. If \overline{X} is ordinary, as in the hypotheses of the proposition, then we may equally well restrict to the smaller open affine neighbourhood

Ω̂_{Λ}^{ord} ≃ Spf W [T, (T^{p} − T)^{−1}]^{∨},  (3.24)

via the open immersion

Ω̂_{Λ}^{ord} → Ω̂_{[Λ,Λ′]}, induced by T_{0} ↦ T, T_{1} ↦ pT^{−1}.

The point X ∈ Ω̂_{Λ}^{ord}(A) then corresponds to the same element t ∈ A as before, which in this case now satisfies t ∈ A^{×}. In any case, by carefully tracing through the construction of the charts (3.22) and (3.24) as they are constructed in [BC], we find that there exists isomorphisms

𝔻(X/A)_{0} ≃ Λ ⊗_{o_{k,p}} A,  𝔻(X′/A′)_{0} ≃ Λ ⊗_{o_{k,p}} A′  (3.25)

such that (i) the 0-th graded piece of the Hodge filtration for X is identified with

(F_{X})_{0} = span_{A}{v_{0} ⊗ 1 − v_{1} ⊗ t} ⊂ 𝔻(X/A)_{0} ≃ Λ ⊗ A,

where {v_{0}, v_{1}} is the fixed basis of Λ as in (3.15); (ii) the element 𝔻(β_{X′})(f_{1}) is identified with b ⊗ 1 ∈ Λ ⊗ A; (iii) the vertical map in (3.20) is the natural map

Λ ⊗ A → Λ ⊗ A′, l ⊗ a ↦ l ⊗ a′,

where a ↦ a′ is the projection A → A′. Now suppose 𝔟 is any ideal of A contained in J, and let B := A/𝔟. Let t_{B} denote the image of t in B and X_{B} the corresponding B-valued point of D. We may view X_{B} as a deformation of X′, which by Grothendieck-Messing theory corresponds to the direct summand

F_{X_{B}} = F_{X} ⊗_{A} B ⊂ 𝔻(X′/B),

and in particular we have

(F_{X_{B}})_{0} = span_{B}{v_{0} ⊗ 1 − v_{1} ⊗ t_{B}} ⊂ 𝔻(X′/B)_{0} = Λ ⊗ B.

Then in light of (3.21), and noting that b is o_{k,p}-antilinear, we have

X_{B} ∈ Z(b)(B) ⟺ 𝔻(β_{X′}/B)(f_{1}) ⊗ 1 ∈ (F_{X_{B}})_{0}
⟺ b ⊗ 1 ∈ span{v_{0} ⊗ 1 − v_{1} ⊗ t_{B}} in Λ ⊗ B.

Recall that we had written b = a_{0}v_{0} + a_{1}v_{1}. Hence the last condition is equivalent to

a_{0}t_{B} + a_{1} = 0 in B.

Applying the necessity of this condition to the case 𝔟 = J, so B = A/J, we find that

f = a_{0}t + a_{1} ∈ J.

Suppose now that \overline{X} is ordinary. By the sufficiency of the above condition, the map Spf(A/(f)) → Ω̂_{Λ}^{ord} factors through Z(b). But by definition J is the smallest ideal of A such that Spf(A/J) → Ω̂_{Λ}^{ord} factors through Z(b), and so J = (f); this concludes the proof in this case. When \overline{X} is superspecial, an identical argument implies that Z(b) is again cut out by f in O_{D,x}.

Case 2: b ∈ 𝕍_{ϕ}^{+} with b = φ^{+}b ∈ Λ.

Here it is necessary to restrict our attention to the ordinary setting. Let r = r(b, Λ) := max{n | p^{−n}b ∈ Λ}, and write

b = p^{r}b_{0} = p^{r}(α_{0}v_{0} + α_{1}v_{1}),

where b_{0} = α_{0}v_{0} + α_{1}v_{1} ∈ Λ − pΛ, and so in particular at least one of α_{0}, α_{1} is a unit in o_{k,p}^{×}.

As a first step, we shall prove that every W_{r+1} := W/(p^{r+1}) valued point of Ω̂_{Λ}^{ord} belongs to the special cycle Z(b). To this end, suppose (X, ρ_{X}) ∈ Ω̂_{Λ}^{ord}(W_{r+1}), and let \overline{X} ∈ Ω_{Λ}^{ord}(𝔽) denote its reduction modulo p. Let

\overline{M} = M(\overline{X}) = \overline{M}_{0} ⊕ \overline{M}_{1}

denote the Dieudonné module of \overline{X}, endowed with the grading induced by the action of o_{k,p}. As the projection W_{r+1} → 𝔽 has kernel generated by p, it is equipped with a PD structure. Hence, via Grothendieck-Messing theory, the point X corresponds to an O_{B,p}-stable summand

F ⊂ 𝔻(\overline{X}/W_{r+1}) = \overline{M} ⊗_{W} W_{r+1} = \overline{M}/p^{r+1}\overline{M},

such that

F ⊗_{W_{r+1}} 𝔽 = V \overline{M}/p\overline{M}.

By Proposition 3.7, we have \overline{X} ∈ Z(b)(𝔽), and so the map β := ρ_{X}^{−1}∘b: 𝕐 → \overline{X} induces a morphism of Dieudonné modules

β: M(𝕐) = W · f_{0} ⊕ W · f_{1} → \overline{M}.

By definition,

β(f_{0}) = b ∈ Λ ⊗ W = \overline{M}_{0}, β(f_{1}) = V β(f_{0}) = V b ∈ \overline{M}_{1}.

The morphism β also induces a map on crystals

𝔻(β/W_{r+1}) : M(𝕐)⊗_{W} W_{r+1} = 𝔻(𝕐 /W_{r+1}) → 𝔻(\overline{X}/W_{r+1}) = \overline{M} ⊗_{W} W_{r+1}.

Recall that the direct summand

F_{𝕐} ⊂ M(𝕐) ⊗_{W} W_{r+1}

corresponding to the lift 𝕐_{W_{r+1}} of 𝕐 is simply the rank-1 module F_{𝕐} = span_{W_{r+1}}{f_{1} ⊗ 1}. Hence, by Grothendieck-Messing, we have

X ∈ Z(b)(W_{r+1}) ⟺ 𝔻(β/W_{r+1})(f_{1}) ∈ F_{1}
⟺ (V b) ⊗ 1 ∈ F_{1} in \overline{M}_{1} ⊗ W_{r+1}.  (3.26)

We shall show that this latter condition always holds for any X ∈ Ω̂_{Λ}^{ord}(W_{r+1}). Consider the element b_{0} = p^{−r}b ∈ Λ \ pΛ. If V b_{0} ∈ p\overline{M}_{1}, then

V b ⊗ 1 = p^{r} · V b_{0} ⊗ 1 ∈ p^{r+1} · (\overline{M}_{1} ⊗ W_{r+1}) = \{0\},

and so (3.26) holds trivially. If on the other hand V b_{0} ∈ \overline{M}_{1} − p\overline{M}_{1}, then

span_{𝔽}(V b_{0} + p\overline{M}_{1}) = V \overline{M}_{0}/p\overline{M}_{1} = F_{1} ⊗_{W_{r+1}} 𝔽 .

In other words, the image of V b_{0} in \overline{M}_{1}/p\overline{M}_{1} is a basis vector for F_{1} ⊗ 𝔽. Hence, there exists α ∈ \overline{M}_{1} ⊗ W_{r+1} such that

V b_{0} ⊗ 1 + pα ∈ F_{1},

and so

p^{r}(V b_{0} ⊗ 1) + p^{r+1}α = V b ⊗ 1 ∈ F_{1},

as required. Thus, we have proven

Z(b) ∩ Ω̂_{Λ}^{ord}(W_{r+1}) = Ω̂_{Λ}^{ord}(W_{r+1}).  (3.27)

We now determine the local equation of Z(b) ∩ Ω̂_{Λ}^{ord} in Ω̂_{Λ}^{ord} ≃ Spf W [T, (T^{p} − T)^{−1}]^{∨}, assuming the intersection is non-empty. Note that as Λ = Λ^{♯} is a lattice of type 0, then by Remark 3.9, we must have ord_{p}q^{+}(b) > 0; otherwise, Z(b) would meet the special fibre only at a type-2 lattice. Set

b′ := Π_{𝕏}^{−1} ∘ b ∈ 𝕍_{ϕ}^{−},

so that ord_{p}q^{−}(b′) = ord_{p}q^{+}(b) − 1 ≥ 0, and note

φ^{−}b′ = b′(f_{1}) = Π_{𝕏}^{−1}b(f_{1}) = V Π_{𝕏}^{−1}b(f_{0})
= ϵ(b)
= p^{r}(α′_{0}v_{0} + α′_{1}v_{1}),

since by assumption the basis {v_{0}, v_{1}} was taken to be ϵ-invariant. Furthermore, we have evident inclusions

Z(b′) ⊂ Z(b) ⊂ Z(p · b′).

If I ⊂ W [T, (T^{p} − T)^{−1}]^{∨} denotes the ideal defining Z(b) ∩ Ω̂_{Λ}^{ord}, then

(p^{r+1}(α′_{0}T + α′_{1})) ⊂ I ⊂ (p^{r}(α′_{0}T + α′_{1})) , which follows by applying the result from Case 1 to the two antilinear homomorphisms b′ and p · b′. Recall that we want to prove that I is generated by f := p^{r+1}(α′_{0}T +α′_{1}). Thus, it suffices to prove that I ⊂ (p^{r+1}(α′_{0}T + α′_{1})). To this end, suppose g ∈ I. We write

g = p^{r}(α′_{0}T + α′_{1})g_{0},

and we need to show that p divides g_{0}. Suppose not. Then the reduction modulo p of (α′_{0}T + α′_{1})g_{0} is a non-zero rational function over 𝔽 and so there exists t ≠ 0 ∈ 𝔽 such that

(α′_{0}t + α′_{1})g_{0}(t) ≠ 0 ∈ 𝔽 .

Choose any preimage t̃∈ W_{r+1}^{×} of t. Then the map

W [T, (T^{p} − T)^{−1}]^{∨} → W_{r+1}, T ↦ t̃

does not factor through W [T, (T^{p} − T)^{−1}]^{∨}/I, which contradicts the assertion (3.27). Hence, we have that p divides g_{0}, and so

I = (p^{r+1}(α′_{0}T + α′_{1})) = (p(a′_{0}T + a′_{1}))

as required.

For a type 2 lattice Λ′, we describe the analogous result: choose an ϵ-invariant basis {w_{0}, w_{1}} for Λ′ such that q(w_{0}) = q(w_{1}) = 0 and h(w_{0}, w_{1}) = p^{−1}δ. Suppose b ∈ 𝕍_{ϕ}^{±}, and let b = φ^{±}b denote the corresponding vector. If Z(b) ∩ Ω̂_{Λ′}^{ord} ≠ ∅, then by Proposition 3.7, we must have b ∈ Λ′, and so we may write

b = a_{0}w_{0} + a_{1}w_{1}, a_{0}, a_{1} ∈ o_{k,p}.

The proof of the following proposition is completely analogous to Proposition 3.10, and is therefore omitted.

Proposition 3.11. Let the notation be as in the previous paragraph. Then if Z(b) ∩ Ω̂_{Λ′}^{ord} ≠ ∅, we have

Z(b) ∩ Ω̂_{Λ′}^{ord} ≃ Spf W [T, (T^{p} − T)^{−1}]^{∨}/(f),

where

f = \begin{cases}a_{0} + a_{1}T & if b ∈ 𝕍_{ϕ}^{+} \\ a′_{0} + a′_{1}T & if b ∈ 𝕍_{ϕ}^{−} . \end{cases}

Lemma 3.12. For b ∈ C, write

ord_{p}(q(b)) = \begin{cases}2t, & if ord_{p}(q(b)) is even \\ 2t − 1, & otherwise. \end{cases}

For any vertex lattice Λ, let

r(b, Λ) := max{r ∈ ℤ | p^{−r}b ∈ Λ};  (3.28)

note that here we do not assume b ∈ Λ, and so r(b, Λ) may be negative. Finally, let Λ_{⊙} denote the unique vertex lattice containing p^{−t}b, as in Lemma 3.8. Recall that Λ_{⊙} is type 0 (resp. type 2) if ord_{p}q(b) is even (resp. odd). Then we have the formula

r(b, Λ) = \begin{cases}t − ⌊\frac{d(Λ,Λ_{⊙})}{2}⌋, & ord_{p}q(b) even \\ t − ⌊\frac{d(Λ,Λ_{⊙})+1}{2}⌋, & ord_{p}q(b) odd. \end{cases}

Here d(Λ, Λ_{⊙}) is the distance function on B, the Bruhat-Tits tree.

Proof. By scaling by a power of p, it suffices to prove this lemma in the case t = 0; that is, we may assume that either ord_{p}q(b) = 0 or ord_{p}q(b) = −1. We proceed by induction on d = d(Λ, Λ_{⊙}). If d(Λ, Λ_{⊙}) = 0, i.e. Λ = Λ_{⊙}, then r(b, Λ_{⊙}) = 0 by the definition of Λ_{⊙}. Next, suppose we have proven the claim for all lattices L with 0 ≤ d(L, Λ_{⊙}) ≤ d, and let Λ be a lattice with d(Λ, Λ_{⊙}) = d. We shall prove the desired formula holds for all the neighbours of Λ. There are four cases here to check, as Λ can be either type 0 or 2, and ord_{p}q(b) can be even or odd. For example, suppose that Λ = Λ^{♯} and ord_{p}q(b) = 0, so d = d(Λ, Λ_{⊙}) is even. There exists an o_{k,p} basis {v_{0}, v_{1}} for Λ with (v_{0}, v_{0}) = (v_{1}, v_{1}) = 0, and (v_{0}, v_{1}) = −(v_{1}, v_{0}) = δ. With respect to this basis, a complete list of the p + 1 neighbours of Λ in the Bruhat-Tits tree are described as follows:

Λ′_{∞} := span_{o_{k,p}}{p^{−1}v_{0}, v_{1}}
Λ′_{α} := span_{o_{k,p}}{v_{0}, p^{−1}αv_{0} + p^{−1}v_{1}}

as α ∈ ℤ_{p} ranges over a complete set of representatives for 𝔽_{p} = ℤ_{p} /p ℤ_{p}. If d = 0, the claim holds for all the neighbours of Λ = Λ_{⊙} by inspection. Otherwise, without loss of generality we may assume that d(Λ′_{∞}, Λ_{⊙}) = d − 1, and so for all the other neighbours, d(Λ′_{α}, Λ_{⊙}) = d + 1. We may write

b = p^{r}(a_{0}v_{0} + a_{1}v_{1}) = p^{r} (pa_{0} · (p^{−1}v_{0}) + a_{1} · v_{1})
= p^{r+1} (a_{0} · (p^{−1}v_{0}) + p^{−1}a_{1} · v_{1})

where r = r(b, Λ) = −⌊d/2⌋. Since d is even, the induction hypothesis applied to Λ and Λ′_{∞} yields

r(b, Λ′_{∞}) = −⌊(d − 1)/2⌋ = −⌊d/2⌋ + 1 = r(b, Λ) + 1 = r + 1,

and so we must have a_{0} ∈ o_{k,p}^{×} and p|a_{1}. By inspecting the remaining neighbours Λ′_{α} of Λ, we immediately see

r(b, Λ′_{α}) = r = −⌊\frac{d + 1}{2}⌋,

as required. The remaining cases all follow in the same manner.

We now determine the equations of the special cycles Z(b) in the local ring at a superspecial point x ∈ ℙ_{Λ}(𝔽) ∩ ℙ_{Λ′}(𝔽), for a pair of neighbouring vertex lattices Λ and Λ′ of type 0 and 2 respectively. Recall that we have a (formal) affine open neighbourhood of x

Ω̂_{[Λ,Λ′]} ≃ Spf (W [T_{0}, T_{1}, (T_{0}^{p−1} − 1)^{−1}, (T_{1}^{p−1} − 1)^{−1}]/(T_{0}T_{1} − p))^{∨} ,

where the point x corresponds to the maximal ideal 𝔪_{x} = (T_{0}, T_{1}). Without loss of generality, we suppose that there is a basis {v_{0}, v_{1}} for Λ, such that (v_{0}, v_{1}) = −(v_{1}, v_{0}) = δ, (v_{0}, v_{0}) = (v_{1}, v_{1}) = 0, and

Λ = span{v_{0}, v_{1}},  Λ′ = span{p^{−1}v_{0}, v_{1}}.  (3.29)

Proposition 3.13. Suppose b ∈ 𝕍_{ϕ}^{±}, with b = φ^{±}b ∈ C the corresponding vector. Suppose x ∈ Z(b)(𝔽) is a superspecial point, with x ∈ ℙ_{Λ}(𝔽) ∩ ℙ_{Λ′}(𝔽) as above. If we set

r = r(b, Λ), r′ = r(b, Λ′),

as in (3.28), then the equation for Z(b) in the local ring O_{D,x} is given by

\begin{cases}(T_{0})^{r′}(T_{1})^{r} = 0, & if b ∈ 𝕍_{ϕ}^{−} \\ (T_{0})^{r′}(T_{1})^{r+1} = 0, & if b ∈ 𝕍_{ϕ}^{+} . \end{cases}

Proof. We begin with the case b ∈ 𝕍_{ϕ}^{−}. With respect to a basis {v_{0}, v_{1}} of Λ as in (3.29), we may write

b = p^{r}(α_{0}v_{0} + α_{1}v_{1})
= p^{r} (pα_{0} · (p^{−1}v_{0}) + α_{1}v_{1})  (3.30)

where, as usual, r = r(b, Λ), and at least one of α_{0} or α_{1} is a unit in o_{k,p}. Then in light of the choice of basis (3.29), we have

r′ = \begin{cases}r, & if a_{1} ∈ o_{k,p}^{×} \\ r + 1, & if p divides α_{1}. \end{cases}

By the proof of Proposition 3.10, Case (i), the special cycle Z(b) is defined at x by the vanishing of the element

f := p^{r}(α_{0}T_{0} + α_{1}) ∈ O_{D,x}.

Now consider the case r′ = r (i.e α_{1} ∈ o_{k,p}^{×}). Then the factor α_{0}T_{0} + α_{1} is a unit in O_{D,x} and so Z(b) is given by

p^{r} = (T_{0})^{r} · (T_{1})^{r} = 0.

If on the other hand r′ = r + 1, then p divides α_{1} and α_{0} ∈ o_{k,p}^{×}. Thus

p^{r}(α_{0}T_{0} + α_{1}) = p^{r}T_{0} (α_{0} + (α_{1}/p)T_{1}) = p^{r}T_{0} · u

with u = (α_{0} + (α_{1}/p)T_{1}) ∈ O_{D,x}^{×}, and so the cycle Z(b) is given by

(T_{0})^{r+1}(T_{1})^{r} = (T_{0})^{r′}(T_{1})^{r} = 0,

as required. The proof for a homomorphism b ∈ 𝕍_{ϕ}^{+} is completely analogous.

Theorem 3.14. Suppose b ∈ 𝕍_{ϕ}^{±}, with ord_{p}q^{±}(b) ≥ 0 and write

ord_{p}q^{±}(b) = \begin{cases}2t, & if ord_{p}q^{±}(b) is even, \\ 2t − 1, & if ord_{p}q^{±}(b) is odd. \end{cases}

Let b := φ^{±}b ∈ C be the corresponding vector, and write b = p^{k}b_{⊙}, where ord_{p}q(b_{⊙}) is either 0 or −1. Then by Lemma 3.8, there is a unique vertex lattice Λ_{⊙} (the “central lattice”) such that b_{⊙} ∈ Λ_{⊙} − pΛ_{⊙}. Finally, for any vertex lattice Λ we define

m(b, Λ) := \begin{cases}0, & if b ∉ Λ \\ t − ⌊d(Λ, Λ_{⊙})/2⌋, & if b ∈ Λ and ord_{p}q^{±}(b) is even, \\ t − ⌊(d(Λ, Λ_{⊙}) + 1)/2⌋, & if b ∈ Λ and ord_{p}q^{±}(b) is odd. \end{cases}

(3.31)

Then we have the equality of cycles on D:

Z(b) = Z(b)^{hor} + ∑_{Λ} m(b, Λ) ℙ_{Λ},

where Z(b)^{hor} ≃ Spf W is a horizontal divisor meeting the special fibre of D at a single ordinary special point in the component ℙ_{Λ_{⊙}}.

Proof. To start, we have the following relations:

m(b, Λ) = \begin{cases}0, & if b ∉ Λ \\ r(b, Λ) + 1, & if b ∈ Λ, Λ^{♯} = Λ \\ r(b, Λ), & if b ∈ Λ, Λ^{♯} = pΛ \end{cases}  for b ∈ 𝕍_{ϕ}^{+}, b = φ^{+}b

and

m(b, Λ) = \begin{cases}0, & if b ∉ Λ \\ r(b, Λ), & if b ∈ Λ \end{cases}  for b ∈ 𝕍_{ϕ}^{−}, b = φ^{−}b

which are easily verified by comparing the definition of m(b, Λ) above with Lemma 3.12. The proof of this theorem amounts to collating the information contained in the local equations given by Propositions 3.10, 3.11, and 3.13. We shall illustrate this argument in the case b ∈ 𝕍_{ϕ}^{−} with ord_{p}q^{−}(b) = 2t even; the remaining cases follow in an identical manner. Suppose b = φ^{−}b ∈ C is the vector corresponding to b ∈ 𝕍_{ϕ}^{−}, so by assumption ord_{p}q(b) = ord_{p}q^{−}(b) = 2t is also even. In particular, we have b_{⊙} = p^{−t}b. Let Λ denote a type 0 lattice, and choose a basis {v_{0}, v_{1}} such that h(v_{0}, v_{1}) = δ and q(v_{0}) = q(v_{1}) = 0. If we write

b = p^{r}(α_{0}v_{0} + α_{1}v_{1}) = p^{r}b_{0},

where r = r(b, Λ) and b_{0} ∈ Λ − pΛ as usual, then by Proposition 3.10, the intersection Z(b) ∩ Ω̂_{Λ}^{ord} with the ordinary locus Ω̂_{Λ}^{ord} of Ω̂_{Λ} is defined by the vanishing of the element

p^{r}(α_{0}T + α_{1}) ∈ W [T, (T^{p} − T)^{−1}]^{∨}.  (3.32)

Note that by Lemma 3.12,

ord_{p}q(b_{0}) = 0 ⟺ b_{0} = b_{⊙} ⟺ r = r(b, Λ) = t ⟺ Λ = Λ_{⊙}.

By the choice of basis {v_{0}, v_{1}}, we also have

q(b_{0}) = δ (α_{0}α′_{1} − α′_{0}α_{1}) .

If Λ ≠ Λ_{⊙}, so that ord_{p}q(b_{0}) > 0, then α_{0}α_{1}^{p} ≡ α_{0}^{p}α_{1} mod p. A short calculation implies that the term (α_{0}T + α_{1}) is a unit in W [T, (T^{p} − T)^{−1}]^{∨}. Hence, for Λ ≠ Λ_{⊙}, we have that Z(b) ∩ Ω̂_{Λ}^{ord} is determined by the equation

p^{r} = 0.

As the component ℙ_{Λ}^{ord} is given by the equation p = 0, we then obtain the equality of cycles

Z(b) ∩ Ω̂_{Λ}^{ord} = m(b, Λ) ℙ_{Λ}^{ord},  for Λ of type 0, Λ ≠ Λ_{⊙}.  (3.33)

On the other hand, if Λ = Λ_{⊙}, we have α_{0}, α_{1} ∈ o_{k,p}^{×}, and so the cycle defined by (3.32) is

Z(b) ∩ Ω̂_{Λ_{⊙}}^{ord} = Z(b)^{hor} + m(b, Λ_{⊙}) ℙ_{Λ_{⊙}}^{ord}  (3.34)

where

Z(b)^{hor} := Spf W [T, (T^{p} − T)^{−1}]^{∨}/(α_{0}T + α_{1}) ≃ Spf W.

Now suppose Λ′ is a type 2 lattice. We choose a basis {w_{0}, w_{1}} of Λ′ such that h(w_{0}, w_{1}) = p^{−1}δ and q(w_{0}) = q(w_{1}) = 0. Writing

b = p^{r′} (α_{0}w_{0} + α_{1}w_{1}) ,

with r′ = r(b, Λ′) and b_{0} := α_{0}w_{0} + α_{1}w_{1}, we have

q(b_{0}) = p^{−1}δ (α_{0}α′_{1} − α′_{0}α_{1}) .

By Proposition 3.11, the cycle Z(b) ∩ Ω̂_{Λ′}^{ord} is given by the vanishing of the element

p^{r′}(α′_{0} + α′_{1}T) ∈ W [T, (T^{p} − T)^{−1}]^{∨}.  (3.35)

Since we have assumed at the outset that ord_{p}q(b) is even, it follows that ord_{p}q(b_{0}) > −1. As before, this implies that the factor (α′_{0} + α′_{1}T) is a unit, and so the cycle Z(b) ∩ Ω̂_{Λ′}^{ord} is given by the equation p^{r′} = 0. Hence, we have the equality of cycles

Z(b) ∩ Ω̂_{Λ′}^{ord} = r′ ℙ_{Λ′}^{ord} = m(b, Λ′) ℙ_{Λ′}^{ord},  for Λ′ type 2.  (3.36)

Combining (3.33), (3.34) and (3.36), we have

Z(b)^{ord} = Z(b)^{hor} + ∑_{Λ} m(b, Λ) ℙ_{Λ}^{ord},

where Z(b)^{ord} denotes the restriction of Z(b) to the ordinary locus of D (i.e. the complement of the superspecial points). Now suppose x is a superspecial point lying in the intersection ℙ_{Λ} ∩ ℙ_{Λ′}, for a type 0 lattice Λ and its type 2 neighbour Λ′. Then Proposition 3.13 tells us that in a neighbourhood of x, the special cycle Z(b) is determined by the equation

(T_{0})^{r′} · (T_{1})^{r} = 0.

Recall that in such a neighbourhood, the components ℙ_{Λ} and ℙ_{Λ′} are given by the equations T_{1} = 0 and T_{0} = 0 respectively, and so meet Z(b) with multiplicities r = m(b, Λ) and r′ = m(b, Λ′) respectively. Therefore, we have

Z(b) = Z(b)^{hor} + ∑_{Λ} m(b, Λ) ℙ_{Λ}

as required. We have proven the theorem in the case of an anti-linear special homomorphism b ∈ 𝕍_{ϕ}^{−} with ord_{p}q^{−}(b) even; the other cases all follow in a completely analogous manner.

3.3 Relation to local orthogonal special cycles

In this section, we briefly recall definition of the local orthogonal special cycles constructed by Kudla and Rapoport [KR1], and relate them to the unitary special cycles of the previous section.

Definition 3.15. Suppose

j ∈ End_{O_{B,p}}(𝕏)_{ℚ_{p}} := End_{O_{B,p}}(𝕏) ⊗_{ℤ_{p}} ℚ_{p}

is an O_{B,p}-linear quasi-endomorphism of 𝕏. Define Z^{o}(j)^{♯} to be the closed formal subscheme of D which represents the following moduli problem: if S ∈ Nilp, we take Z^{o}(j)^{♯}(S) to be the locus of points (X, ι_{X}, ρ_{X}) ∈ D(S) such that the quasi-morphism

ρ_{X}^{−1} ∘ j ∘ ρ_{X} : X × \overline{S} → X × \overline{S}

lifts to an endomorphism of X. We also define Z^{o}(j) to be the Cohen-Macauleyfication of Z^{o}(j)^{♯}, namely the closed subscheme of Z^{o}(j)^{♯} defined by the ideal sheaf of sections with finite support, cf. [KR1, §4].

These cycles (and their global counterparts, discussed in the next section) have been studied extensively by Kudla-Rapoport [KR1] and Kudla-Rapoport-Yang [KRY]. Of immediate interest to us is the following special case of a result of Kudla and Rapoport, which is the counterpart to Theorem 3.14.

Proposition 3.16 ([KR1] Proposition 4.5). Suppose j ∈ End_{O_{B,p}}(𝕏)_{ℚ_{p}} such that j^{2} = u^{2}p^{2α}δ^{2} for some u ∈ ℤ_{p}^{×} (so in particular, T r(j) = 0, and ℚ_{p}(j) is isomorphic to k_{p}). Then Z^{o}(j) is a divisor on D. Moreover, let Λ_{0} be the unique vertex lattice such that j(Λ_{0}) = p^{α}Λ_{0}. For any vertex lattice Λ, define

μ^{o}(j, Λ) := max { α − d(Λ, Λ_{0}), 0 } .

Then there is an equality of cycles on D:

Z^{o}(j) = Z^{o}(j)^{hor} + ∑_{Λ} μ^{o}(j, Λ)ℙ_{Λ},

where Z^{o}(j)^{hor} is a disjoint sum of two divisors, each of which is isomorphic to Spf W and meets the special fibre of D at an ordinary special point in the component ℙ_{Λ_{0}}.

Suppose γ ∈ End_{O_{B,p}}(𝕏)_{ℚ_{p}}. Then γ induces an endomorphism of N(𝕏)_{0}, the 0-graded component of the rational Dieudonné module of 𝕏, and commutes with the operators F and V. Therefore γ defines an o_{k,p}-linear endomorphism on C := N(𝕏)_{0}^{V^{−1}F}, which, abusing notation, we also denote by γ. Now we observe that if j is as in Proposition 3.16, then as an endomorphism of C, its characteristic polynomial splits. In particular, we may find an eigenvector b_{0} ∈ C with eigenvalue p^{α}δ. By scaling, we may assume ord_{p}q(b_{0}) is either 0 or −1. Note that j also commutes with the Galois-semilinear operator ϵ := V^{−1} ∘ Π_{𝕏}, and so

(j ∘ ϵ)(b_{0}) = (ϵ ∘ j)(b_{0}) = −p^{α}δ · ϵ(b_{0}).

In other words, b_{0} and ϵ(b_{0}) are eigenvectors for j, with eigenvalues p^{α}δ and −p^{α}δ respectively. Moreover, the “central” lattice Λ_{0} attached to j, as in Proposition 3.16 (ii), is Λ_{0} = span{b_{0}, ϵ(b_{0})}. In anticipation of the next section, we make the following definition: for j as above, set

ν_{p}(j) = ν_{p}(j, ϕ) := \begin{cases}1, & if ∃ an eigenvector b ∈ C with ord_{p}q(b) odd \\ p, & otherwise. \end{cases} (3.37)

In other words, we have ν_{p}(j) = 1 if and only if there exists a homomorphism b ∈ 𝕍_{ϕ}^{+} such that

j ∘ b = p^{α} · (b ∘ i_{𝕐}(δ)) , with ord_{p}q^{+}(b) even.

The next two theorems relate the orthogonal and unitary special cycles on D.

Theorem 3.17. Suppose j ∈ End_{O_{B,p}}(𝕏)_{ℚ_{p}}, with j^{2} = u^{2}p^{2α}δ^{2} for some α > 0 and u ∈ ℤ_{p}^{×}. Abbreviate ν = ν_{p}(j), and fix b_{0} ∈ C an eigenvector with eigenvalue p^{α}δ such that q(b_{0}) = p^{−1}ν.

(i) If α is even, define b^{+} ∈ 𝕍_{ϕ}^{+} and b^{−} ∈ 𝕍_{ϕ}^{−} by the relations:

φ^{+}(b^{+}) = ν^{−1} · p^{α/2} · b_{0}, φ^{−}(b^{−}) = p^{α/2} · b_{0}.

(ii) If α is odd, define b^{+} and b^{−} by the relations:

φ^{+}(b^{+}) = p^{\frac{α−1}{2}} · b_{0}, φ^{−}(b^{−}) = ν^{−1} · p^{\frac{α+1}{2}} · b_{0}

Then we have an equality of cycles on D

Z^{o}(j) = Z(b^{+}) + Z(b^{−}).  (3.38)

Remark 3.18. The key feature of this formula is that in every case, exactly one of the special homomorphism b^{±} appearing on the right hand side has norm p^{α}, while the other has norm p^{α−1} (up to units in ℤ_{p}^{×}). ⋄

Proof. Note that the central lattices for the cycles Z^{o}(j), Z(b^{+}), and Z(b^{−}) are all the same lattice, namely Λ_{⊙} := span_{o_{k,p}}{b, ϵ(b)}. One can easily verify that in every case in the statement of the theorem, we have

m(b^{+}, Λ) + m(b^{−}, Λ) = α − ⌊d(Λ, Λ_{⊙})/2⌋ − ⌊(d(Λ, Λ_{⊙}) + 1)/2⌋
= α − d(Λ, Λ_{⊙})
= μ^{o}(j, Λ)

for any vertex lattice Λ with d(Λ, Λ_{⊙}) ≤ α. Indeed, the vectors b^{+} and b^{−} were scaled precisely so that this relation holds. On the other hand, if d(Λ, Λ_{⊙}) > α, then neither side of (3.38) meets the component ℙ_{Λ}. Hence, any component ℙ_{Λ} always occurs with the same multiplicity on both sides of (3.38). It therefore suffices to show the horizontal components of both sides, which live in the open affine Ω̂_{Λ_{⊙}}^{ord}, are equal. Suppose Λ_{⊙} is type 0, i.e.

ord_{p}q(b_{0}) = 0 = ord_{p}ν_{p}(j).

Let {v_{0}, v_{1}} denote an ϵ-invariant basis for Λ_{⊙}, with q(v_{0}) = q(v_{1}) = 0, and h(v_{0}, v_{1}) = δ, and write

b_{0} = a_{0}v_{0} + a_{1}v_{1}, ϵ(b_{0}) = a′_{0}v_{0} + a′_{1}v_{1}.

Note that q(b_{0}) = −q(ϵ(b_{0})) = δ(a_{0}a′_{1} − a′_{0}a_{1}) is a unit in ℤ_{p}^{×}, by assumption. Let j_{0} := p^{−α}j. Then (j_{0})^{2} = δ^{2}, and by [KR1, Proposition 3.2], we have

Z^{o}(j)^{hor} = Z^{o}(j_{0}).

As an endomorphism of C, the element j_{0} is determined by the fact that b_{0} and ϵ(b_{0}) are eigenvectors with eigenvalues δ and −δ respectively. It is therefore defined by the matrix

[j] = \frac{δ}{a_{0}a′_{1} − a′_{0}a_{1}} (\begin{matrix}a_{0}a′_{1} + a′_{0}a_{1} & −2n(a_{0}) \\ 2n(a_{1}) & −a_{0}a′_{1} − a′_{0}a_{1} \end{matrix}) ∈ M_{2}(ℤ_{p})

with respect to the basis {v_{0}, v_{1}}. Now in the affine neighbourhood

Ω̂_{Λ_{⊙}}^{ord} ≃ Spf W [T, (T^{p} − T)^{−1}]^{∨},

the proof of Proposition 3.10 says that the cycle Z(b^{+})^{hor} + Z(b^{−})^{hor} is given by the vanishing of

(a_{0}T + a_{1}) · (a′_{0}T + a′_{1}) = n(a_{0})T^{2} + (a_{0}a′_{1} + a′_{0}a_{1})T + n(a_{1}).  (3.39)

On the other hand, by [KR1, Equation (3.5)], the equation for Z^{o}(j_{0}) in Ω̂_{Λ_{⊙}}^{ord} is given by

\frac{−2δ}{a_{0}a′_{1} − a′_{0}a_{1}} (n(a_{0})T^{2} + (a_{0}a′_{1} + a′_{0}a_{1})T + n(a_{1})) ,

which differs from (3.39) by a scalar in ℤ_{p}^{×}, and hence defines the same divisor. If Λ_{⊙} = span{b_{0}, ϵ(b_{0})} is a type 2 lattice, fix an ϵ-invariant basis {w_{0}, w_{1}} for Λ_{⊙}, such that h(w_{0}, w_{1}) = p^{−1}δ and q(w_{0}) = q(w_{1}) = 0. As before, we write

b_{0} = a_{0}w_{0} + a_{1}w_{1}, ϵ(b_{0}) = a′_{0}w_{0} + a′_{1}w_{1},

and note that a_{0}a′_{1} − a′_{0}a_{1} ∈ o_{k,p}^{×}. The cycle Z(b^{+})^{hor} + Z(b^{−})^{hor} in Ω̂_{Λ_{0}}^{ord} is then given by the vanishing of the element

(a_{0} + a_{1}T) · (a′_{0} + a′_{1}T) = n(a_{0}) + (a_{0}a′_{1} + a′_{0}a_{1})T + n(a_{1})T^{2}.

Turning to the special endomorphism j, we note that it acts on C via the matrix

[j] = \frac{δ}{a_{0}a′_{1} − a′_{0}a_{1}} (\begin{matrix}a_{0}a′_{1} + a′_{0}a_{1} & −2n(a_{0}) \\ 2n(a_{1}) & −a_{0}a′_{1} − a′_{0}a_{1} \end{matrix})

with respect to the basis {w_{0}, w_{1}}. One can then check (e.g. by exploiting the action of GL_{2}(ℚ_{p}) on D cf. [KR1, §1]), that the equation for Z^{o}(j_{0}) in this case is given by

\frac{−2δ}{a_{0}a′_{1} − a′_{0}a_{1}} (n(a_{0}) + (a_{0}a′_{1} + a′_{0}a_{1})T + n(a_{1})T^{2}) ,

and hence defines the divisor Z(b^{+})^{hor} + Z(b^{−})^{hor}, as required.

We also have the corresponding theorem in the case α = 0; note that all the cycles involved are horizontal.

Theorem 3.19. Suppose j ∈ End_{O_{B,p}}(𝕏)_{ℚ_{p}} with j^{2} = u^{2}δ^{2} with u ∈ ℤ_{p}^{×}. Let b_{0} ∈ C denote an eigenvector with eigenvalue δ, and suppose that q(b_{0}) = p^{−1}ν_{p}(j, ϕ), so ord_{p}q(b_{0}) is 0 or −1. Then if ν_{p}(j, ϕ) = 1, and we define b_{1}, b_{2} ∈ 𝕍_{ϕ}^{+} by the formulas

b_{1} = φ^{+}(b_{0}), b_{2} = φ^{+}(ϵ(b_{0})),

then

Z^{o}(j) = Z(b_{1}) + Z(b_{2}).  (3.40)

Similarly, if ν_{p}(j, ϕ) = p and we define b_{1}, b_{2} ∈ 𝕍_{ϕ}^{−} by

b_{1} = φ^{−}(b_{0}), b_{2} = φ^{−}(ϵ(b_{0})),

then

Z^{o}(j) = Z(b_{1}) + Z(b_{2}).  (3.41)

Proof. The proof is completely analogous to the proof of Theorem 3.17, and is omitted.

3.4 p-adic uniformizations

Recall that we have fixed maps

τ_{0} : o_{k}/p → 𝔽  and  τ_{0} : o_{k,p} → W.

In this section, when we take the 𝔽- or W -valued points of a stack defined over o_{k}, we shall implicitly do so via the o_{k}-structure induced by τ_{0}. If X denotes any of the stacks we have defined so far, we let X̃ denote the base change to W of the formal completion of its fibre at p. The p-adic uniformizations we are about to describe relate these completions to the moduli spaces of p-divisible groups discussed in the previous section.

We begin with the Shimura curve C_{B}. Fix a pair \underline{A} = (A, ι_{A}) ∈ C_{B}(𝔽), such that the p-divisible group

\underline{A}[p^{∞}] = (A[p^{∞}], ι_{A} ⊗ ℤ_{p}) ∈ D(𝔽),

together with the induced O_{B,p}-action, is equal to the “base-point” (𝕏, ι_{X}) that we had fixed in defining the Drinfeld moduli space D, cf. Definition 3.1. Let B′ = End_{O_{B}}(A)⊗_{ℤ}ℚ. Then B′ is a quaternion algebra over ℚ whose invariants differ from those of B at exactly p and ∞. We let H′ = (B′)^{×}, viewed as an algebraic group over ℚ, and define

H′(ℚ)^{⋆} :={h′ ∈ H′(ℚ) | ord_{p}Nrd′(h′) = 0},

where Nrd′ is the reduced norm on B′. Let K^{p} ⊂ H′(𝔸_{f}^{p}) denote the image of (Ô_{B}^{p})^{×} under the natural identification B′(𝔸_{f}^{p}) ≃ B^{op}(𝔸_{f}^{p}), and set

Γ′ := K^{p} ∩ H′(ℚ)^{⋆}.

Then Γ′ acts on the Drinfeld upper-half plane D as follows: if \underline{X} = (X, ι_{X}, ρ_{X}) ∈ D(S) for some S ∈ Nilp, and γ ∈ Γ′, then we set

γ · \underline{X} := (X, ι_{X}, γ_{p} ∘ ρ_{X}),

where γ_{p} denotes the image of γ in B′ ⊗_{ℚ} ℚ_{p} = End_{O_{B,p}}(𝕏) ⊗_{ℤ_{p}} ℚ_{p}. We then obtain the following p-adic uniformization of C_{B}:

Theorem 3.20. There is an isomorphism of formal stacks over Nilp:

C̃_{B} ≃ [Γ′\ D] .  (3.42)

Proof. This is a restatement of [BC, Theorem III.5.3], see also [KR1, §8]. Primarily to set up notation for the sequel, we indicate the idea of the proof. We begin by fixing an O_{B}-linear isomorphism

η_{A} : T a^{p}(A) ⟶^{∼} Ô_{B}^{p},

where T a^{p}(A) := ∏_{ℓ≠p} T a_{ℓ}(A) is the prime-to-p Tate module, and Ô_{B}^{p} := O_{B} ⊗ℤ̂^{p} viewed as an O_{B}-module via left-multiplication. Let (A, ι) ∈ C_{B}(S), for some S ∈ Nilp. Then there exists an O_{B}-linear quasiisogeny

ψ_{A}: A → A  (3.43)

such that (i) the induced map on p-divisible groups (over \overline{S} = S ×_{W} 𝔽 )

ψ_{A}[p^{∞}]\overline{S}: A[p^{∞}] × \overline{S} → A[p^{∞}] × \overline{S} = 𝕏 ×\overline{S}

is a quasi-isogeny of height 0, and (ii) the composition

η_{A} ∘ ψ_{A}^{p}: T a^{p}(A)_{ℚ} → B(𝔸_{f}^{p})

maps T a^{p}(A) isomorphically onto Ô_{B}^{p}. Noting that any two quasi-isogenies satisfying both these properties necessarily differ by an element of Γ′, it follows that the map

C̃_{B} → [Γ′\ D], (A, ι) ↦ [A[p^{∞}], ι ⊗ ℤ_{p}, ψ_{A}[p^{∞}]\overline{S}]

is an isomorphism.

We have a similar statement for the orthogonal special cycles ( Definition 2.3).

Theorem 3.21. For n ∈ ℤ_{>0}, let

Ω^{o}(n) :=

{ξ ∈ B′(ℚ) | T rd′(ξ) = 0, ξ^{2} = −n, and η_{A} ∘ T a^{p}(ξ) ∘ η_{A}^{−1} ∈ End(Ô_{B}^{p})} ,

(3.44)

and note that Γ′ acts on Ω^{o}(n) by conjugation. Then there is an isomorphism

Z̃^{o}(n) ≃ [Γ′ \ ∐_{ξ∈Ω^{o}(n)} Z^{o}(ξ[p^{∞}])]

as formal stacks over W. Here ξ[p^{∞}] ∈ End_{O_{B}}(𝕏)_{ℚ_{p}} is the endomorphism of 𝕏 = A[p^{∞}] induced by ξ, and Z^{o}(ξ[p^{∞}]) is a local orthogonal special cycle as in Definition 3.15.

Proof. Rewrite [KR1, (8.17)] in terms of the uniformization Theorem 3.20.

We now turn to the p-adic uniformization of the unitary special cycles, following [KR3, §6]. Recall that we had fixed an embedding τ_{0} : o_{k}/(p) → 𝔽 = 𝔽_{p}^{alg}, which lifts uniquely to an embedding τ_{0} : o_{k} → W = W (𝔽); via these maps, we may view both 𝔽 and W as o_{k}-algebras. Fix a triple \underline{E} = (E, i_{E}, λ_{E}) ∈ ℰ^{+}(𝔽). We may also identify the p-divisible group \underline{E}[p^{∞}], with its extra data, with the triple \underline{𝕐} = (𝕐, i_{𝕐}, λ_{𝕐}) of the previous section. We may further assume, by replacing E with an isogenous elliptic curve if necessary, that there is an o_{k}-linear isomorphism

η_{E} : T a^{p}(E) ⟶^{∼} ô_{k}^{p}  (3.45)

such that the pullback of the symplectic form

(a, b) ↦ tr(ab′(\sqrt{Δ})^{−1})

on ô_{k}^{p} is equal to the Weil pairing

e_{E} : T a^{p}(E)_{ℚ} × T a^{p}(E)_{ℚ} → 𝔸_{f}^{p}(1) ≃ 𝔸_{f}^{p} . defined by λ_{E}. The two base points \underline{E} = (E, i_{E}, λ_{E}) and \underline{A} = (A, ι_{A}) allow us to define the global analogues of the spaces of special homomorphisms of the previous section, as follows. For any optimal embedding ϕ : o_{k} ↪ O_{B}, let

V_{ϕ}^{+} := {β ∈ Hom(E, A)_{ℚ} | β ∘ i_{E}(a) = ι_{A} (ϕ(a)) ∘ β for all a ∈ o_{k,p}}

and V_{ϕ}^{−} := {β ∈ Hom(E, A)_{ℚ} | β ∘ i_{E}(a) = ι_{A} (ϕ(a′)) ∘ β for all a ∈ o_{k,p}} .

We view these spaces as k-vector spaces via the action

a · β := β ∘ i_{E}(a), for all a ∈ k, β ∈ V_{ϕ}^{±}.

Recall that for any optimal embedding ϕ, we had defined a (non-principal) polarization λ_{A,ϕ} on A, as in (2.1). The spaces V_{ϕ}^{±} can then be equipped with Hermitian forms h^{±}, defined by the formulas:

h^{+}(β_{1}, β_{2}) :=λ_{E}^{−1} ∘ β_{2}^{∨} ∘ λ_{A,ϕ} ∘ β_{1} ∈ End_{o_{k}}(E)_{ℚ} ≃ k,  β_{1}, β_{2} ∈ V_{ϕ}^{+} (3.46)

and

h^{−}(β_{1}, β_{2}) :=λ_{E}^{−1} ∘ β_{1}^{∨} ∘ λ_{A,ϕ} ∘ β_{2} ∈ End_{o_{k}}(E)_{ℚ} ≃ k,  β_{1}, β_{2} ∈ V_{ϕ}^{−} (3.47)

respectively. Let q^{±}(β) = h^{±}(β, β) denote the associated quadratic forms. Finally, we recall the decomposition ℰ = ℰ^{+} ∐ ℰ^{−}, which induces a decomposition of the (global) cycles

Z(m, ϕ) = Z^{+}(m, ϕ) ∐ Z^{−}(m, ϕ).

We may now state their p-adic uniformization:

Theorem 3.22. Suppose m ∈ ℤ_{>0}. For any fractional ideal 𝔞 in k, and any optimal embedding ϕ : o_{k} → O_{B}, let

Ω^{±}(m, 𝔞, ϕ) :=

\{β ∈ V_{ϕ}^{±} | q^{±}(β) = m · \frac{p^{ord_{p}N(𝔞)}}{N(𝔞)} , η_{A} ∘ β^{p} ∘ η_{E}^{−1}(𝔞̂^{p}) ⊂ Ô_{B}^{p}\}, (3.48)

where, for any β ∈ V_{ϕ}^{±}, we denote by β^{p}: T a^{p}(E)_{ℚ} → T a^{p}(A)_{ℚ} the induced map on rational prime-to-p Tate modules. Note Γ′ acts on these sets by composition, and we let o_{k}^{×} act via the k-vector space structure on V_{ϕ}^{±}. Then

Z̃^{±}(m, ϕ) ≃ [(Γ′ × o_{k}^{×})\ ( ∐_{[𝔞]∈Cl(k)} ∐_{β∈Ω^{±}(m,𝔞,ϕ)} Z(β[p^{∞}]))] , (3.49)

where β[p^{∞}] ∈ 𝕍_{ϕ}^{±} is the quasi-morphism of p-divisible groups induced by β. Here 𝔞 ranges any set of representatives of the class group Cl(k) of k.

Proof. This is a reformulation of [KR3, Proposition 6.3] in the present context.

Remark 3.23. Note that the right hand side of (3.49) is independent of the choices of representatives 𝔞 ∈ Cl(k). Indeed, for any a ∈ k^{×}, we have a bijection

Ω^{±}(m, a · 𝔞, ϕ) ⟶^{∼} Ω^{±}(m, 𝔞, ϕ),  β ↦ p^{−ord_{p}n(a)/2}a · β.  (3.50)

However, the cycles Z(β[p^{∞}]) on D depend only on (i) the image of β[p^{∞}] in ℙ(𝕍_{ϕ_{0}}^{±}) (which determines the horizontal part), and (ii) the p-adic valuation of q^{±}(β), (which determines the vertical part). By construction, both of these are the same for elements β appearing on either side of (3.50).

4 Proof of the main theorem

4.1 The (formal) Shimura lift

The Shimura lift is a classical operation that takes modular forms of halfintegral weight to modular forms of even weight. In its original formulation [Shim], the lift of a modular form F is realized by writing down a formal qexpansion involving the Fourier coefficients of F, and then proving that it is indeed the q-expansion of a modular form with the desired properties. This recipe inspires the following definition:

Definition 4.1 (The formal Shimura lift). Let M be any ℤ-module, and

F = ∑_{n≥0} a(n) q^{n} ∈ M⟦q⟧

a formal power series with coefficients in M. Suppose that

(i) κ ≥ 3 is an odd integer and λ := (κ − 1)/2; (ii) N and t are positive integers with t squarefree; (iii) and χ is a Dirichlet character modulo 4N. We define a new character

χ_{t}(a) := χ(a) (\frac{−1}{a})^{(κ−1)/2} (\frac{t}{a}) ,

where (·/·) denotes Shimura’s modification of the Kronecker symbol, cf. [Cip, Appendix A]. Then the Shimura lift of F, with respect to the parameters (κ, N, t, χ), is by definition the formal power series

Sh(F) := ∑_{m≥0} b(m) q^{tm} ∈ M⟦q⟧ ⊗ ℂ  (4.1)

whose coefficients are given as follows. For m > 0, we set

b(m) := ∑_{n|m} χ_{t}(n) n^{(κ−3)/2} a (t\frac{m^{2}}{n^{2}} ) .  (4.2)

The constant term b(0) is given as follows: let

χ̌_{t}(a) = ∑_{h=0}^{4Nt−1} χ_{t}(h) exp(2πiah/4Nt)

denote the Gauss sum, and set

b(0) = a(0) \frac{(−1)^{λ}}{4} (2πi)^{λ}(π)^{−2λ}(Nt)^{λ−1} Γ(λ) ∑_{m>0} m^{−λ}χ̌_{t}(m).  (4.3)

Note that if χ^{2} = 1 then b(m) ∈ M for all m > 0.

As alluded to above, if F is the q-expansion of a holomorphic modular form of weight κ/2, level Γ_{0}(4N) and nebencharacter χ, then for each squarefree integer t, the Shimura lift Sh(F) is the q-expansion of a modular form of weight κ − 1 for Γ_{0}(4Nt) with character χ^{2}, cf. [Cip, Proposition 2.17].

4.2 The Shimura lift formula in fibres of bad reduction

In this section, which is the technical heart of the paper, we show that a relation in the spirit of (4.2) holds for the orthogonal and unitary special cycles in a formal neighbourhood of the fibre at p for p|D_{B} by comparing their p-adic uniformizations. Let 𝔽 = \overline{𝔽}_{p}, and fix a base point \underline{A} ∈ C_{B}(𝔽) as in Section 3.4. Our first aim is to relate the sets Ω^{o}(n) and Ω^{±}(m, 𝔞, ϕ) in the case of interest, when the squarefree part of n is equal to |Δ|. Suppose ξ ∈ Ω^{o}(|Δ|a^{2}), for some a ∈ ℤ_{>0}. Then, by assumption, the endomorphism

η_{A}^{−1} ∘ ξ^{p} ∘ η_{A} ∈ End_{O_{B}}(Ô_{B}^{p})

is given by right-multplication by some finite adele (x_{ℓ})_{ℓ≠p} ∈ Ô_{B}^{p}, such that (x_{ℓ})^{2} = a^{2}Δ. Therefore, for every ℓ ≠ p, we obtain an embedding

φ_{ℓ} : k_{ℓ} = k ⊗_{ℚ} ℚ_{ℓ} → B_{ℓ}, a\sqrt{Δ} ↦ x_{ℓ}.

We define the conductor c = c(ξ) of ξ to be the smallest (rational) positive integer such that for all ℓ ≠ p, we have

φ_{ℓ} (ℤ_{ℓ}[c\sqrt{Δ}]) ⊂ O_{B}_{,ℓ} .

In other words, c(ξ) is the smallest integer such that φ_{ℓ} maps the unique o_{k}- order of conductor c(ξ) into O_{B}_{,ℓ}, for all ℓ ≠ p. We note that since

φ_{ℓ} (ℤ_{ℓ}[a\sqrt{Δ}]) = ℤ_{ℓ} + ℤ_{ℓ} x_{ℓ} ⊂ O_{B}_{,ℓ},

we have that c(ξ) divides a, and by valuation considerations it is easy to see (c(ξ), D_{B}) = 1. In particular, we obtain a disjoint decomposition

Ω^{o}(|Δ|a^{2}) = ∐_{\begin{matrix}c|a \\ (c,D_{B})=1 \end{matrix}} Ω^{o}(|Δ|a^{2}, c),  (4.4)

where Ω^{o}(|Δ|a^{2}, c) denotes the subset of elements ξ ∈ Ω^{o}(|Δ|a^{2}) with c(ξ) = c. Note also that if ξ ∈ Ω^{o}(|Δ|a^{2}, c) and t is any integer, then the conductor of t · ξ is again c, and one checks easily that we have a bijection

Ω^{o}(|Δ|a^{2}, c) ⟶^{∼} Ω^{o}(|Δ|a^{2}t^{2}, c),  ξ ↦ t · ξ.  (4.5)

Now suppose ϕ: o_{k} → O_{B} is an optimal embedding, and ℓ ≠ p is a prime dividing D_{B}; recall our standing assumption that such a prime is inert in k. Reducing modulo ℓ, we obtain two maps

\overline{φ}_{ℓ}, \overline{ϕ}_{ℓ}: o_{k,ℓ}/(ℓ) → O_{B}_{,ℓ} /(θ),

where θ ∈ O_{B} is a fixed element such that θ^{2} = −D_{B}. As both the source and target of the maps are isomorphic to the field 𝔽_{ℓ^{2}}, the two maps \overline{ϕ}_{ℓ} and \overline{φ}_{ℓ} are either equal, or they differ by the Frobenius automorphism on 𝔽_{ℓ^{2}}. This observation allows to define the Frobenius type away from p of ξ as follows:

ν^{p}(ξ, ϕ) := ∏_{\begin{matrix}ℓ|D_{B} \\ ℓ≠p \end{matrix}} ν_{ℓ}(ξ, ϕ),  where ν_{ℓ}(ξ, ϕ) := \begin{cases}1, & if \overline{ϕ}_{ℓ} = \overline{φ}_{ℓ} \\ ℓ, & otherwise. \end{cases}  (4.6)

Recall that we also had the notion of a Frobenius type

ν_{p}(ξ, ϕ) := ν_{p}(ξ[p^{∞}], ϕ_{0})  (4.7)

at p, which was defined in (3.37) in terms of the parity of the p-valuation of the norm of any eigenvector of the induced map ξ[p^{∞}] on Dieudonné modules. For an optimal embedding ϕ, and an element β ∈ V_{ϕ}^{+}, we may also define the notion of a conductor, as follows. Let

(h_{ℓ})_{ℓ≠p} := η_{A} ∘ β^{p} ∘ η_{E}^{−1}(1) ∈ B(𝔸_{f}^{p}),

and for each ℓ ≠ p, define an embedding φ′_{ℓ} : k_{ℓ} → B_{ℓ} by the formula

φ′_{ℓ}(\sqrt{Δ}) = (h_{ℓ})^{−1} · ϕ(\sqrt{Δ}) · h_{ℓ}.

As before, we define the conductor c(β) of β to be the smallest integer c such that φ′_{ℓ}(o_{c}) ⊂ O_{B}_{,ℓ} for all ℓ ≠ p, where o_{c} = ℤ[c\sqrt{Δ}] is the unique order of conductor c. Note that this quantity depends on the choice of embedding ϕ.

Lemma 4.2. Suppose β ∈ V_{ϕ}^{+}, and let h = (h_{ℓ})_{ℓ≠p} := η_{A} ∘β^{p} ∘η_{E}^{−1}(1) ∈ B(𝔸_{f}^{p}). Then

(i) q^{+}(β) = Δ Nrd_{B(𝔸_{f}^{p})}(h) ∈ 𝔸_{f}^{p}. (ii) If β ∈ Ω^{+}(m, 𝔞, ϕ), then the conductor c(β) divides m.

Proof. (i) Recalling our fixed trivialization 𝔸_{f}^{p}(1) ≃ 𝔸_{f}^{p}, consider the diagram

T a^{p}(E)_{ℚ} × T a^{p}(E)_{ℚ}  ⟶^{β^{p}×β^{p}}  T a^{p}(A)_{ℚ} × T a^{p}(A)_{ℚ}  ⟶^{e_{A}} 𝔸_{f}^{p}(1) ≃ 𝔸_{f}^{p}
η_{E}×η_{E}↓  ^{η_{A}×η_{A}}↓  ∥
𝔸_{k,f}^{p} × 𝔸_{k,f}^{p} \underrightarrow{(x,y)↦(ϕ(x)h,ϕ(y)h)} B(𝔸_{f}^{p}) × B(𝔸_{f}^{p}) \underrightarrow{(x,y)↦T rd(xy^{ι}ϕ(\sqrt{Δ}))} 𝔸_{f}^{p}

where e_{A} is the Weil pairing on A defined by the polarization λ_{A,ϕ}. It is an immediate consequence of the definitions that both squares commute. Let

e_{E} : T a^{p}(E)_{ℚ} × T a^{p}(E)_{ℚ} → 𝔸_{f}^{p}

denote the Weil pairing on E defined by λ_{E}. Then on the one hand, by taking adjoints, we have

e_{A} (β^{p}(x), β^{p}(y)) = q^{+}(β) · e_{E}(x, y), x, y ∈ T a^{p}(E)_{ℚ},

while on the other hand, if we write s = η_{E}(x) and t = η_{E}(y), then the commutative diagram above tells us that

e_{A} (β^{p}(x), β^{p}(y)) = T rd [ ϕ(s)h · (ϕ(t)h)^{ι} · ϕ(\sqrt{Δ})]
= Nrd(h) · T rd (ϕ(s · t′ · \sqrt{Δ}))
= Nrd(h) Δ e_{E}(x, y),

where the last line follows by the choice of η_{E}, as in (3.45). This proves (i). (ii) Suppose β ∈ Ω^{+}(m, 𝔞, ϕ). It suffices to check that

m · (h_{ℓ})^{−1}ϕ(\sqrt{Δ})h_{ℓ} ∈^{?} O_{B}_{,ℓ}, for all ℓ ≠ p.

Choose a_{ℓ} ∈ k_{ℓ}^{×} such that 𝔞 ⊗ ℤ_{ℓ} = a_{ℓ} · o_{k,ℓ}. Then by the definition of Ω^{+}(m, 𝔞, ϕ), we have h_{ℓ} ∈ ϕ(a_{ℓ}^{−1}) · O_{B}_{,ℓ}, and in particular,

(h_{ℓ})^{ι} ϕ(\sqrt{Δ}) h_{ℓ} ∈ \frac{1}{N(𝔞)} O_{B}_{,ℓ} .

Hence, by combing part (i) of the lemma with the assumption

q^{+}(β) = m · \frac{p^{ord_{p}N(𝔞)}}{N(𝔞)} ,

we have

m · (h_{ℓ})^{−1}ϕ(\sqrt{Δ})h_{ℓ} = p^{−ord_{p}(N(𝔞))} · Δ · N(𝔞) · [(h_{ℓ})^{ι} ϕ(\sqrt{Δ}) h_{ℓ}] ∈ O_{B}_{,ℓ}

as required.

Lemma 4.3. Let β ∈ Ω^{+}(m, 𝔞, ϕ). Then there is a unique element ξ = ξ(β) ∈ Ω^{o}(|Δ|m^{2}) such that ξ ∘ β = m\sqrt{Δ} · β. Moreover, the conductor c(ξ) of ξ is equal to c(β).

Proof. Note that each ξ ∈ B′(ℚ) defines a k-linear endomorphism [ξ] of V_{ϕ}^{+} by composition:

[ξ] · β := ξ ∘ β.

Moreover, it follows immediately from definitions that

q^{+}([ξ] · β, [ξ] · β) = Nrd′(ξ) · q^{+}(β, β) for all β ∈ V_{ϕ}^{+}.

Hence, if β ∈ Ω^{+}(m, 𝔞, ϕ) and ξ_{1}, ξ_{2} both satisfy [ξ_{i}] · β = m\sqrt{Δ} · β, then Nrd′(ξ_{1} − ξ_{2}) = 0, so ξ_{1} = ξ_{2}, as B′ is division. This proves uniqueness. To show existence, we choose an element ϑ ∈ O_{B} with ϑ^{ι} = −ϑ, and such that for all a ∈ k, we have ϑϕ(a) = ϕ(a′)ϑ. Then we may define a quasi-isogeny

ψ_{β}: E × E → A, (x, y) ↦ β(x) + ι_{A}(ϑ) · β(y).

Note that for all a ∈ o_{k}, we have

ι_{A} (ϕ(a)) · ψ_{β}(x, y) = ψ_{β} (i_{E}(a) · x, i_{E}(a′) · y)  (4.8)
and  ι_{A} (ϑ) · ψ_{β}(x, y) = ψ_{β}(ϑ^{2}y, x).  (4.9)

We may then define an element ξ ∈ End(A)_{ℚ} by the formula

ξ := m · ψ_{β} ∘ (i_{E}(\sqrt{Δ}), i_{E}(\sqrt{Δ})) ∘ ψ_{β}^{−1}.

It follows from (4.8) and (4.9) that ξ commutes with ι_{A}(ϑ), and ι_{A}(ϕ(a)) for all a ∈ o_{k}, so in fact ξ ∈ B′(ℚ) = End_{O_{B}}(A)_{ℚ}, and it is easily seen that

ξ ∘ β = m · β ∘ i_{E}(\sqrt{Δ}) = m\sqrt{Δ} · β.  (4.10)

It is also straightforward to verify that ξ satisfies the conditions (3.44) defining Ω^{o}(m^{2}|Δ|), as well as the claim regarding conductors.

Proposition 4.4. Fix a set of representatives {𝔞_{1}, …, 𝔞_{h}} for the class group Cl(k) of k, such that each 𝔞_{i} is relatively prime to (p). Consider the map

f: ∐_{i=1}^{h} Ω^{+}(m, 𝔞_{i}, ϕ) → Ω^{o}(|Δ|m^{2}), β ↦ ξ(β)

where ξ(β) is the unique element which has β as an eigenvector with eigenvalue m\sqrt{Δ}, as in Lemma 4.3. For any ξ ∈ Ω^{o}(|Δ|m^{2}), let ν^{p} = ν^{p}(ξ, ϕ) (resp. ν_{p} = ν_{p}(ξ[p^{∞}], ϕ)) denote its Frobenius type away from (resp. at) p, and c = c(ξ) its conductor. Then we have

# (f^{−1}(ξ)) = |o_{k}^{×}| · ρ (\frac{m}{c|Δ|ν_{p}ν^{p}} ) ,

where for any rational number N, ρ(N) denotes the number of integral ideals of k of norm N. In particular, if m/c|Δ|ν_{p}ν^{p} is not an integer, then the fibre f^{−1}(ξ) is empty.

Proof. Let ξ ∈ Ω^{o}(|Δ|m^{2}). Viewing ξ as a k-linear endomorphism of V_{ϕ}^{+}, it has two distinct eigenvalues ±m\sqrt{Δ} and so there certainly exists some β_{0} ∈ V_{ϕ}^{+} such that

ξ · β_{0} = m\sqrt{Δ} · β_{0}.

Moreover, any other eigenvector with the same eigenvalue differs from β_{0} by a scalar in k^{×}. Hence, we have

#f^{−1}(ξ) = # ∐_{i=1}^{h} {β ∈ Ω^{+}(m, 𝔞_{i}, ϕ) | ξ(β) = ξ}

= # ∐_{i=1}^{h} \{a ∈ k^{×} | q^{+}(aβ_{0}) = \frac{m}{N(𝔞_{i})}, η_{A} ∘ β_{0}^{p} ∘ η_{E}^{−1}(a𝔞̂_{i}^{p}) ⊂ Ô_{B}^{p}\}
=|o_{k}^{×}| · #{𝔞 ⊂ k a fractional ideal |
N(𝔞) = \frac{m}{q^{+}(β_{0})}, η_{A} ∘ β_{0}^{p} ∘ η_{E}^{−1}(𝔞̂^{p}) ⊂ Ô_{B}^{p}\}. (4.11)

As before, set

h = (h_{ℓ})_{ℓ≠p} := η_{A} ∘ β_{0}^{p} ∘ η_{E}^{−1}(1),  and  x = (x_{ℓ})_{ℓ≠p} := η_{A} ∘ ξ^{p} ∘ η_{A}^{−1}(1).

Then the condition ξ · β_{0} = m\sqrt{Δ} · β_{0} implies

x_{ℓ} = m · h_{ℓ}^{−1} ϕ(\sqrt{Δ}) h_{ℓ}, for all ℓ ≠ p.

For each ℓ ≠ p, let φ_{ℓ}: k_{ℓ} → B_{ℓ} be the embedding determined by the relation

φ_{ℓ}(\sqrt{Δ}) = m^{−1}x_{ℓ} = h_{ℓ}^{−1} ϕ(\sqrt{Δ}) h_{ℓ}.  (4.12)

We shall translate the conditions on the right hand side of (4.11) to a collection of local ones. First, suppose that ℓ is a prime not dividing D_{B}. We may fix an isomorphism O_{B}_{,ℓ} ≃ M_{2}(ℤ_{ℓ}) that identifies

ϕ(\sqrt{Δ})  with  (\begin{matrix}& 1 \\ Δ \end{matrix});  (4.13)

note that this is possible when ℓ = 2 on account of the assumption that |Δ| is even. Let c = c(ξ) be the conductor of ξ, and define

w(c)_{ℓ} := (\begin{matrix}c \\ & 1 \end{matrix}) ∈ O_{B}_{,ℓ} .

Then the map

φ_{c}: k_{ℓ} → B_{ℓ}, φ_{c}(\sqrt{Δ}) = w(c)_{ℓ}^{−1} ϕ(\sqrt{Δ}) w(c)_{ℓ}

is a local embedding of conductor c; that is, we have

φ_{c}(k_{ℓ}) ∩ O_{B}_{,ℓ} = φ_{c} ( ℤ_{ℓ}[c\sqrt{Δ}] ) .

By the definition of the conductor of ξ, the same is true for the embedding φ_{ℓ} : k_{ℓ} → B_{ℓ} as in (4.12). However, by [Vig, Theoreme II.3.2], any two embeddings of conductor c necessarily differ by an inner automorphism determined by an element of (O_{B}_{,ℓ})^{×}. Furthermore, for x, y ∈ B_{ℓ}^{×},

Ad_{x^{−1}} ∘ ϕ = Ad_{y^{−1}} ∘ ϕ  ⟺  x = ϕ(a)y for some a ∈ k_{ℓ}^{×}.

Hence, for each ℓ not dividing D_{B}, we may write

h_{ℓ} = ϕ(a_{ℓ}) · w(c)_{ℓ} · u_{ℓ}  (4.14)

for some a_{ℓ} ∈ k_{ℓ}^{×} and u_{ℓ} ∈ (O_{B}_{,ℓ})^{×}. Note moreover that h_{ℓ} ∈ (O_{B}_{,ℓ})^{×} for almost all ℓ. Now consider a prime ℓ|D_{B}, ℓ ≠ p. Fix a uniformizer Π_{ℓ} ∈ O_{B}_{,ℓ} such that Π_{ℓ}ϕ(a) = ϕ(a′)Π_{ℓ} for all a ∈ k_{ℓ}. Recalling our assumption that ℓ is inert in k, we may write

h_{ℓ} = ϕ(a_{ℓ}) · (Π_{ℓ})^{ϵ_{ℓ}},  (4.15)

for some a_{ℓ} ∈ k_{ℓ}^{×} and ϵ_{ℓ} ∈ {0, 1}. One checks that for the reductions

\overline{φ}_{ℓ}, \overline{ϕ} ∈ Hom (o_{k,ℓ}/(ℓ), O_{B}_{,ℓ} /(Π_{ℓ})) ,

we have

\overline{φ}_{ℓ} = \overline{ϕ} ⟺ ϵ = 0,

where φ_{ℓ} is the embedding determined by h_{ℓ} as in (4.12). Hence, by definition of the Frobenius type (4.6),

ϵ_{ℓ} = ord_{ℓ} (ν_{ℓ}(ξ, ϕ)) .

At this point, we have amassed a list of elements (a_{ℓ}) ∈ (𝔸_{k,f}^{p})^{×}, as they appear in (4.14) and (4.15). We supplement this list with an element a_{p} ∈ k_{p}^{×} defined as follows: note that by the definition of the Frobenius type at p (cf. (4.7) and (3.37)), we have

ord_{p} (ν_{p}(ξ, ϕ)) ≡ ord_{p}q^{+}(β_{0}) (mod 2).

We then set

a_{p} := p^{^{\frac{1}{2}}(ord_{p}q^{+}(β_{0}) − ord_{p}ν(ξ,ϕ))}.

Let 𝔞_{0} denote the fractional ideal defined by (a_{ℓ}) ∈ 𝔸_{k,f}^{×}.

Abbreviating ν_{ℓ} = ν_{ℓ}(ξ, ϕ), the product formula for ℚ implies

q^{+}(β_{0}) = (∏_{ℓ≠p} |q^{+}(β_{0})|_{ℓ}^{−1}) |q^{+}(β_{0})|_{p}^{−1}

= (∏_{ℓ≠p} |Δ|_{ℓ}^{−1} · |Nrd(h_{ℓ})|_{ℓ}^{−1}) |q^{+}(β_{0})|_{p}^{−1}  [by Lemma 4.2(i)]

= ( ∏_{ℓ∤D_{B}} |Δ|_{ℓ}^{−1}|n(a_{ℓ})|_{ℓ}^{−1} · |Nrd(w(c)_{ℓ})|_{ℓ}^{−1})

× ( ∏_{\begin{matrix}ℓ|D_{B} \\ ℓ≠p \end{matrix}} |n(a_{ℓ})|_{ℓ}^{−1} · ν_{ℓ}) |n(a_{p})|_{p}^{−1} · ν_{p}

= |Δ| · N(𝔞_{0}) (∏_{ℓ≠p} |Nrd(w(c)_{ℓ})|_{ℓ}^{−1} · ν_{ℓ}) · (ν_{p})
= |Δ| · N(𝔞_{0}) · c · ν^{p}(ξ, ϕ) · ν_{p}(ξ, ϕ),

where we have used the facts: (i) Nrd(w(c)_{ℓ}) = c for all ℓ and (ii) |Δ| is relatively prime to D_{B}. At long last, we return to the quantity we wish to compute. Note that for a fractional ideal 𝔞 appearing in the right hand side of (4.11), we have the equivalence

η_{A} ∘ β^{p} ∘ η_{E}^{−1}(𝔞̂^{p}) ⊂ Ô_{B}^{p}  ⟺  ϕ(𝔞̂^{p}) ⊂ Ô_{B}^{p} · h^{−1}.

This in turn is equivalent to the collection of local statements, for all ℓ ≠ p:

ϕ(𝔞_{ℓ}) ⊂ O_{B}_{,ℓ} · h_{ℓ}^{−1} = \begin{cases}O_{B}_{,ℓ} · Π_{ℓ}^{−ν_{ℓ}}ϕ(a_{ℓ})^{−1}, & if ℓ|D_{B} \\ O_{B}_{,ℓ} ·w(c)_{ℓ}^{−1} · ϕ(a_{ℓ})^{−1}, & if ℓ ∤ D_{B}. \end{cases}

Hence, replacing the ideals 𝔞 appearing in (4.11) by 𝔞_{0} · 𝔞, we obtain

#f^{−1}(ξ) = |o_{k}^{×}| · #\{𝔞 ⊂ k a fractional ideal | N(𝔞) = \frac{m}{c|Δ|ν^{p}ν_{p}} ,

ϕ(𝔞_{ℓ}) ⊂ O_{B}_{,ℓ} ·w(c)_{ℓ}^{−1} for ℓ ∤ D_{B}, and ϕ(𝔞_{ℓ}) ⊂ O_{B}_{,ℓ} Π_{ℓ}^{−ν_{ℓ}} for ℓ|D_{B}}.

(4.16)

To conclude the proof, we show that an ideal 𝔞 appearing on the right hand side above is necessarily integral. Note that for ℓ|D_{B}, including the case ℓ = p, the condition on the norm of 𝔞 implies ϕ(𝔞_{ℓ}) ⊂ O_{B}_{,ℓ}. If ℓ ∤ D_{B}, then with respect to the isomorphism O_{B}_{,ℓ} ≃ M_{2}(ℤ_{ℓ}) as in (4.13), we compute

O_{B}_{,ℓ} ·w(c)_{ℓ}^{−1} = \{(\begin{matrix}c^{−1}x & y \\ c^{−1}w & z \end{matrix}) , x, y, w, z ∈ ℤ_{ℓ}\} .

Recall that for a, b ∈ ℚ_{ℓ},

ϕ(a + b\sqrt{Δ}) = (\begin{matrix}a & b \\ bΔ & a \end{matrix}) .

Therefore,

(ϕ(k_{ℓ}) ∩ O_{B}_{,ℓ} w(c)_{ℓ}^{−1}) ⊂ ϕ (ℤ_{ℓ}[\sqrt{Δ}]) ⊂ O_{B}_{,ℓ}

and so for all fractional ideals 𝔞 appearing in (4.16), and all finite primes ℓ, we have ϕ(𝔞_{ℓ}) ⊂ O_{B}_{,ℓ}. As ϕ: o_{k} → O_{B} is an optimal embedding, it follows that 𝔞 ⊂ o_{k} is integral.

Before stating the main result of this section, we need a few more lemmas.

Lemma 4.5. There exists a prime q that is split in k, and an element

ϖ ∈ End(E)_{ℚ}, −Nm(ϖ) = ϖ^{2} = −pq,

such that ϖ ∘ i_{E}(a) = i_{E}(a′) ∘ ϖ for all a ∈ o_{k}.

Proof. See [Mann, p. 144]. The idea is that since End(E)_{ℚ} is the quaternion algebra ramified at exactly p and ∞, the existence of such an element ϖ is equivalent to the existence of a prime q such that

(−pq, Δ)_{ℓ} = \begin{cases}−1, & if ℓ ∈ \{∞, p\} \\ 1, & otherwise, \end{cases}

where (·, ·)_{ℓ} is the Hilbert symbol. This imposes a finite set of congruence conditions on q, for which (infinitely many) solutions exists by Dirichlet’s theorem, and furthermore such a solution is necessarily split in k.

Lemma 4.6. Suppose ϕ: o_{k} → O_{B} is optimal, and let ϕ′ denote the conjugate embedding. Then ϕ and ϕ′ are not O_{B}^{×}-equivalent.

Proof. Write ϕ′ = Ad_{t}∘ϕ for some t ∈ B^{×}, which is always possible by Noether-Skolem. Let ℓ be a prime dividing D_{B}; then there is a uniformizer Π_{ℓ} such that

ϕ′ = Ad_{Π_{ℓ}} ∘ ϕ ∈ Hom(o_{k,ℓ}, O_{B}_{,ℓ}).

Hence Π_{ℓ}^{−1} · t ∈ ϕ(k_{ℓ}^{×}), and so ord_{ℓ}Nrd(t) is necessarily odd. In particular, t ∉ O_{B}^{×}.

Lemma 4.7. Suppose m ∈ ℤ_{>0} is a positive integer, ϕ_{1} : o_{k} → O_{B} is an optimal embedding, and 𝔞 is a fractional ideal of k. Suppose further that t ∈ O_{B} such that Nrd(t) divides gcd(D_{B}, m). Note that t normalizes O_{B}, and in particular

ϕ_{2} := Ad_{t^{−1}} ∘ ϕ_{1}

is again an optimal embedding. Then we have a bijection

Ω^{±}(m, 𝔞, ϕ_{1}) ⟶^{∼} Ω^{±}(m/Nrd(t), 𝔞, ϕ_{2}), β ↦ ι_{A} (t^{−1}) ∘ β.

Proof. Suppose β ∈ Ω^{±}(m, 𝔞, ϕ_{1}), and set β′ := ι_{A} (t^{−1}) ∘ β. We first verify

β′ ∈^{?} Ω^{±}(m/Nrd(t), 𝔞, ϕ_{2}).

The condition on the norm of β′ follows immediately from definitions, and so we only need to check that the inclusion

η_{A} ∘ (β′)^{p} ∘ η_{E}^{−1} (𝔞̂^{p}) ⊂^{?} Ô_{B}^{p}  (4.17)

holds. Write 𝔞̂^{p} = (a_{ℓ}) · ô_{k}^{p} for some prime-to-p idele (a_{ℓ}), and set

h = (h_{ℓ})_{ℓ≠p} := η_{A} ∘ (β)^{p} ∘ η_{E}^{−1}((a_{ℓ})) ∈ Ô_{B}^{p},

and

h′ = (h′_{ℓ}) := η_{A} ∘ (β′)^{p} ∘ η_{E}^{−1}((a_{ℓ})).

Note that to prove (4.17), it suffices to show that h′ ∈ Ô_{B}^{p}. Recall that we had chosen η_{A} : T a^{p}(A) → Ô_{B}^{p} to be an O_{B}-linear isomorphism, where the action of O_{B} on Ô_{B}^{p} is given by left-multiplication. Hence we have

h′ = t^{−1} · h ∈ t^{−1} · Ô_{B}^{p}.

For primes ℓ not dividing Nrd(t), note that t^{−1} · O_{B}_{,ℓ} = O_{B}_{,ℓ}. On the other hand, suppose that ℓ divides Nrd(t); in particular, ℓ divides gcd(D_{B}, m). Then, by Lemma 4.2,

ord_{ℓ}Nrd(h_{ℓ}) = ord_{ℓ}(m) ≥ 1.

As B_{ℓ} is division, and ord_{ℓ}Nrd(t) = 1, it follow that h′_{ℓ} = t^{−1} · h_{ℓ} ∈ O_{B}_{,ℓ} by valuation considerations. Thus h′_{ℓ} ∈ O_{B}_{,ℓ} for all ℓ ≠ p, as required, and so we have shown that the assignment β ↦ ι_{A}(t^{−1}) ∘ β indeed defines a map Ω^{±}(m, 𝔞, ϕ_{1}) → Ω^{±}(m/Nrd(t), 𝔞, ϕ_{2}). By a similar argument, there is an inverse map

Ω^{±}(m/Nrd(t), 𝔞, ϕ_{2}) → Ω^{±}(m, 𝔞, ϕ_{1}), β′ ↦ β := ι_{A}(t) ∘ β′,

which concludes the proof of the lemma.

Recall that Z̃^{±}(m, ϕ) denotes the formal completion of Z^{±}(m, ϕ) along its fibre at p, and that we have p-adic uniformizations

Z̃^{±}(m, ϕ) ≃ [o_{k}^{×} × Γ′ \ ∐_{[𝔞]} ∐_{β∈Ω^{±}(m,𝔞,ϕ)} Z(β[p^{∞}])]  (4.18)

cf. Theorem 3.22.

Lemma 4.8. Let m ∈ ℤ_{>0} be a positive integer, ϕ_{1} : o_{k} → O_{B} an optimal embedding, and 𝔞 a fractional ideal of k. Suppose t ∈ O_{B} such that Nrd(t) divides gcd(D_{B}/p, m), and let ϕ_{2} := Ad_{t^{−1}} ∘ ϕ_{1}. Then we have an equality of cycles on C̃_{B}:

Z̃^{±}(m, ϕ_{1}) = Z̃^{±}(m/Nrd(t), ϕ_{2}).

In particular, the cycle Z̃^{±}(m, ϕ) only depends on the equivalence class [ϕ] ∈ Opt/ O_{B}^{×} of ϕ.

Proof. Let β ∈ Ω^{±}(m, 𝔞, ϕ_{1}), set β′ = ι_{A}(t^{−1}) ∘ β, and let b = β[p^{∞}] and b′ = β′[p^{∞}] denote the corresponding maps of p-divisible groups. Recall that for a scheme S ∈ Nilp, with special fibre \overline{S} = S × 𝔽, we had defined the S-points of the cycle Z(b) on D to be the locus of points (X, ι_{X}, ρ_{X}) ∈ D(S) such that the map

ρ_{X}^{−1} ∘ b: 𝕐\overline{S} → X\overline{S}

lifts to a map 𝕐_{S} → X. Similarly, the cycle Z(b′) parametrizes tuples (X, ι_{X}, ρ_{X}) such that ρ_{X}^{−1} ∘ b′: 𝕐\overline{S} → X\overline{S} lifts to a map 𝕐_{S} → X. Note that the image of t in B_{p} in fact lies in (O_{B,p})^{×} by valuation considerations, and the endomorphism

ι_{X}(t^{−1})\overline{S} : X × \overline{S} → X × \overline{S}

evidently admits a lift to S, namely ι_{X}(t^{−1}) ∈ End_{S}(X)^{×}. Combining these observations with the fact that the quasi-isogeny ρ_{X} is assumed to be O_{B,p}- linear, we have that

ρ_{X}^{−1} ∘ b′ = ρ_{X}^{−1} ∘ ι_{𝕏}(t^{−1})\overline{S} ∘ b lifts ⟺ ι_{X}(t^{−1})\overline{S} ∘ ρ_{X}^{−1} ∘ b lifts
⟺ ρ_{X}^{−1} ∘ b lifts.

Hence, we find that as cycles on D, we have Z(b) = Z(b′). The lemma then follows from Lemma 4.7 and the p-adic uniformization (4.18).

We now state and prove our key result:

Theorem 4.9. Let m ∈ ℤ_{>0}. Then we have the following equalities of cycles:

(i) If |Δ| does not divide m, then Z̃(m, ϕ) = 0. (ii) If m = |Δ|m′, then we have

∑_{[ϕ]∈Opt/ O_{B}^{×}}Z̃(m, ϕ) + Z̃ (\frac{m}{gcd(D_{B}, m)}, ϕ)

= 2h(k) ∑_{\begin{matrix}α|m′ \\ (α,D_{B})=1 \end{matrix}} χ_{k}(α) Z̃^{o} (|Δ|\frac{(m′)^{2}}{α^{2}} ) ,  (4.19)

where χ_{k} is the quadratic character associated to k, cf. (1.4).

Proof. (i) Suppose |Δ| does not divide m. Then for all possible values of c, ν_{p}, and ν^{p}, the quantity m/c|Δ|ν_{p}ν^{p} is not an integer, and so by Proposition 4.4,

Ω^{+}(m, 𝔞_{i}, ϕ) = ∅

for any fractional ideal 𝔞_{i} and any embedding ϕ. Hence we also have

Ω^{−}(m, 𝔞_{i}, ϕ) = Ω^{+}(m, 𝔞_{i}, ϕ′) = ∅,

where ϕ′ is the conjugate embedding. By the p-adic uniformization (4.18), it follows that Z̃^{±}(m, ϕ) = ∅, and so as a cycle, Z̃(m, ϕ) = 0. (ii) Suppose |Δ| divides m, and ϕ: o_{k} → O_{B} is an embedding with conjugate ϕ′. By Lemma 4.6, the embeddings ϕ and ϕ′ are O_{B}^{×}-inequivalent. Moreover, for any n we have Z^{+}(n, ϕ) = Z^{−}(n, ϕ′) essentially by definition. Summing over classes of optimal embeddings, it follows that

∑_{[ϕ]∈Opt/ O_{B}^{×}} Z̃(n, ϕ) = 2 ∑_{[ϕ]∈Opt/ O_{B}^{×}} Z̃^{+}(n, ϕ)

= 2 ∑_{[ϕ]∈Opt/ O_{B}^{×}} Z̃^{−}(n, ϕ).  (4.20)

We proceed by cases: Case 1: ord_{p}(m) > 0: For any cycle Z on D, let [Z] denote the corresponding

cycle on C̃_{B} = [Γ′\ D], and for convenience, let

A := ∑_{[ϕ]} Z̃(m, ϕ) + Z̃(m/gcd(m, d), ϕ)

denote the cycle we wish to compute (the left hand side of (4.19)). Write gcd(m, D_{B}) = p · μ, and fix an element t ∈ O_{B} such that Nrd(t) = μ. Suppose ϕ_{1} : o_{k} → O_{B} is an optimal embedding. By Lemma 4.8,

Z̃^{−}(m/p, ϕ_{1}) = Z̃^{−} (\frac{m}{gcd(m, D_{B})}, Ad_{t^{−1}} ∘ ϕ_{1}) ,

and so, upon taking the sum over all optimal embeddings and using (4.20),

A = 2 ∑_{[ϕ]∈Opt/ O_{B}^{×}} Z̃^{+}(m, ϕ) + Z̃^{−} (\frac{m}{gcd(m, D_{B})}, ϕ)

= 2 ∑_{[ϕ]} Z̃^{+}(m, ϕ) + Z̃^{−}(m/p, Ad_{t} ∘ ϕ)

= 2 ∑_{[ϕ]} Z̃^{+}(m, ϕ) + Z̃^{−}(m/p, ϕ).

Now suppose β ∈ Ω^{+}(m, 𝔞, ϕ) and set

β′ := β ∘ ϖ,

where ϖ ∈ End(E) satisfies ϖ^{2} = −pq for some split prime q, and ϖ ∘ i_{E}(a) = i_{E}(a′) ∘ ϖ for all a ∈ o_{k}, cf. Lemma 4.5. Then it is easily verified that

β′ ∈ Ω^{−}(m/p, 𝔞′ · 𝔮, ϕ),

where 𝔞′ is the conjugate of 𝔞, and 𝔮 is one of the ideals above q. Furthermore, let ξ = ξ(β) ∈ Ω^{o}(m^{2}|Δ|) denote the unique element such that β is an eigenvector with eigenvalue m\sqrt{Δ}, cf. Lemma 4.3. Then β′ is also an eigenvector, with eigenvalue −m\sqrt{Δ}. Moreover, the corresponding maps β[p^{∞}] and β′[p^{∞}] on p-divisible groups are (up to scalars in ℤ_{p}^{×}) precisely the special homomorphisms b^{+} and b^{−} appearing in Theorem 3.17, with ξ[p^{∞}] playing the role of j. The conclusion of that theorem then reads

Z^{o}(ξ[p^{∞}]) = Z(β[p^{∞}]) + Z(β′[p^{∞}]),

as cycles on D. Hence

A = 2 ∑_{[ϕ]} ∑_{[𝔞]} \frac{1}{|o_{k}^{×}|} \{ ∑_{\begin{matrix}β∈Ω^{+}(m,𝔞,ϕ) \\ mod Γ′ \end{matrix}}[Z(β[p^{∞}])] + ∑_{\begin{matrix}β′∈Ω^{−}(m/p,𝔞,ϕ) \\ mod Γ′ \end{matrix}}[Z(β′[p^{∞}])]\}

= 2 ∑_{[ϕ]} ∑_{[𝔞]} \frac{1}{|o_{k}^{×}|} \{ ∑_{\begin{matrix}β∈Ω^{+}(m,𝔞,ϕ) \\ mod Γ′ \end{matrix}}[Z(β[p^{∞}])] + ∑_{\begin{matrix}β′∈Ω^{−}(m/p,𝔮𝔞′,ϕ) \\ mod Γ′ \end{matrix}}[Z(β′[p^{∞}])]\}

= 2 ∑_{[ϕ]} ∑_{[𝔞]} \frac{1}{|o_{k}^{×}|} \{ ∑_{\begin{matrix}β∈Ω^{+}(m,𝔞,ϕ) \\ mod Γ′ \end{matrix}}[Z^{o}(ξ[p^{∞}])]\} where ξ = ξ(β).  (4.21)

Given integers c, ν_{p}, and ν^{p} such that c|m, ν_{p} ∈ {1, p} and ν^{p}|(D_{B}/p), we set

Ω^{o}(|Δ|m^{2},c, ν, ϕ) :=
{ξ ∈ Ω^{o}(|Δ|m^{2}) | c(ξ) = c, ν_{p}(ξ, ϕ) = ν_{p}, ν^{p}(ξ, ϕ) = ν^{p}};

that is, the set of elements of Ω^{o}(|Δ|m^{2}) whose conductor and Frobenius types relative to ϕ are as specified, and where, for ease of notation, we have written ν = ν_{p} · ν^{p}. Note that the action of Γ′ by conjugation preserves these sets. By Proposition 4.4, we may continue:

A = 2 ∑_{[ϕ]} ∑_{\begin{matrix}c|m \\ (c,D_{B})=1 \end{matrix}} ∑_{ν|D_{B}} ρ (\frac{m}{|Δ|cν^{p}ν_{p}} ) \{ ∑_{\begin{matrix}ξ∈Ω^{o}(m^{2}|Δ|,c,ν,ϕ) \\ mod Γ′ \end{matrix}}[Z^{o}(ξ[p^{∞}])]\} .

Writing m = |Δ|m′, only terms with c dividing m′ contribute to the above sum. Morevover, recall that all primes dividing D_{B} are inert in k by assumption. Let ν^{∗} be the unique integer dividing D_{B} such that ord_{ℓ}(m′/ν^{∗}) is even for all ℓ dividing D_{B}: only terms involving ν = ν^{∗} can contribute to the above sum, since o_{k} has no ideals with norm an odd power of an inert prime. Thus

A = 2 ∑_{\begin{matrix}c|m′ \\ (c,D_{B})=1 \end{matrix}} ρ (\frac{m′}{cν^{∗}} ) \{∑_{[ϕ]} ∑_{\begin{matrix}ξ∈Ω^{o}(m^{2}|Δ|,c,ν^{∗},ϕ) \\ mod Γ′ \end{matrix}}[Z^{o}(ξ[p^{∞}])]\} .

Note that by [Vig, Corollaire III.5.12],

#Opt/ O_{B}^{×} = h(k) · 2^{o(D_{B})},

where o(D_{B}) denotes the number of prime factors of D_{B}. Now consider the action of the normalizer N_{B^{×}}(O_{B}) on the set Opt/ O_{B}^{×}, acting by conjugation. For a fixed element ξ ∈ Ω^{o}(m^{2}|Δ|), the various values of the Frobenius types ν(ξ, ϕ), as ϕ varies in an N(O_{B})-orbit of optimal embeddings, will cover all 2^{o(D_{B})} possibilities. Thus, for fixed ξ, there are exactly h(k) classes [ϕ] of optimal embeddings such that

ν(ξ, ϕ) = ν^{p}(ξ, ϕ) · ν_{p}(ξ, ϕ) = ν^{∗}.

Hence, it follows that

A = 2h(k) ∑_{\begin{matrix}c|m′ \\ (c,D_{B})=1 \end{matrix}} ρ (\frac{m′}{cν^{∗}} ) ∑_{\begin{matrix}ξ∈Ω^{o}(m^{2}|Δ|) \\ c(ξ)=c \\ mod Γ′ \end{matrix}}[Z^{o}(ξ[p^{∞}])].

Next, we claim that for any integer N > 0, we may write

ρ(N) = ∑_{a|N} χ_{k}(a).

Indeed, both sides of the above formula are multiplicative, and for N = ℓ^{n} a prime power, the fomula can immediately be verified by considering the cases ℓ split, inert, and ramified separately. Moreover, as ord_{ℓ}(m′/cν^{∗}) is even for all ℓ|D_{B}, it follows that

ρ (\frac{m′}{cν^{∗}} ) = ∑_{\begin{matrix}a|(m′/cν^{∗}) \\ (a,D_{B})=1 \end{matrix}} χ_{k}(a) = ∑_{\begin{matrix}a|(m′/c) \\ (a,D_{B})=1 \end{matrix}} χ_{k}(a).

Substituting, we obtain

A = 2h(k) ∑_{\begin{matrix}c|m′ \\ (c,D_{B})=1 \end{matrix}} ∑_{\begin{matrix}a|(m′/c) \\ (a,D_{B})=1 \end{matrix}} χ_{k}(a) ∑_{\begin{matrix}ξ∈Ω^{o}(m^{2}|Δ|) \\ c(ξ)=c \\ mod Γ′ \end{matrix}}[Z^{o}(ξ[p^{∞}])]

= 2h(k) ∑_{\begin{matrix}a|m′ \\ (a,D_{B})=1 \end{matrix}} χ_{k}(a) · ∑_{\begin{matrix}c|(m′/a) \\ (c,D_{B})=1 \end{matrix}} ∑_{\begin{matrix}ξ∈Ω^{o}(m^{2}|Δ|) \\ c(ξ)=c \\ mod Γ′ \end{matrix}}[Z^{o}(ξ[p^{∞}])].  (4.22)

Applying (4.5), we have that for each a|m′ with (a, D_{B}) = 1,

∑_{\begin{matrix}c|(m′/a) \\ (c,D_{B})=1 \end{matrix}} ∑_{\begin{matrix}ξ∈Ω^{o}(|Δ|m^{2}) \\ c(ξ)=c \\ mod Γ′ \end{matrix}}[Z^{o}(ξ[p^{∞}])]

= ∑_{\begin{matrix}c|(m′/a) \\ (c,D_{B})=1 \end{matrix}} ∑_{\begin{matrix}ξ∈Ω^{o}(|Δ|m^{′2}/a^{2}) \\ c(ξ)=c \\ mod Γ′ \end{matrix}} [Z^{o} (a|Δ| · ξ[p^{∞}])] .

= ∑_{\begin{matrix}c|(m′/a) \\ (c,D_{B})=1 \end{matrix}} ∑_{\begin{matrix}ξ∈Ω^{o}(|Δ|m^{′2}/a^{2}) \\ c(ξ)=c \\ mod Γ′ \end{matrix}} [Z^{o} (ξ[p^{∞}])]  (since a|Δ| ∈ ℤ_{p}^{×})

= ∑_{\begin{matrix}ξ∈Ω^{o}(|Δ|m^{′2}/a^{2}) \\ mod Γ′ \end{matrix}} [Z^{o} (ξ[p^{∞}])]  (by (4.4))

= Z̃^{o} (|Δ|\frac{(m′)^{2}}{a^{2}} ) .

Substituting this back into (4.22) concludes the proof of the theorem in the case ord_{p}(m) > 0. Case 2: ord_{p}(m) = 0. As the proof is along similar lines as the previous case, we shall only indicate the necessary modifications. Fix an element t ∈ O_{B} with Nrd(t) = p·gcd(m, D_{B}), and such that the image of t in O_{B,p} is the uniformizer Π as in Section 2. Let ϖ ∈ End(E)_{ℚ} be as in Lemma 4.5, so that in particular ϖ^{2} = −pq for a split prime q. Then if β ∈ Ω^{+}(m, 𝔞, ϕ), it follows that

β′ := ι_{A}(t) ∘ β ∘ ϖ^{−1} ∈ Ω^{−}(\frac{m}{gcd(m, D_{B})}, 𝔮^{−1} · 𝔞′, Ad_{t} ∘ ϕ)

= Ω^{+} (\frac{m}{gcd(m, D_{B})}, 𝔮^{−1} · 𝔞′, Ad_{t} ∘ ϕ′) ,

where 𝔮 is one of the prime ideals above q, and ϕ′ is the conjugate of ϕ. Let ξ = ξ(β) ∈ Ω^{o}(|Δ|m^{2}) denote the special endomorphism corresponding to β as in Lemma 4.3. If j = ξ[p^{∞}] is the corresponding map on p-divisible groups, then β[p^{∞}] and β′[p^{∞}] are equal (up to scaling by ℤ_{p}^{×}) to the elements b_{1} and b_{2} described in Theorem 3.19, and so

Z^{o}(ξ[p^{∞}]) = Z(β[p^{∞}]) + Z(β′[p^{∞}]).

Therefore, we have

(\begin{matrix}left hand \\ side \\ of (4.19) \end{matrix}) = ∑_{[ϕ]} Z̃(m, ϕ) + Z̃(m/gcd(m, D_{B}), ϕ)

= 2 ∑_{[ϕ]} Z̃^{+}(m, ϕ) + Z̃^{+}(m/gcd(m, D_{B}), ϕ) [by (4.20)]

= 2 ∑_{[ϕ]} ∑_{[𝔞]} \frac{1}{|o_{k}^{×}|}\{ ∑_{\begin{matrix}β∈Ω^{+}(m,𝔞,ϕ) \\ mod Γ′ \end{matrix}}[Z(β[p^{∞}])]

+ ∑_{\begin{matrix}β′∈Ω^{+}(m/(m,D_{B}),𝔞,ϕ) \\ mod Γ′ \end{matrix}}[Z(β′[p^{∞}])]\}

= 2 ∑_{[ϕ]} ∑_{[𝔞]} \frac{1}{|o_{k}^{×}|}\{ ∑_{\begin{matrix}β∈Ω^{+}(m,𝔞,ϕ) \\ mod Γ′ \end{matrix}}[Z(β[p^{∞}])]

+ ∑_{\begin{matrix}β′∈Ω^{+}(m/(m,D_{B}),𝔮^{−1}𝔞′,Ad_{t}∘ϕ′) \\ mod Γ′ \end{matrix}}[Z(β′[p^{∞}])]\}

= 2 ∑_{[ϕ]} ∑_{[𝔞]} \frac{1}{|o_{k}^{×}|}\{ ∑_{\begin{matrix}β∈Ω^{+}(m,𝔞,ϕ) \\ mod Γ′ \end{matrix}}[Z^{o}(ξ[p^{∞}])]\},

where in the last line ξ = ξ(β). The proof proceeds from this point exactly as in the previous case, cf. (4.21).

4.3 The main theorem and applications

Let ω denote the relative dualizing sheaf of C_{B}, which we view as a divisor class in the Chow group CH^{1}(C_{B}), and choose any divisor K in this class. Define the orthogonal generating series

Φ^{o}(τ) := −K + ∑_{n>0} Z^{o}(n) q_{τ}^{n} ∈ Div(C_{B})⟦q_{τ} ⟧,  (4.23)

and consider the base change

Φ_{/o_{k}}^{o}(τ) := −K_{/o_{k}} + ∑_{n>0} Z^{o}(n)_{/o_{k}} q_{τ}^{n} ∈ Div(C_{B/o_{k}})⟦q_{τ} ⟧.

Let χ_{k} = (·/Δ) denote the quadratic character attached to k, and χ′_{k} denote its induction to level 4D_{B}|Δ|. Define the Gauss sum

χ̌′_{k}(a) := ∑_{h=0}^{4D_{B}|Δ|−1} χ′_{k}(h) exp(2πiah/4D_{B}|Δ|),

and the ‘L-function’

L(s, χ̌′_{k}) := ∑_{m>0} m^{−s} χ̌′_{k}(m),

which is analytic (as written) in the half-plane ℜ(s) > 0. We then define the unitary generating series:

Φ^{u}(τ) := \frac{i}{2π} L(1, χ̌′_{k}) · K_{/o_{k}} + \frac{1}{2h(k)} ∑_{m>0} ∑_{[ϕ]} [Z(m, ϕ) + Z^{∗}(m, ϕ)] q_{τ}^{m} (4.24)
∈ Div(C_{B/o_{k}})⟦q_{τ} ⟧ ⊗_{ℤ} ℚ

where, as we recall,

Z^{∗}(m, ϕ) = Z(m/ gcd(m, D_{B}), ϕ),

and the sum on [ϕ] is over the set Opt/ O_{B}^{×} of optimal embeddings taken up to O_{B}^{×}-conjugacy. Note that

\frac{i}{2π} L(1, χ̌′_{k}) = −\frac{h(k)}{|o_{k}^{×}|} ∏_{ℓ|D_{B}} (\frac{2ℓ^{2} + ℓ − 1}{ℓ^{2}}) ∈ ℚ,

which can be seen by evaluating the L-function on the left via Dirichlet’s class number formula.

Theorem 4.10 (Main theorem). Suppose Δ < 0 is squarefree and even, and every prime dividing D_{B} is inert in k. Then we have an equality

Sh(Φ_{/o_{k}}^{o})(τ) = Φ^{u}(τ) ∈ Div(C_{B/o_{k}})⟦q_{τ} ⟧ ⊗ ℚ  (4.25)

of formal generating series, where Sh is the formal Shimura lift with parameters κ = 3, N = D_{B}, t = |Δ| and χ = 1 in the notation of Definition 4.1.

Proof. We apply the formulas (4.2) and (4.3) for the formal Shimura lift. Note that the constant terms match by design, and so – keeping in mind the shift by t = |Δ| in the exponent in (4.1) – it suffices to prove that

(i) if |Δ| does not divide m, then ∑_{[ϕ]} Z(m, ϕ) = 0; (ii) and if m = m′|Δ|, then

∑_{[ϕ]∈Opt/ O_{B}^{×}}Z(m, ϕ) + Z (\frac{m}{gcd(D_{B}, m)}, ϕ)

= 2h(k) ∑_{\begin{matrix}α|m′ \\ (α,D_{B})=1 \end{matrix}} χ_{k}(α) Z^{o} (|Δ|\frac{(m′)^{2}}{α^{2}} )_{/o_{k}} ,

in Div(C_{B/o_{k}}). By [KRY, Proposition 3.4.5], each orthogonal special cycle decomposes as

Z^{o}(n)_{/o_{k}} = Z^{o}(n)^{hor} + ∑_{q|D_{B}} Z^{o}(n)_{q}^{ver},

where Z^{o}(n)^{hor} is the closure of the generic fibre in Z^{o}(n)_{/o_{k}}, and Z^{o}(n)_{q}^{ver} is the sum of the vertical irreducible components supported in characteristic q. By Proposition 2.6, we have an identical decomposition for the unitary cycles. Now for each q|D_{B}, Theorem 4.9 implies that (i) and (ii) hold for the vertical components at q. Moreover, since C_{B/o_{k}} is proper over Spec(o_{k}), the same proposition implies (i) and (ii) hold in the generic fibre, by Grothendieck’s existence theorem [EGA3, Thm. 5.1.4]; more precisely, if we choose a prime p|D_{B}, then the existence theorem asserts that the desired relations hold over o_{k,p}, and then we observe that o_{k,p} is faithfully flat over the localization (o_{k})_{(p)}.

Our next step is to recall how certain quantities involving the orthogonal special cycles Z^{o}(n) arise as Fourier coefficients of actual modular forms of weight 3/2, as in [KRY, §4]. First, we consider the ‘rational degree’ generating series

Φ_{deg}^{o}(τ) := − deg_{k}(ω) + ∑_{n>0} deg_{k} Z^{o}(n) q_{τ}^{n} ∈ ℚ⟦q_{τ} ⟧;

obtained by taking the degrees of the generic fibres. Next, we may consider the generating series

Φ_{/k}^{o}(τ) = −ω_{/k} + ∑_{n>0} Z^{o}(n)_{/k} q_{τ}^{n} ∈ CH_{ℂ}^{1}(C_{B}_{/k})⟦q_{τ} ⟧

formed by taking generic fibres and then passing to the Chow group CH_{ℂ}^{1}(C_{B}_{/k}) = CH^{1}(C_{B}_{/k}) ⊗_{ℤ} ℂ. Finally, let 𝒴 denote an irreducible component of the fibre (C_{B/o_{k}})_{p} of the Shimura curve, where p|D_{B} is a prime of bad reduction. For a closed substack Z of C_{B/o_{k}}, we define the pairing

⟨Z, 𝒴 ⟩ := 2 log(p) · χ(O_{𝒴} ⊗^{𝕃} O_{Z}),

where χ is the stack version of the Euler-Poincaré characteristic, which takes into account the automorphism groups of points, cf. [DeRa, §VI.4]. We then form the generating series

Φ_{𝒴}^{o} (τ) := −⟨ω, 𝒴 ⟩ + ∑_{n>0} ⟨Z^{o}(n), 𝒴 ⟩ q_{τ}^{n}  (4.26)

We may form the analogues Φ_{deg}^{u}(τ), Φ_{/k}^{u}(τ) and Φ_{𝒴}^{u} (τ) for the unitary cycles, by applying the appropriate maps to the coefficients.

Corollary 4.11. (i) Each of Φ_{deg}^{o}(τ), Φ_{/k}^{o}(τ) and Φ_{𝒴}^{o} (τ) is (the q-expansion of ) a holomorphic modular form of weight 3/2, level 4D_{B} and trivial character. (ii) Under the hypotheses of Theorem 4.10, the Shimura lift of each of these modular forms is equal to its unitary counterpart. In particular, the series Φ_{deg}^{u}, Φ_{/k}^{u} and Φ_{𝒴}^{u} are the q-expansions of modular forms of weight 2.

Proof. (i) The modularity of these three generating series is proved in [KRY, §4]; specifically, see Equation (4.2.12), Theorem 4.5.1 and Theorem 4.3.4 there. (ii) This follows immediately by applying the appropriate map to both sides of (4.25).

Remark 4.12. Consider the more natural generating series

Φ_{ϕ}^{naive} = C + \frac{1}{2h(k)} ∑_{n>0} Z(n, ϕ)q^{n} ∈ Div(C_{B/o_{k}})⟦q⟧ ⊗ ℚ

which omits the Z^{∗} terms appearing in Φ^{u}. On account of the general philosophy of such generating series, one may be tempted to conjecture that for an appropriate constant term C, the series Φ_{ϕ}^{naive} is already ‘modular’ (to fix ideas, we may take this to mean that the application of a linear functional which factors through the Chow group yields the q-expansion of a modular form). If this is indeed the case, we may explain the modularity of Φ^{u} in the following way. For an integer d, let U_{d} and B_{d} denote the following operators on formal q-expansions: if F = ∑_{n≥0} a(n)q^{n}, then

U_{d}(F) = ∑ a(dn)q^{n}, B_{d}(F) = ∑ a(n)q^{dn}.

When F is the q-expansion of a modular form, these operators are induced by the maps U_{d} = U_{d}^{1} and B_{d} described in [Li]; both are maps between spaces of modular forms that preserve the weight, but may change the level. Set

φ_{d} := B_{d} ∘ (1 − U_{d}) , φ_{d}(F) = ∑ (a(n) − a(dn)) q^{dn}

and note that if (d, d′) = 1, then φ_{d} and φ_{d′} commute. Next, let P(D_{B}) denote the set of primes dividing D_{B}, and for a non-empty subset I = {p_{1}, …, p_{n}} ⊂ P(D_{B}), we define

φ_{I} := φ_{p_{1}} ∘ φ_{p_{2}} ∘ ⋯ ∘ φ_{p_{n}}.

Then a direct calculation reveals that at the level of formal generating series,

Φ^{u} = C′ + ∑_{[ϕ]} (2 + ∑_{I⊂P(D_{B})} φ_{I} ) (Φ_{ϕ}^{naive}),

where

C′ = \frac{i}{2π} L(1, χ̌′_{k}) · K − 2 · #|Opt/ O_{B}^{×} | · C.

In particular, if C happens to take on the value

C = \frac{i}{4π} · \frac{L(1, χ̌′_{k})}{#|Opt/ O_{B}^{×} |} · K = −\frac{2^{−o(D_{B})}}{2|o_{k}^{×}|} ∏_{ℓ|D_{B}} (\frac{2ℓ^{2} + ℓ − 1}{ℓ^{2}}) · K,

then C′ = 0; this would imply that whenever the application of a suitable functional to the coefficients of Φ_{ϕ}^{naive} yields a modular form, the same is true for Φ^{u}. It would be interesting to know if this implication can be reversed, and also if there is a geometric interpretation to the value of C given above. ⋄

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Siddarth Sankaran

Mathematisches Institut der Universität Bonn Endenicher Allee 60 53115 Bonn, Germany sankaran@math.uni-bonn.de