Fake CM and the Stable Model of X_{0}(Np^{3})

Robert Coleman^{1} and Ken McMurdy

Received: September 11, 2005 Revised: October 5, 2006

Abstract. We complete the determination of the stable model of X_{0}(Np^{3}), p ≥ 5, (N, p) = 1 begun in [CMc] and compute the inertial action on the stable reduction of X_{0}(p^{3}).

2000 Mathematics Subject Classification: Primary 11G18; Secondary 14G22, 11G07 Keywords and Phrases: stable reduction, modular curve

1 Introduction

In [CMc] we found a stable model for the modular curve, X_{0}(p^{3}), over the ring of integers in ℂ_{p}, for a prime p ≥ 13. The stable models of X_{0}(p) and X_{0}(p^{2}) were previously known, due to work of Deligne-Rapoport and Edixhoven (see [CMc, §1] for a more complete list of relevant results). Finding a stable model for X_{0}(p^{n}) for n > 3 remains an open problem, although a conjectural stable model for X_{0}(p^{4}) is given in [M2, §5].

The results and main ideas of the argument used in [CMc] are summarized below in Section 2. Nevertheless, we still refer to [CMc] frequently, and do recommend that it be read first. Indeed, the purpose of this paper is to refine and extend those results. First, we prove results which enable us to define our model over an explicit finite extension of ℚ_{p}, and to compute the inertia action on the stable reduction. More precisely, we show that a stable model for X_{0}(p^{3}) can be defined over any field over which a stable model for X_{0}(p^{2}) exists, and which contains the j-invariants of all elliptic curves whose formal groups have endomorphism rings isomorphic to ℤ_{p}[p\sqrt{−p}] or ℤ_{p}[p\sqrt{−Dp}] for D a nonsquare (mod p). Such elliptic curves, whose formal groups have endomorphism rings bigger than ℤ_{p}, are said to have fake CM. In Section 4, we show that (real) CM points are dense in these fake CM points. Thus we are able to apply the theory of CM elliptic curves when we determine, in Section 5, an explicit field of definition for our model. Once this is done, we compute the action of the inertia group on the stable reduction (in Section 6). This uses the results of Sections 3-5 and the fact (which we show) that the formal groups of elliptic curves with fake CM are relative Lubin-Tate groups as in [dS2]. As a consequence, we show that the extension of ℚ_{p}^{nr} found by Krir in [K], over which the Jacobian of X_{0}(p^{3}) has semi-Abelian reduction, is minimal.

1Supported by NSF grant DMS-0401594

Documenta Mathematica · Extra Volume Coates (2006) 261–300

We also extend the results of [CMc] in two other ways. In order to do the explicit analysis in [CMc], it was necessary to have an approximation formula for the forgetful map, π_{f} : X_{0}(p) → X(1), over some supersingular annulus. Such a formula followed from a result of de Shalit (recalled in Section 2) for any region corresponding to a supersingular elliptic curve A/𝔽_{p} whose j-invariant, j(A), does not equal 0 or 1728. By a result of Everett Howe (see [CMc, §10]), one always has such an A as long as p ≥ 13. So the only nontrivial cases which were left open were the three specific primes: p = 5, 7, and 11. This shortcoming of our construction could be resolved by either generalizing de Shalit’s result or by adding level structure to the more symmetric deformation space of formal groups studied by Gross-Hopkins in [GH]. We handle the open cases here, however, by applying explicit known formulas (in Section 7). It is our hope that these calculations not only deal with the remaining open cases, but also serve to make the constructions of [CMc] more concrete and understandable. Finally, in Section 8 we extend the result of [CMc] by adding tame level, i.e. we compute the stable reduction of X_{0}(Np^{3}) when (N, p) = 1. This is done by first viewing X_{0}(Np^{3}) as the fiber product of X_{0}(N) and X_{0}(p^{3}) over X(1). We construct semi-stable maps (as in [C2]) which extend both forgetful maps, and prove a lemma which implies that the product of semi-stable maps is semi-stable in this case. Then we compute the reductions of the components of X_{0}(Np^{3}) by crossing pairs of components in X_{0}(N) and X_{0}(p^{3}) which have the same image in X(1). Two specific examples are then worked out in some detail.

2 Stable Reduction of X_{0}(p^{3}) for p ≥ 13

In this section we summarize the content of [CMc] and in particular the construction of the stable model of X_{0}(p^{3}) for p ≥ 13. The goal will be to present the main ideas, along with the specific details which pertain directly to the results in this paper.

2.1 Foundations

Fake CM and the Stable Model of X_{0}(Np^{3}) 263

Over ℂ_{p}, we may think of points on the modular curve, X_{0}(p^{n}), as corresponding to pairs (E, C) where E/ℂ_{p} is an elliptic curve and C is a cyclic subgroup of order p^{n}. One way of studying the p-adic geometry of X_{0}(p^{n}) is to study regions where the pair, (E, C), has prescribed properties. The most basic distinction which one can make is whether E has ordinary (including multiplicative) or supersingular reduction, and the geometry of the ordinary region of X_{0}(p^{n}) is well understood. Indeed, if E is an elliptic curve with ordinary reduction, we define the canonical subgroup K(E) to be the p-power torsion of E(ℂ_{p}) in the kernel of reduction. For each a, b ≥ 0 with a + b = n, we then have rigid subspaces of the ordinary locus of X_{0}(p^{n}) given by

X_{a b} := { (E, C) : |C ∩ K(E)| = p^{a} }.

Then X_{a b} is an affinoid disk when ab = 0. Otherwise, it is shown in [C1, §1] that X_{a b} is the disjoint union of two irreducible affinoids, X_{a b}^{±}, which reduce to the Igusa curve, Ig(p^{c}), where c = min{a, b}. This curve is studied in [Ig] and classifies pairs, (E, α), where E/\overline{𝔽}_{p} is an elliptic curve and α : μ_{p^{c}} ↪ E is an embedding.

The supersingular locus is not as well understood, but there are a number of tools which can provide a line of attack. One of the most important is the theory of the canonical subgroup for curves with supersingular reduction, for which we take [B, §3, §4] as our primary reference. When E/ℂ_{p} has supersingular reduction, one can still define the canonical subgroup of order p^{n}, H_{n}(E), to be the cyclic subgroup of order p^{n} which is (p-adically) closest to the origin. For each E with supersingular reduction, however, there is a largest n for which H_{n}(E) exists, and we denote this subgroup by K(E). The size of K(E) is then completely determined by the valuation of the Hasse invariant of Ē. Denoting this valuation by h(E), from [B, Thm 3.3, Def 3.4] we have

|K(E)| > p^{n} ⟺ h(E) < p^{1−n}/(p + 1).

The theory of canonical subgroups is intimately connected to the geometry of the supersingular region of X_{0}(p). For a fixed supersingular elliptic curve, A/𝔽_{p^{2}}, we let W_{A}(p^{n}) be the subspace of X_{0}(p^{n}) consisting of pairs (E, C) where Ē ≅ A. It is well-known (from [DR, §VI 6.16], for example) that W_{A}(p) is an annulus of width i(A) = |Aut(A)|/2. Furthermore, one can choose a parameter x_{A} on this annulus, which identifies it with 0 < v(x_{A}) < i(A), and such that

v(x_{A}(E, C)) = \begin{cases}i(A)h(E), & if |C ∩ K(E)| = p \\ i(A)(1 − h(E/C)), & if |C ∩ K(E)| = 1. \end{cases}

Inside the annulus, W_{A}(p), there are two circles of fundamental importance. The “too-supersingular circle,” denoted TS_{A}, is where

v(x_{A}(E, C)) = (p/(p + 1))i(A)

or (equivalently) K(E) is trivial. The self-dual circle, SD_{A}, consists of all pairs (E, C) where C is potentially self-dual, equivalently those points where v(x_{A}(E, C)) = i(A)/2. When A/𝔽_{p}, this circle is fixed by the Atkin-Lehner involution, w_{1} (recalled below), and hence can be called the “Atkin-Lehner circle.”

Another tool for the analysis of the supersingular region of X_{0}(p^{n}) is Woods Hole Theory [WH], which essentially says that lifting an elliptic curve is equivalent to lifting its formal group. More precisely, if R_{p} ⊆ ℂ_{p} is the ring of integers, we have the following theorem.

Theorem 2.1. The category of elliptic curves over R_{p} is equivalent to the category of triples (F, A, α), where F/R_{p} is a formal group, A/𝔽̄_{p} is an elliptic curve, and α : F̄ → Â is an isomorphism. A morphism between two triples, (F, A, α) and (F′, A′, β), is either the 0 map or a pair (σ, τ), where σ : F → F′ and τ : A → A′ are isogenies such that the following diagram commutes.

F̄ ⟶^{σ̄} F̄′
α↓  ↓^{β}
Â ⟶_{τ̂} Â′

The theorem is used in two specific ways in [CMc]. First of all, for any two supersingular elliptic curves, A and A′, there is an isogeny ϕ : A → A′ whose degree is prime to p and which therefore passes to an isomorphism on formal groups. By taking (F, A, α) to (F, A′, ϕ̂ ∘ α), we can define a surjection of W_{A}(p^{n}) onto W_{A′}(p^{n}) as long as i(A) = 1 (see [CMc, §4.1]). Note that here we have added level structure to Theorem 2.1 in the obvious way. So this implies that all of the supersingular regions are nearly isomorphic, which enables us to analyze W_{A}(p^{n}) under the simplifying assumptions that A/𝔽_{p} and j(A) ≠ 0, 1728 (as long as p ≥ 13, by the result of Howe). In particular, much of our explicit analysis depends on an approximation formula for the forgetful map from the annulus, W_{A}(p), to the disk, W_{A}(1). For A/𝔽_{p} with j(A) ≠ 0, 1728, such a formula was essentially found by de Shalit in [dS1, §3]. Let π_{f} : W_{A}(p) → W_{A}(1) denote the forgetful map, and w_{1} : W_{A}(p) → W_{A}(p) the Atkin-Lehner involution, given by π_{f}(E, C) = E and w_{1}(E, C) = (E/C, E[p]/C) respectively. We reformulate de Shalit’s result as the following theorem.

Theorem 2.2. Let R = W (𝔽_{p^{2}}) and A/𝔽_{p} be a supersingular curve with j(A) ≠ 0, 1728. There are parameters s and t over R which identify W_{A}(1) with the disk B(0, 1) and W_{A}(p) with the annulus A(p^{−1}, 1), and series, F (T), G(T) ∈ T R[[T]], such that

(i) w_{1}^{∗}(t) = κ/t for some κ ∈ R with v(κ) = 1. (ii) π_{f}^{∗}s = F (t) + G(κ/t), where

(a) F′(0) ≡ 1 (mod p), and

(b) G(T) ≡ (F (T))^{p} (mod p).

The other way we use Woods Hole Theory is by letting Aut(Â) act on W_{A}(p^{n}) in the obvious way (here, as in Theorem 2.1, Â denotes the formal group of A). From [T], we can identify End𝔽̄_{p}(Â) with B := ℤ_{p}[i, j, k], where i^{2} = −r (a non-residue), j^{2} = −p, and ij = −ji = k. When A/𝔽_{p}, j can also be identified with the Frobenius endomorphism. The action of B^{∗} ≅ Aut(Â) on W_{A}(1) commutes with the Gross-Hopkins period map, Φ, which can be viewed as a map from W_{A}(1) to P^{1} whenever j(A) ≠ 0, 1728. Furthermore, for α, β ∈ ℤ_{p}[i] and ρ = α + jβ ∈ B^{∗}, Gross-Hopkins show in [GH, §25] that the action of B^{∗} on P^{1} is given explicitly by

ρ(t) = \frac{−pβ̄ + ᾱt}{α + βt}.

It is important to note here that the action of B^{∗} on W_{A}(1) is then only completely determined by the explicit formula of Gross-Hopkins for B^{∗}- invariant subspaces on which Φ is an injection. Fortunately, the Atkin-Lehner circle, or rather π_{f}(SD_{A}), is such a subspace and is identified with the circle described by v(t) = 1/2. So as an immediate consequence, the action of B^{∗} induces a faithful action of

B^{∗}/ℤ_{p}^{∗}(1 + jB) ≅ μ_{p^{2}−1}/μ_{p−1}

on \overline{SD}_{A} (still when j(A) ≠ 0, 1728). Also, on SD_{A} the involution w_{1} can be identified with j in the above sense. We use this in [CMc, §4.2] to show that an involution on SD_{A} can be defined by w_{ρ} := ρ∘w_{1}, for any ρ = a+bi+dk ∈ B^{∗} (this subset of B^{∗} is called B′).

Remark 2.3. An affinoid X defined over a complete subfield of ℂ_{p} has a canonical reduction over the ring of integers, which is what we mean by \overline{X}. Later, we adopt the convention of un-bolding affinoid names to refer to associated components of the stable reduction. Thus, whenever both make sense, \overline{X} and X are birational but not isomorphic.

2.2 Stable Model Construction

Our approach to constructing a stable model is purely rigid-analytic, in the sense that we actually construct a stable covering by wide open spaces. This equivalent notion is explained in detail in [CMc, §2]. Roughly, the wide open subspaces in a semi-stable covering intersect each other in disjoint annuli, and have underlying affinoids with (almost) good reduction. Each component in the stable reduction is (almost) the reduction of one of these underlying affinoids, and the annuli of intersection reduce to the ordinary double points where components intersect.

With this rigid analytic reformulation in mind, our strategy for constructing the stable model of X_{0}(p^{3}) is basically to construct nontrivial components explicitly and then prove that nothing else interesting can happen (this is done, in part, with a total genus argument). In addition to the components in the ordinary region, we use the above tools to construct three distinct types of components in the supersingular region of X_{0}(p^{3}) corresponding to any fixed supersingular elliptic curve, A/𝔽_{p}, with j(A) ≠ 0 or 1728. First we consider the affinoid, Y_{A} := π_{ν}^{−1}(TS_{A}) ⊆ W_{A}(p^{2}), where π_{ν} : X_{0}(p^{2}) → X_{0}(p) is given by π_{ν}(E, C) = (E/C[p], C/C[p]). We show in [CMc, §5] that Y_{A} can be identified with the rigid space,

T_{A} := { (x, y) ∈ TS_{A} × TS_{A} | x ≠ y, π_{f}(x) = π_{f}(y) }.

Then by applying Theorem 2.2 we compute the reduction of Y_{A} explicitly to be y^{2} = x^{p+1} + 1. This affinoid, Y_{A}, reduces to the supersingular component which Edixhoven found in [E1, Thm 2.1.1]. It can also be pulled back to X_{0}(p^{3}) via π_{f} and π_{ν} (defined as above) to obtain nontrivial components of W_{A}(p^{3}) (these pullbacks of Y_{A} are denoted by E_{1 A} and E_{2 A}). However, there are other nontrivial components as well. Analogous to the above construction, let Z_{A} := π_{1 1}^{−1}(SD_{A}) ⊆ W_{A}(p^{3}), where π_{1 1} = π_{f} ∘ π_{ν}. Then Z_{A} can be identified with

S_{A} := { (x, y) ∈ C_{A} × C_{A} | τ_{f}(x) = w_{1} ∘ τ_{f}(y) }.

Here C_{A} ⊆ W_{A}(p) is the circle whose points correspond to pairs, (E, C), where h(E) = 1/2 and C ≠ H_{1}(E). Then τ_{f} : C_{A} → SD_{A} is the degree p map which replaces C with H_{1}(E). The above reformulation of de Shalit’s analysis is again sufficient to explicitly compute the reduction of Z_{A} (in [CMc, §8]), which is given by

X^{p+1} + X^{−(p+1)} = Z^{p}.

Finally, we show that each of the 2(p + 1) singular residue classes of Z_{A} contains an affinoid which reduces to the curve, y^{2} = x^{p} − x. We do this by constructing a family of involutions on Z_{A}, given by w̃_{ρ}(x, y) = (ρy, ρ̄x) (for ρ ∈ B′) and compatible with the w_{ρ}’s in the sense that π_{1 1} ∘ w̃_{ρ} = w_{ρ} ∘ π_{1 1}. Thus, fixed points of w̃_{ρ} lie over fixed points of w_{ρ}. Each singular residue class of Z_{A} is shown to be a connected wide open with one end, on which one of these involutions acts with p fixed points. To finish the argument, we show in [CMc, §8.2] that the quotient by w̃_{ρ} of such a residue class is a disk, in which the images of the p fixed points are permuted by an automorphism of order p (reducing to a translation). It is then straightforward analysis to prove that any such wide open is basic (as in [CMc, §2]), with an underlying affinoid that reduces to y^{2} = x^{p} − x.

Remark 2.4. We show in [CMc, Prop 4.9] that the fixed points of w_{ρ} correspond to pairs, (E, C), where E has fake CM by ℤ_{p}[\sqrt{−p}] or ℤ_{p}[\sqrt{−Dp}] (and C = H_{1}(E)). So this is where fake CM enters into the arithmetic of our stable model.

The last step in our stable model construction is to form an admissible covering of X_{0}(p^{3}) by wide open neighborhoods of the nontrivial affinoids that we know about. Once again, any supersingular region corresponding to j(A) = 0 or 1728, or for which j(A) ∉ 𝔽_{p}, is dealt with by applying an appropriate surjection from W_{A′}(p^{n}) onto W_{A}(p^{n}). We then total up our lower bounds for the genera of all of these wide opens (and the Betti number of the graph associated to our covering), and compare this with the genus of X_{0}(p^{3}). Since the two are equal, we are able to conclude from [CMc, Proposition 2.5] that we haven’t missed anything. Thus we have the following theorem.

Theorem 2.5. The stable reduction of X_{0}(p^{3}) for p ≥ 13 consists of six ordinary components (reductions of the X_{a b}^{±}) and a “necklace” of components, for each supersingular elliptic curve A/𝔽_{p^{2}}, whose graph is given below in Figure 1. The reductions of E_{1 A} and E_{2 A} are isomorphic to y^{2} = x^{(p+1)/i(A)} + 1, and Z_{A} has 2(p + 1)/i(A) singular residue classes with underlying affinoids that reduce to y^{2} = x^{p} − x.

[figure: \overline{Z}_{A}; ordinary; \overline{E}_{1,A}; .; .; .; .; .; .; .; .; .; .; ordinary; \overline{E}_{2,A}]

Figure 1: Partial Graph of the Stable Reduction of X_{0}(p^{3})

3 Fake CM

Let K be a complete subfield of ℂ_{p} with ring of integers R. Then we say that an elliptic curve, E/R, has fake CM if End_{R}(Ê) ≠ ℤ_{p}, and potential fake CM if this happens over ℂ_{p}. We showed in [CMc] that curves with certain types of fake CM can be used to understand the geometry of X_{0}(p) and X_{0}(p^{3}). In particular, let R be the set of rings of integers in quadratic extensions of ℚ_{p}, and let S ∈ R be the ring of integers in a ramified extension. Then by [CMc, Prop 4.9], curves E with potential fake CM by S are precisely those for which (E, H_{1}(E)) is fixed by some involution w_{ρ} (for ρ ∈ B′, as in Section 2). Moreover, by [CMc, Prop 7.4], any fixed point of some involution, w̃_{ρ}, is obtained from one of these by a non-canonical p-isogeny.

In this section we further investigate properties of curves which have fake CM by some S ∈ R. In particular, we focus on the ways in which the fake endomorphism ring can embed into B ≅ End(Â) (via Woods-Hole theory), when A is supersingular and E corresponds to a point of W_{A}(1). First we show that all subrings of B which are isomorphic to the same S ∈ R are B^{∗} conjugate, and hence (using results from [G]) that all the curves in W_{A}(1) with fake CM by the same ramified S make up a B^{∗} orbit. Then we suppose that (E, C) is fixed by the involution w_{ρ}, for some ρ ∈ B′, and give alternative characterizations of the image of End(Ê) in B in terms of ρ.

3.1 Fake CM Curves and Orbits of B^{∗}

With notation as in Section 2.1, we fix a supersingular elliptic curve A/𝔽_{p^{2}} and an isomorphism between End(Â) and B = ℤ_{p}[i, j, k]. Then B^{∗} ≅ Aut(Â) acts on W_{A}(1) by ρ(F, α) = (F, ρ ∘ α). It is immediate that this restricts to an action of B^{∗} on the subset of W_{A}(1) corresponding to curves E with fake CM by a fixed S ∈ R. We want to describe the orbits of this (restricted) action.

Lemma 3.1. If S_{1} and S_{2} are subrings of B which are isomorphic to S, there is a ρ ∈ B^{∗} such that S_{2} = ρ^{−1}S_{1}ρ.

Proof. We can assume without loss of generality that S_{1} = ℤ_{p}[ι], where ι = i, j or k. Note that for each of these ι, and for any α, we have

T r(αι) = 0 ⇒ ια = ᾱι.

Suppose first that S_{1} = ℤ_{p}[i]. Since S_{1} and S_{2} are isomorphic, there must be an α ∈ S_{2} such that α^{2} = −r. Hence we have N(α) = r and T r(α) = 0. Now set γ = α/i ∈ B, from which it follows that N(γ) = 1 (and therefore γ^{−1} = γ̄). Finally, choose ϵ = ±1 so that ρ := 1 + ϵγ is a unit. Then using T r(ρi) = 0 we calculate:

\frac{ρiρ̄}{N(ρ)} = (ρ̄)^{−1}ρi = (ρ̄)^{−1}ργ^{−1}α = (ρ̄)^{−1}(ϵ + γ^{−1})α = ϵα.

In other words, ρiρ^{−1} = ϵα, and therefore ρ^{−1}S_{2}ρ = ℤ_{p}[i] = S_{1}.

Now suppose that S_{1} = ℤ_{p}[j]. In this case there must be an α ∈ S_{2} such that α^{2} = −p, and hence α = bi + cj + dk, for some b, c, d ∈ ℤ_{p} such that

−b^{2}r − c^{2}p − d^{2}rp = −p.

Thus, we see that p|b. So b = (ej)j for some e ∈ ℤ_{p}, and α = γj where γ := ek + c + di ∈ B. Again take ρ = 1 ± γ. The remaining case, when S_{1} = ℤ_{p}[k], is similar.

Corollary 3.2. When S is ramified, any two formal S-module structures,

σ_{1}, σ_{2} : S → B = End(Â),

are conjugate in the sense that there is a ρ ∈ B^{∗} with

ρ^{−1}σ_{1}(s)ρ = σ_{2}(s) ∀s ∈ S.

Proof. From Lemma 3.1, there exist γ_{1}, γ_{2} ∈ B^{∗} such that γ_{i}^{−1}σ_{i}(S)γ_{i} = ℤ_{p}[ι] where ι = j or k. Note that iιi^{−1} = −ι in either case. Therefore we obtain two distinct automorphisms of S (over ℚ_{p}) by taking

s → σ_{2}^{−1}(ρ^{−1}σ_{1}(s)ρ),

where ρ is either γ_{1}γ_{2}^{−1} or γ_{1}i^{−1}γ_{2}^{−1}. One of these automorphisms must be the identity, which proves the corollary.

Theorem 3.3. Suppose that a := (E, C) and b := (E′, C′) are points in W_{A}(p) such that E and E′ have (potential) fake CM by S (ramified), and such that C and C′ are either both canonical or both not. Then a = ρb for some ρ ∈ B^{∗}.

Proof. Let E = (F, α) and E′ = (F′, β). By the lemma, there is a ρ ∈ B^{∗} such that

α\overline{End(F)}α^{−1} = (ρβ)\overline{End(F′)}(ρβ)^{−1}.

Moreover, by the corollary, we can choose ρ so that (F, α) and (F′, ρ ∘ β) are two liftings of the same formal S-module structure on Â (in the sense of [G]). Hence by [G, Prop 2.1], we have ρ(E′) = E.

Now, if C and C′ are canonical, it is immediate that a = ρb for this same ρ. So suppose that C and C′ are both non-canonical. Then the isomorphism between (F′, ρ∘β) and (F, α) at least takes C′ to some non-canonical subgroup D ⊆ F. But Aut(F) transitively permutes the non-canonical subgroups by Remark 4.11 of [CMc]. Therefore we may choose an automorphism σ with σ(D) = C, and thus we have a = ρ_{1}b for ρ_{1} = (α ∘ σ ∘ α^{−1})ρ.

Remark 3.4. If E is defined over W (𝔽_{p^{2}}), and Ē ≅ A for some supersingular A with A defined over 𝔽_{p} or with j(A) ≠ 0 or 1728, then E has fake CM. Indeed, the Frobenius endomorphism of Ē over 𝔽_{p^{2}} is [±p]Ē. Since this endomorphism lifts to E, Ê is a Lubin-Tate formal group.

For example, suppose that p = 2 and E is given by y^{2} + 2xy − Ay = x^{3}, where A^{3} = 1. Then in characteristic 2, we have [2](x, y) = (Ax^{4}, y^{4}). So if A ≠ 1, we don’t know if E has fake CM 2-adically.

3.2 Embeddings of Fake Endomorphism Rings

Now suppose that A is defined over 𝔽_{p} and that j(A) ≠ 0, 1728. Recall (from [CMc, §4.2]) that for any ρ ∈ B′, the involution of SD_{A} given by w_{ρ} = ρ ∘ w_{1} has two fixed points. Let x = (E, C) = (F, α, C) be one of the them. As in the previous section, Woods Hole theory gives us an embedding of End(F) into B:

α_{∗}End(F) := α^{−1}\overline{End(F)}α ⊆ End(Â) = B.

In this section, we use the embedding to reprove the result that E has fake CM by the ring of integers in a ramified quadratic extension of ℚ_{p}. We also give alternate descriptions of the embedding which depend only on ρ, in particular showing that the fake endomorphism rings of both fixed points embed onto the same subring of B.

Definition 3.5. For ρ = a + bi + cj + dk ∈ B, we let ρ′ = a − bi + cj − dk.

Lemma 3.6. (i) For all ρ ∈ B, ρj = jρ′. (ii) B′ = { ρ ∈ B^{∗} | ρρ′ ∈ ℤ_{p}^{∗} } (iii) If ρ_{1}, ρ_{2} ∈ B^{∗}, (ρ_{1}ρ_{2})′ = ρ′_{1}ρ′_{2}. (iv) If ρ ∈ B′, ρρ′ = ρ′ρ.

Proposition 3.7. Let ρ ∈ B′, and let x := (E, C) = (F, α, C) be fixed by w_{ρ}. Then α_{∗}End(F) = ℤ_{p}[γ], where γ = ρj and hence γ^{2} ∈ pℤ_{p}^{∗}.

Proof. This is basically proven in [CMc, Prop 4.9], although we repeat the argument here. By Theorem 2.1 (and the fact that the only degree p endomorphisms of A are ±j) we can choose isogenies,

β : \overline{F/C} ⟶^{∼} Â ι_{C} : F → F/C,

such that E/C = (F/C, β), and such that (ι_{C}, j) represents the natural isogeny from E to E/C. In fact, ι_{C} can be taken to be the natural map.

Now, the fact that ρ(E/C) = E implies that there is an isomorphism, σ : F/C → F, such that ρ ∘ β = α ∘ σ̄. So let π_{0} = σ ∘ ι_{C} ∈ End(F), and then take γ = α\overline{π}_{0}α^{−1}. Then γ ∈ α_{∗}End(F) by definition, γ = ρj by commutativity, and from Lemma 3.6 we have

γ^{2} = ρjρj = −pρρ′ ∈ −pℤ_{p}^{∗}.

Furthermore, since this implies that ℤ_{p}[γ] is a maximal order, it must be all of α_{∗}End(F).

Corollary 3.8. Let x = (F, α, C) be fixed by w_{ρ} for ρ ∈ B′, and let K = ℚ_{p}(\sqrt{−p}, \sqrt{−Dp}) for D a quadratic non-residue (mod p). Then x is defined over K, and

End(F) = End_{K}(F) ≅ ℤ_{p}[\sqrt{−ρρ′p}].

Proof. The fixed points of w_{ρ} are defined over K, by the explicit formula for w_{ρ} (given in [CMc, Eq 3]). Therefore, F/C and the natural map, ι_{C} : F → F/C, are defined over K. Hence, the endomorphism, π_{0} (as in Proposition 3.7), is defined over K.

Proposition 3.9. If ρ ∈ B′ and x := (F, α, C) is fixed by w_{ρ}, then

α_{∗}End(F) = S_{ρ} := {τ ∈ B: ρτ′ = τρ}.

Proof. One direction is easy. In particular, from the previous proposition, everything in α_{∗}End(F) can be written as a + bγ. This is in S_{ρ} since

ρ(a + bρj)′ = aρ + bρρ′j = (a + bρj)ρ.

For the other direction, Lemma 3.6 implies that S_{ρ} is at least a ring. We want to show that S_{ρ} ⊆ α_{∗}End(F). So first choose a τ ∈ S_{ρ}^{∗}. From the fact that ρ ∘ w_{1} = w_{1} ∘ ρ′ on SD_{A} (basically just ρj = jρ′, see [CMc, Cor 4.6]), we have

w_{ρ}(τx) = ρτ′w_{1}x = τρw_{1}x = τx,

which means that τx is one of the two fixed points of w_{ρ}. Suppose first that τx = x, i.e., (F, α) ≅ (F, τ ∘ α). Then by Theorem 2.1, there is a σ ∈ End(F) such that

α ∘ σ̄ = τ ∘ α,

and hence τ ∈ α_{∗}End(F). We conclude that if τ ∈ S_{ρ}^{∗}, at least τ^{2} ∈ α_{∗}End(F). But then, since (1 ± τ)^{2} = 1 ± 2τ + τ^{2}, it follows that τ ∈ α_{∗}End(F). Finally, if c ∈ S_{ρ}, one of either 1 + c or 1 − c must be in S_{ρ}^{∗}. Thus, S_{ρ} ⊆ α_{∗}End(F).

Corollary 3.10. If F and G are formal groups corresponding to the two fixed points of w_{ρ}, End(F) is canonically isomorphic to End(G).

Proof. Let x = (F, α, C) and y = (G, β, C′) be the two fixed points of w_{ρ}. Then from either proposition, we have

α_{∗}End(F) = β_{∗}End(G).

So α_{∗} and β_{∗} identify End(F) and End(G) with the same subring of B.

Remark 3.11. Let x = (F, α, C) and y = (G, β, C′) be the two fixed points of w_{ρ}, for ρ ∈ B′ (as above). Let S be the ring of integers in the ramified quadratic extension of ℚ_{p} for which End(F) ≅ End(G) ≅ S. Then by [G, Prop 2.1], x and y are the two canonical liftings of the two S-module structures on Â with image α_{∗}End(F) = β_{∗}End(G).

4 Real CM

In this section, we shift our focus to elliptic curves E/R which have real CM, i.e. for which End_{R}(E) ≠ ℤ. Our main result is that, inside SD_{A}, real CM points are dense in the set of fake CM points. The strategy is to use Woods Hole theory and the fact that End(A) is dense in End(Â). First we make B into a topological ring in the usual way, by defining

||ρ|| = max{|h(ρ)|: h ∈ Hom_{ℤ_{p}}(B, ℤ_{p})}.

Then from the explicit formula of Gross-Hopkins (see [CMc, §4.2] or Section 2.1), the action,

B^{∗} × SD_{A} → SD_{A},

is continuous with respect to both variables.

Now assume that A/𝔽_{p}, and let K = ℚ_{p}(\sqrt{−p}, \sqrt{−Dp}) and R = O_{K}. Thus the fake CM curves corresponding to points of SD_{A} are all defined and have fake CM over R by Corollary 3.8. Then real CM points are dense in these fake CM points in the following sense.

Theorem 4.1. Choose S ∈ R ramified. Then points of SD_{A} corresponding to elliptic curves, E/R, for which End_{R}(E)⊗ℤ_{p} ≅ S are dense in those for which End_{R}(Ê) ≅ S. In fact, if (F, α) has fake CM and ϵ ∈ ℝ^{+}, there exist ρ ∈ B^{∗} such that ||ρ − 1|| < ϵ and (F, ρα) ≠ (F, α) has real CM.

Proof. In general, when E = (F, α) is defined over R with residue field k, E has CM over R if and only if α_{∗}End_{R}(F) ∩ End_{k}(A) ≠ ℤ in End_{k}Â. In fact,

End_{R}(F, α) ≅ α_{∗}End_{R}F ∩ End_{k}A.

This follows from Theorem 2.1 if R = R_{p}, and an argument for more general R can be made via crystalline cohomology. In our case, k = 𝔽_{p^{2}}, and since A/𝔽_{p} is supersingular, this guarantees that End_{k}(A) is dense in End_{k}(Â) = End(Â).

So suppose (F, α) is defined over R (as above) and has fake CM by S ∈ R (ramified) and α_{∗}S = ℤ_{p}[γ]. Fix an ϵ > 0. Then there exists δ > 0 such that for all g ∈ End(Â) with ||g − γ|| < δ, there exists ρ ∈ B^{∗} with ||ρ − 1|| < ϵ and

ρSρ^{−1} = ℤ_{p}[g].

This follows from the construction of Theorem 3.3, since δ can be chosen so that ℤ_{p}[g] ≅ ℤ_{p}[γ] for all ||g−γ|| < δ. In particular, we may then choose g ∈ End(A) with ℤ_{p}[g] ≠ ℤ_{p}[γ], since End(A) is dense in End(Â). Then (F, ρ ∘ α) has CM because

(ρ ∘ α)_{∗}End_{R}F = {ρ ∘ α ∘ γ ∘ (ρ ∘ α)^{−1} : γ ∈ End_{R}F } = ρSρ^{−1}.

Therefore, g ∈ (ρ ∘ α)_{∗}End_{R}(F) ∩ End(A).

Corollary 4.2. Let A be any supersingular elliptic curve over 𝔽_{p^{2}}. Then points corresponding to elliptic curves E with CM by an order of discriminant pM with (p, M) = 1 fill out a μ_{2(p+1)/i(A)} orbit of G_{m} ≅ \overline{SD}_{A}. Two such curves correspond to points in the same μ_{(p+1)/i(A)} orbit if and only if M_{1}M_{2} is a square (mod p).

Proof. First suppose that A/𝔽_{p} and j(A) ≠ 0, 1728. Recall that curves with fake CM by S (as above) correspond to fixed points of the involutions w_{ρ} for ρ ∈ B′ by [CMc, Prop 4.10]. Remark 4.8 of [CMc] says that such points fill out a μ_{2(p+1)} orbit of G_{m} ≅ \overline{SD}_{A}, and that B^{∗} acts like μ_{p+1}. Now we have Theorem 3.3 which says that curves with the same fake endomorphism ring are B^{∗} translates. So this proves the analogous statement for fake CM curves, and by Theorem 4.1 the statement about real CM curves then follows.

Now suppose that j(A) = 0 or 1728. Remark 4.8 is based on the explicit formula for the action of B^{∗} on the deformation space, X_{K}, for the formal group Â. When j(A) = 0 or 1728, W_{A}(1) can be identified with the quotient of X_{K} by a faithful action of Aut(A)/ ± 1, in a way which is compatible with the natural embedding of Aut(A) into B^{∗}. So basically, we can use the same argument as above for the circle of X_{K} which lies over SD_{A}, and then apply the degree i(A) map. Similarly, if A is not defined over 𝔽_{p}, we can choose some A_{0}/𝔽_{p} and then apply an isomorphism between W_{A_{0}}(p) and W_{A}(p) as in [CMc, §4.1].

Remark 4.3. When A/𝔽_{p}, a canonical choice of parameter on SD_{A} is given by (j(E) − Teich(j(A)))/\sqrt{p^{i(A)}}.

Question: If E and E′ both satisfy the above conditions, the residue class of

\frac{j(E′) − Teich(j(A))}{j(E) − Teich(j(A))} (mod \sqrt{p})

is the residue class of a \frac{p+1}{i(A)}-th root of unity. Which one and when is it 1?

Documenta Mathematica · Extra Volume Coates (2006) 261–300

4.1 Heegner Points

By a Heegner point on X_{0}(N) we mean a pair (E, C) where E is a CM elliptic curve and C is a cyclic subgroup of order N such that End(E) ≅ End(E/C). Let X_{0}(p^{n}) denote a stable model for X_{0}(p^{n}). In this section, we discuss the placement of Heegner points on X_{0}(p^{n}), beginning with those Heegner points which lie in the ordinary region.

Let R_{i}(D) denote the order of discriminant p^{i}D in ℚ[\sqrt{p^{i}D}] where D < 0, and (D, p) = 1. Suppose End(E) ≅ R_{i}(D). Then E has ordinary reduction if and only if (\frac{D}{p}) = 1 and i is even.

In order to study ordinary Heegner points, we interpret the irreducible affinoids, X_{a b}^{±}, which make up the ordinary locus as in [C1]. Recall that X_{a b} (for a, b ≥ 0 and a + b = n) was defined in Section 2.1 as the affinoid in X_{0}(p^{n}) whose points correspond to pairs (E, C) where E is ordinary and |C ∩K(E)| = p^{a}. The first author showed (see [C1, §2] or [CMc, §3.2]) that for a ≥ b this is equivalent to the affinoid whose points correspond to pairs, (E, P), where E is ordinary and P is a certain pairing from K_{a}(E) := K(E) ∩ E[p^{a}] onto μ_{p^{b}}. Furthermore, let C_{a b} denote the set of isomorphism classes of pairings from ℤ/p^{a}ℤ onto μ_{p^{b}} (which has two elements when a ≥ b ≥ 1). Then for any β ∈ C_{a b} the subspace, X_{a b}^{β} ⊆ X_{a b}, consisting of those pairs for which P ∈ β, is an irreducible affinoid which reduces to Ig(p^{a}). Now, using the Atkin-Lehner involution, the remaining irreducible affinoids (for a < b) in the ordinary locus can be defined by

X_{a b}^{β} = w_{n}X_{b a}^{( \frac{−1}{p})β}.

(Note: This is a slight change from the notation of [C1].) Ordinary points of X_{0}(p^{n}) all have smooth reduction on one of these components, and we will show that there are in fact infinitely many Heegner points on each.

Lemma 4.4. For any b ≥ 0, there are infinitely many Heegner points on X_{b b}.

Proof. Points of X_{b b} can also be thought of as triples, (E, C_{1}, C_{2}), where E is an ordinary elliptic curve and C_{i} is a cyclic subgroup of order p^{b} such that C_{1} ∩ C_{2} = (0) and C_{i} ∩ K(E) = (0). If we let ι_{C} denote the natural map from E → E/C, then the triple, (E, C_{1}, C_{2}), just corresponds to the pair, (E/C_{1}, C(C_{1}, C_{2})), where

C(C_{1}, C_{2}) := ker(ι_{C_{2}} ∘ ι̌_{C_{1}}) ⊆ E/C_{1}.

Now, choose any ordinary elliptic curve, E, with CM by R_{2i}(D), and then choose C_{1} and C_{2} (as above) so that End(E/C_{1}) ≅ End(E/C_{2}) ≅ R_{2(i+b)}(D). If i > 0, any choice of C_{1} and C_{2} (as above) will do. If i = 0, one also needs C_{i} to be disjoint from the kernel of the Verschiebung lifting (which is always possible if p > 2). Then (E/C_{1}, C(C_{1}, C_{2})) is a Heegner point on X_{b b}.

There are various maps between ordinary affinoids which can now be used (along with Lemma 4.4) to construct Heegner points on every X_{a b}^{β}. First of all, w_{n} takes Heegner points of X_{a b} to Heegner points of X_{b a} by definition. Secondly, the group ℤ_{p}^{∗} acts through (ℤ/p^{b}ℤ)^{∗} on X_{a b} via τ_{r}: (E, P) ↦ (E, P^{r}). Moreover, τ_{r} fixes X_{a b}^{β} (i.e. preserves the class of the pairing in C_{a b}) if and only if τ_{r} is a square. Finally, we have a natural isomorphism, α_{a b}: X_{b b} → X_{a b}, which takes the pair (E, P) to the pair (E, P′) for

P′(R, S) = P(p^{a−b}R, p^{a−b}S).

We now investigate the effect of these maps on Heegner points.

Lemma 4.5. Let F be a fixed ordinary elliptic curve. Then (ℤ/p^{b}ℤ)^{∗} acts transitively on the set of points of the form (F, C) which lie in X_{b,b}.

Proof. Let B_{b}(F) denote this set. Then points of B_{b}(F) correspond to triples (E, C_{1}, C_{2}) as above where E = F/K_{b}(F) and C_{1} = F [p^{b}]/K_{b}(E). There are p^{b−1}(p − 1) such triples. The lemma follows because (ℤ/p^{b}ℤ)^{∗} acts faithfully on B_{b}(F).

Lemma 4.6. If (F, C) is a Heegner point on X_{b b} and End(F) = R_{2b}(D) then α_{a b}(F, C) is a Heegner point.

Proof. The point (F, C) is (E/C_{1}, C(C_{1}, C_{2})), where E = F/p^{b}C, C_{1} = ι_{p^{b}C}(F [p^{b}]) and C_{2} = ι_{p^{b}C}(C). In this case, (p, disc(End(E))) = 1. Let ϕ^{c}: E → E^{σ^{c}} be the lifting of Frobenius. Then α_{a b}(F, C) =

(F, ker(ι_{C_{2}^{σ^{a−b}}} ∘ ϕ^{a−b} ∘ ι_{p^{b}C})),

which is clearly a Heegner point.

Theorem 4.7. There are infinitely many Heegner points lying over each ordinary component of X_{0}(p^{n}) for n ≥ 1 and p > 2 (all with smooth reduction).

Proof. By Lemma 4.6, it suffices to guarantee at least one Heegner point, (F, C), on each X_{b,b}^{β} with End(F) ≅ R_{2b}(D). From the proof of Lemma 4.4, such points correspond to triples (E, C_{1}, C_{2}) where End(E) ≅ R_{0}(D). For a fixed F, we must have E = F/K_{b}(F) and C_{1} = F [p^{b}]/K_{b}(E). Then we get a point of X_{b b} by choosing any C_{2} disjoint from C_{1} and K(E), and a Heegner point if C_{2} is also disjoint from the kernel of the Verschiebung lifting.

At this point, the argument is reduced to simple counting. We have a total of p^{b−1}(p − 2) Heegner points in each B_{b}(F). The cardinality of B_{b}(F) is p^{b−1}(p − 1), and from Lemma 4.5 half of these points lie in each X_{b b}^{β}. So since p^{b−1}(p − 2) > p^{b−1}(p − 1)/2 if p > 3, we are done (p = 3 can be handled by Atkin-Lehner).

Heegner points in the supersingular region of X_{0}(p^{n}) are somewhat easier to describe.

Lemma 4.8. Let E be a CM elliptic curve with supersingular reduction, such that p^{m} exactly divides the discriminant of End(E). Then we have

h(E) = \begin{cases}p^{1−k}/(p + 1), & if m = 2k \\ p^{1−k}/2, & if m = 2k − 1. \end{cases}

Furthermore, if End(Ê) = ℤ_{p}[γ] and γ^{2} ∈ ℤ_{p}, we have Ker(γ) ∩ K(E) = K(E) (which has order p^{k}).

Proof. This is an exercise in applying [B, Thm 3.3]. The point is that if E/C ≅ E, we must at least have h(E/C) = h(E).

Theorem 4.9. Let E be a CM elliptic curve with supersingular reduction, such that p^{m} exactly divides the discriminant of End(E). Then (E, C) ∈ X_{0}(p^{n}) (for n > 0) is a Heegner point if and only if m = n and K(E) ⊆ C.

Proof. This follows directly from Lemma 4.8 (and [B, Thm 3.3]). Indeed, if E and m are as above, and C ⊆ E is any cyclic subgroup of order p^{n}, we have

disc(End(E/C)) = \frac{disc(End(E)) · |C|^{2}}{|K(E) ∩ C|^{4}} · \begin{cases}p, & if m is odd and K(E) ⊆ C \\ 1, & otherwise. \end{cases}

Now, when n ≤ 3, the above results make it possible to be very explicit about the placement of Heegner points on X_{0}(p^{n}). On X_{0}(p), the supersingular Heegner points all lie on SD_{A} for some A and have singular reduction (although when j(A) = 1728 they have smooth reduction on the Deligne-Rapoport model from [DR, §VI.6.16]). They also correspond to pairs, (E, C), where E has CM by R_{1}(D) and C = K(E). Heegner points of X_{0}(p^{2}) correspond to those pairs, (E, C), where E has CM by R_{2}(D) (with (\frac{D}{p}) = −1) and K(E) = pC. They all have smooth reduction on the component of X_{0}(p^{2}) which Edixhoven found (and which we call \overline{Y}_{A}). Finally, Heegner points on X_{0}(p^{3}) correspond to pairs where E has CM by R_{3}(D) and K(E) = pC. This implies that they all lie on the affinoid Z_{A}. By Theorem 4.1, there are infinitely many which are fixed by some w̃_{ρ}. Hence, using the discussion at the beginning of [CMc, §8], they have smooth reduction on each of the new components which lie in the singular residue classes of Z_{A}. However, there are also infinitely many supersingular Heegner points of X_{0}(p^{3}) which are not fixed by any w̃_{ρ}, from the preceding theorem and Proposition 7.4 of [CMc] (see also [CMc, Rem 7.5]), and it is unclear where the reductions of these points lie on \overline{Z}_{A}.

5 Field of Definition

Suppose L/K is an unramified extension of local fields. It follows from [DM, Thm 2.4] that an Abelian variety A over K has semi-stable reduction (i.e. has a model with semi-stable reduction over O_{K}) if and only if A_{L} has semi-stable reduction. Also, in the special case where A is the Jacobian of a curve, C/K, and L/K is the maximal unramified extension, A has semi-stable reduction if and only if C_{L} does. It is not true, however, that (in this case) C has semi-stable reduction whenever A does. For example, the Jacobian of X_{0}(p) has a model with semi-stable reduction over ℤ_{p}, while X_{0}(p) may not (for example, when p = 37). This is an important point for us, because Krir determined a field over which the Jacobian of X_{0}(p^{n}) attains stable reduction in [K, Théorème 1]. Indeed, let K = ℚ_{p}(\sqrt{−p}, \sqrt{−Dp}) for D a quadratic non-residue. Then Krir’s result can be stated as follows.

Theorem 5.1 (Krir). The Jacobian of X_{0}(p^{n}) has stable reduction over the class field M_{n} over K of the subgroup of K^{∗} given by

\{a ∈ O_{K}^{∗}: a^{2} ∈ 1 + \sqrt{p^{n−1}}O_{K}\}.

By the above reasoning, it follows that X_{0}(p^{n}) also has a stable model over this same field, M_{n}. However, one can not conclude from this result which extensions of ℚ_{p} are sufficient for X_{0}(p^{n}) to attain stable reduction (and there may not be a minimal such field). What we do in this section is produce a finite extension, F_{3} ⊇ ℚ_{p}, over which our stable model for X_{0}(p^{3}) can be defined, partially using the result of Krir. Fake and real CM also play a role because of the correspondence between w_{ρ} and w̃_{ρ} fixed points and fake CM curves. Our final result is the following.

Theorem 5.2. If 1 ≤ n ≤ 3, the stable model of X_{0}(p^{n}) is defined over the class field F_{n} over K := ℚ_{p}(\sqrt{−p}, \sqrt{−Dp}) of the subgroup of K^{∗} given by

(p^{2a_{n}})^{ℤ}\{a ∈ O_{K}^{∗}: a^{b_{n}} ∈ 1 + \sqrt{p^{n−1}}O_{K}\},

where (a_{n}, b_{n}) = (1, 1) if n = 1, (3, 4) if n = 2, and (3, 2) if n = 3. In particular,

([F_{n}: ℚ_{p}], e(F_{n}/ℚ_{p})) = \begin{cases}(2, 1), & if n = 1; \\ (6(p^{2} − 1), (p^{2} − 1)/2), & if n = 2; \\ (12(p^{2} − 1)p^{2}, (p^{2} − 1)p^{2}) & if n = 3. \end{cases}

5.1 Two Ingredients

One of the main ingredients in our field of definition is the field over which the fixed points of our involutions, w̃_{ρ}, are defined. This field is necessary, by our construction, to obtain good reduction for the underlying affinoids in the singular residue classes of Z_{A}. As real CM curves have been shown to be dense in these points, we are able to apply classical results on CM elliptic curves to determine this field.

Proposition 5.3. Let A be a supersingular curve over 𝔽_{p}. Let F be the smallest field over which all the fixed points in W_{A}(p^{3}) of our involutions w̃_{ρ} are defined. Then,

F = ℚ_{p}(\sqrt{−p}, \sqrt{−Dp}, j(p\sqrt{−p}), j(p\sqrt{−Dp}))

where D ∈ ℤ^{+} is a quadratic non-residue. This is the class field over K := ℚ_{p}(\sqrt{−p}, \sqrt{−Dp}) of the subgroup of K^{∗} given by

(\sqrt{p})^{ℤ}μ_{p^{2}−1}(1 + pO_{K}).

In particular, [F : K] = p^{2}.

Proof. By Theorem 4.1 and Proposition 7.4 of [CMc] we see that F is the field of definition over ℚ_{p} of the set of points (E, C) where E lifts A and has CM by an order whose discriminant is exactly divisible by p (note that here C is not necessarily H_{1}(E)). The proposition now follows from Theorem 5.5 of [S].

Remark 5.4. This field F is the same as that mentioned in Remark 8.1 of [CMc].

We used a surjection from W_{A}(p^{n}) onto W_{A′}(p^{n}), where A and A′ are supersingular elliptic curves over 𝔽_{p^{2}}, to deal with those regions for which j(A′) = 0 or 1728, or for which A′ is not defined over 𝔽_{p}. The surjection can be defined over W (𝔽_{p^{k}}) as long as A and A′ are p-prime isogenous over 𝔽_{p^{k}}. Another ingredient in our determination of a field of definition is the following theorem, that k = 24 always suffices.

Theorem 5.5. Any two supersingular elliptic curves over 𝔽_{p^{2}} are 2-power isogenous over 𝔽_{p^{24}}.

Proof. Suppose A and B are two supersingular elliptic curves over 𝔽_{p^{2}}. It is well known that there exists a 2^{n}-isogeny α: A_{0} := A → A_{n} := B over 𝔽̄_{p} for some n (see [R, Lemma 3.17]). We can factor α as

A_{0}→^{α_{1}}A_{1} ⋯ A_{i−1}→^{α_{i}}A_{i} ⋯ A_{n−1}→^{α_{n}}A_{n}

where A_{i} is an elliptic curve over 𝔽̄_{p} and α_{i} is a 2-isogeny. Furthermore, each A_{i} is supersingular and hence can be defined over 𝔽_{p^{2}}. Thus it suffices to prove the following lemma.

Lemma 5.6. Any two elliptic curves A and B over 𝔽_{p^{k}} which are 2-isogenous over 𝔽̄_{p} are 2-isogenous over 𝔽_{p^{12k}}.

There exists a subgroup C of A(𝔽̄_{p}) of order 2 so that B and A/C are isomorphic over 𝔽̄_{p}. Now, A → A/C is defined over 𝔽_{p^{6k}} because all the points of order 2 on A are defined over the extension of 𝔽_{p^{k}} of degree either 2 or 3. In particular, B ≅ A/C over 𝔽_{p^{12k}} because two elliptic curves with the same j-invariant are isomorphic over the quadratic extension.

5.2 Proof of Theorem 5.2

The case n = 1 follows from [DR, §VI 6.16] and the fact that all supersingular elliptic curves in characteristic p are defined over 𝔽_{p^{2}}. The case n = 2 over ℚ_{p}^{nr} was handled by Edixhoven in [E1, Thm 2.1.1].

When n ∈ {2, 3}, we defined an admissible rigid open cover C_{0}(p^{n}) of X_{0}(p^{n}) in Theorems 5.3 and 9.2 of [CMc] and showed that it was semi-stable over ℂ_{p}. We must show that the cover is defined and semi-stable (as in [CMc, Prop 2.5]) over F_{n}. In particular, we must show that (over F_{n}) each subspace W in the cover is a basic wide open, and that the subspaces intersect each other in the union of annuli.

Recall from [CMc, §3.2] that wide open neighborhoods, W_{a b}^{±}, of the ordinary affinoids, X_{a b}^{±}, can be constructed by considering pairs (E, C) where E is “nearly ordinary.” So we begin by showing that each W_{a b}^{±} is a basic wide open (using essentially the same argument as was used in the proofs of [CMc, Thm 5.3, 9.2]). The affinoid, X_{a b}^{±}, is defined and has good reduction over F_{n} by Lemma 3.6 of [CMc]. Then the intersections, W_{a b}^{±} ∩ W_{A}(p^{n}), are shown to be annuli over F_{n} by choosing an appropriate map to X_{0}(p) and applying Lemma 2.3 about extensions of annuli. Thus each W_{a b}^{±} is a basic wide open over F_{n}. Furthermore, there isn’t anything else to show in the n = 2 case, since

Y_{A} = W_{A}(p^{2}) − ⋃ W_{a b}^{±}

is defined and has good reduction over F_{2} by [CMc, Prop 5.2].

Now suppose that n = 3 and fix a supersingular curve, A/𝔽_{p}, with j(A) ≠ 0 or 1728. By [CMc, Prop 4.2] and Theorem 5.5, it suffices to verify the above conditions for the subspaces which cover W_{A}(p^{3}) for one such A. For convenience, we briefly recall the definitions of these subspaces. Initially, we cover W_{A}(p^{3}) with three subspaces: V_{1}(A), V_{2}(A), and U(A). Each one is π_{1 1}^{−1} of some sub-annulus of W_{A}(p), and they are chosen so that V_{i}(A) is a neighborhood of E_{i A} while U(A) is a neighborhood of Z_{A}. Now, in order to deal with the singular residue classes of Z_{A}, we then refine the cover in the following way. Let S := S(A) be the set of singular residue classes of Z_{A}, and let X_{S} be the underlying affinoid of any S ∈ S. Then we basically remove every X_{S} from U(A) to get a new neighborhood, Û(A), of Z_{A}. Thus the subspaces in C_{0}(p^{3}) which cover W_{A}(p^{3}) are given by:

{V_{1}(A), V_{2}(A), Û(A)} ∪ S(A).

Now, much of the proof of [CMc, Thm 9.2] is still valid, as stated, over F_{3}. For example, by Proposition 8.7 of [CMc] and Proposition 5.3 the elements in S(A) are basic wide opens over F_{3}. Also, V_{i}(A) and U(A) are at least wide opens over F_{3}, because they are residue classes of affinoids which are defined over F_{3} (exactly as in the proof over ℂ_{p}). So the only things which we have to justify are that the affinoids, Z_{A}, E_{1 A}, and E_{2 A} have good reduction over F_{3}, and that V_{i}(A) ∩ U(A) is an annulus over F_{3}. This is where we use Krir.

By Krir’s result we know that the affinoids Z_{A}, E_{1 A} and E_{2 A} have good reduction over M_{3}, and that V_{i}(A) ∩ U(A) is an annulus over M_{3}. Then it follows from Proposition 3.14 of [CMc] that V_{i}(A) ∩ U(A) is an annulus over F_{3}. Also, Z_{A}, E_{1 A}, and E_{2 A} have good reduction over F_{3} because any reduced affinoid which acquires good reduction over an unramified extension must have good reduction. Therefore our cover can be defined and is semi-stable over F_{3}, and hence it corresponds by [CMc, Prop 2.7] to a semi-stable model for the curve over F_{3}.

6 Action of Inertia

If Y/K is a curve, and Y its stable model over ℂ_{p}, there is a homomorphism w_{Y} from

I_{K} := Aut_{cont}(ℂ_{p}/K^{nr}) → Aut(\overline{Y}).

It is characterized by the fact that for each P ∈ Y (ℂ_{p}) and σ ∈ I_{K},

\overline{P^{σ}} = w_{Y} (σ)(\overline{P}).  (1)

We have something similar if Y is a reduced affinoid over K. Namely, we have a homomorphism w_{Y} : I_{K} → Aut(\overline{Y}_{ℂ_{p}}) characterized by (1). This follows from the fact that I_{K} preserves (Y_{ℂ_{p}})^{0} (power bounded elements of A(Y_{ℂ_{p}})) and A(Y_{ℂ_{p}})^{v} (topologically nilpotent elements of A(Y_{ℂ_{p}})). Moreover, inertia action behaves well with respect to morphisms in the following sense.

Lemma 6.1. If f: X → Y is morphism of reduced affinoids over K and σ ∈ I_{K}, then w_{Y} (σ) ∘ f̄ = f̄∘ w_{X}(σ).

For convenience, we let I = I_{ℚ_{p}} and let w be the inertia action (over ℚ_{p}) on

∏_{n≥1} Aut(\overline{X_{0}(p^{n})}).

Also, let m_{n} denote the intersection of all extensions of K^{nr} over which X_{0}(p^{n}) has semi-stable reduction. It is known that m_{n} is the minimal such extension. Clearly m_{n} ⊆ M_{n} but Krir says the extension M_{n} “n’est certainement pas minimale.” In the case of X_{0}(81), this is confirmed in [M2, §4], where a stable model for X_{0}(81) is defined over an extension of ℚ_{3}^{nr} of degree 36 while Krir’s field has ramification index 8 · 3^{4}. From our calculation of the inertia action, however, it will follow that m_{n} = M_{n} for n ≤ 3.

6.1 Inertial action on the ordinary components

For a, b ≥ 0, let X_{a b}^{±} denote the reduction of the ordinary affinoid, (X_{a b}^{±})_{ℂ_{p}}, in the sense of Remark 2.3. Then since X_{a b} is defined over ℚ_{p}, w(σ) must preserve X_{a 0}, X_{0 b} and X_{a b} = X_{a b}^{+} ∐ X_{a b}^{−} (for ab ≠ 0). Also, as explained in [C1, §1] (or the previous section on Heegner points), if a ≥ b, X_{a b} is naturally isomorphic to X_{b a} and to X_{b b}. Therefore, by Lemma 6.1, it suffices to compute the inertial action on X_{b b}.

So recall first that there is an isomorphism between X_{b b}^{β} and Ig(p^{b}) which can be constructed as follows. First we choose a primitive p^{b}-th root of unity, ζ, which represents β in the sense that whenever (E, P) ∈ X_{b b}^{β} and P generates K_{b}(E) we have P(P, P) = ζ^{k^{2}} for some k ∈ (ℤ/p^{b}ℤ)^{∗} (this is explained on page 5 of [C1]). Then we can define an embedding, α_{ζ} : X_{b b}^{β} → X_{1}(p^{b}), given by α_{ζ}(E, P) = (E, p^{b}Q), where Q ∈ E[p^{2b}] such that there exists P ∈ K_{b}(E) with

e_{p^{2b}}(P, Q) = P(P, P) = ζ.

This passes to an isomorphism, X_{b b}^{β} → Ig(p^{b}).

Now, let d be a quadratic non-residue. Identify X_{b b}^{+} with X_{b,b}^{−} by (E, P) → (E, P^{d}), and correspondingly X_{b b} with Ig(p^{b}) × {±1}. Suppose that σ ∈ I and σ(ζ) = ζ^{d^{i}t^{2}}, where i ∈ {0, 1} and t ∈ (ℤ/p^{b}ℤ)^{∗}. Then we have σ(E, P) = (E^{σ}, P^{σ}), where

P^{σ}(σ(A), σ(B)) = σ(P(A, B)).

So if P(P, P) = ζ, it follows that

e_{p^{2b}}(σ(P)/t, σ(Q)/T) = P^{σ}(σ(P)/t, σ(P)/t) = ζ^{d^{i}},

where T ∈ ℤ/p^{2b}ℤ and T ≡ t (mod p^{b}). Identify (the obvious subgroup of) Aut(Ig(p^{n})) with (ℤ/p^{n}ℤ)^{∗}. Then we see that w(σ) acts on X_{b b} as follows.

Proposition 6.2. The inertial action on the ordinary components of X_{0}(p^{n}) is given by

w(σ)|_{X_{b b}} = (t^{−1}, (−1)^{i}).

Corollary 6.3. The field ℚ_{p}^{nr}(μ_{p^{[n/2]}}) is contained in m_{n}.

6.2 Action of Inertia on \overline{X_{0}(p^{2})}

Suppose A is a supersingular elliptic curve over 𝔽_{p^{2}}. Inside the corresponding residue class, W_{A}(p^{2}) ⊆ X_{0}(p^{2}), we have an affinoid Y_{A} defined over W (𝔽_{p^{2}})⊗ ℚ_{p} such that Y_{A} := \overline{Y_{A} ⊗ ℂ_{p}} is the set of non-singular points in a component of the stable reduction of X_{0}(p^{2}). Now we determine the action of I on Y_{A}.

First assume that A is defined over 𝔽_{p} and that j(A) ≠ 0 or 1728 (general case will follow from Lemma 6.1). Let κ be as in Theorem 2.2. We know there are series F (T), G(T) ∈ T ℤ_{p}[[T]] such that Y_{A} is the affinoid

Max (ℚ_{p}⟨a, a^{−1}, b, b^{−1}, x, y⟩/M)

where M is the ideal generated by κ^{p}a = x^{p+1}, κ^{p}b = y^{p+1} and

(F (x) + G(κ/x) − F (y) − G(κ/y)).

Suppose α^{p+1} = κ. Then if K = ℚ_{p}(α), (Y_{A})_{K} is

Max (K⟨u, v, u^{−1}, v^{−1}⟩/(F (α^{p}/u) + G(κu/α^{p}) − F (α^{p}/v) − G(κv/α^{p}))),

where u = α^{p}/x and v = α^{p}/y. It follows that Y_{A} has the equation,

uv(v − u)^{p−1} = 1,

or s^{p+1} = (r^{2} − 1)/4, if we let s = 1/(v − u) and r = (u + v)/(v − u). Now, on one hand we have \overline{σ(u(P))} = \overline{u(P)}. On the other, if σ(α) = ζα for a p + 1-st root of unity, ζ, we have σ(u(P)) = ζ^{p}u(σ(P)). Thus, on Y_{A}, w(σ) is the automorphism (u, v) → (ζu, ζv), or equivalently (r, s) → (r, ζ^{−1}s).

Since we have a finite morphism from Y_{A} to Y_{A′} over W (𝔽̄_{p}) ⊗ ℚ for arbitrary supersingular A′ (an isomorphism when j(A′) ≠ 0 or 1728) we know by Lemma 6.1 the action of I on Y_{A′} for all A′ as long as p ≥ 13. In general, Y_{A} has the equation

s^{(p+1)/i(A)} = (r^{2} − 1)/4

and w(σ) is the automorphism (r, s) → (r, ζ^{−i(A)}s). This also determines the action on E_{i A}, i ∈ {1, 2}, since as explained in Remark 9.3 of [CMc] we have finite degree p morphisms E_{1 A} → Y_{A} and E_{2 A} → Y_{A^{F rob}} with purely inseparable reduction.

Remark 6.4. It follows from the above and Corollary 6.3 that K^{nr}(μ_{p}, α) ⊆ m_{2}. Therefore, since

Gal(K^{nr}(μ_{p}, α)/K^{nr}) ≅ O_{K}^{∗}/\{a ∈ O_{K}^{∗}: a^{2} ∈ 1 + \sqrt{p}O_{K}\},

we see that M_{2} = m_{2}.

6.3 Action of Inertia on \overline{X_{0}(p^{3})}

Suppose A is a supersingular elliptic curve over 𝔽_{p} with B = End(Â). Suppose ρ ∈ B′ and x is a fixed point of \overline{w}_{ρ}. Then we know w_{ρ} has a unique fixed point x̃ := (F, ι, C) in SD(ℂ_{p}) above x. Let C_{x} be the smooth locus of the corresponding component of the stable reduction of X_{0}(p^{3}) (which is affine and hyper-elliptic, with equation y^{2} = x^{p} − x). By Proposition 7.4 of [CMc] the fixed points F_{x} of the hyper-elliptic involution τ_{x} of C_{x} are naturally in 1-1 correspondence with the p non-canonical subgroups of F [p]. So Aut(F) acts on F_{x}.

If L is a finite extension of ℚ_{p}, let Art_{L} denote the Artin map from L^{∗} to Gal(L^{ab}/L). Let D ∈ ℤ^{+} be a quadratic non-residue mod p and K = ℚ_{p}(\sqrt{−p}, \sqrt{−Dp}).

Theorem 6.5. Let N_{F} denote the norm from K^{∗} to ℚ_{p} ⊗ End F. If b ∈ O_{K}^{∗} and Q ∈ F_{x}, then

w(Art_{K}(b))Q = N_{F} (b^{−1})Q.

This makes sense because End F maps naturally into K. Also, if M_{3} is the class field over K of the subgroup of K^{∗} given by (\sqrt{p})^{ℤ}μ_{p^{2}−1}(1 + pO_{K}), then the non-canonical subgroups of F [p] are defined over M_{3} by Proposition 5.3.

Proposition 6.6. Suppose K is an imaginary quadratic field and 𝔭 is a prime ideal of O_{K}. If E is an elliptic curve with good reduction over the ring of integers R of a finite unramified extension L of K_{𝔭} with CM in K then the formal group of E over R is a relative Lubin-Tate group as defined by de Shalit in [dS2].

Lemma 6.7. Suppose E and L are as above and σ is the Frobenius automorphism of L/K_{𝔭}. Then there is an isomorphism of E′ := E/ ker(𝔭 ∩ EndE) with E^{σ} so that the reduction of the natural map α: E → E′ is Frobenius.

Proof. This follows from [S, Thm 5.4]. Indeed, identify ℂ with ℂ_{𝔭}, and take σ to be an automorphism of ℂ which restricts to [s, K] on K^{ab} where s_{𝔩} = 1 for 𝔩 ≠ 𝔭 and (s_{𝔭}) = 𝔭(O_{K})_{𝔭}. Then Shimura’s theorem implies that there exists an isomorphism, ξ: E′ ⟶^{∼} E^{σ}, such that if P is a torsion point on E of order prime to 𝔭, ξ(α(P)) = P^{σ}. Because σ is a lifting of Frobenius and the points of order prime to 𝔭 reduce to infinitely many distinct points of E mod 𝔭, the lemma follows.

Proof. (of proposition) Let β: E → E^{σ} be the isogeny of the above lemma. If T is a parameter at the origin on E, let f(T) = β^{∗}T^{σ} and f(T) = π′T +· · ·. Then f ∈ F_{N_{L/K}(π′)} (notation as in [dS2]), and if p_{1} and p_{2} are the natural projections of E × E onto E, with X = p_{1}^{∗}T and Y = p_{2}^{∗}T, then Ê(X, Y) = F_{f}(X, Y).

Proposition 6.8. If E/R has fake CM, then Ê is a relative Lubin-Tate group.

Proof. Suppose S ∈ R, E = (F, α) and End_{R}F ≅ S. Then, by Theorem 4.1, we know ∃ρ ∈ B^{∗}, such that (F, ρα) has CM.

The theorem now follows from (6.4) of [Iw].

Corollary 6.9. M_{3} = m_{3}.

Proof. It follows from Corollary 6.3 and the remarks at the end of §6.2 that M_{2} = K^{nr}(μ_{p}, p^{1/(p+1)}) ⊆ m_{3}. (In general, it follows from Lemma 2 and Theorem 6 of [BLR, §6] that m_{n} ⊆ m_{n+1}.) The theorem implies that (1 + \sqrt{p}O_{K})/(1+pO_{K}) injects into Gal(m_{3}/K) via the Artin map. Indeed, if K_{1} = ℚ_{p}(\sqrt{p}) and K_{2} = ℚ_{p}(\sqrt{pD}), the map (N_{K_{1}}^{K}, N_{K_{2}}^{K}) from (1+\sqrt{p}O_{K})/(1+pO_{K}) to

(1 + \sqrt{p}O_{K_{1}})/(1 + pO_{K_{1}}) × (1 + \sqrt{pD}O_{K_{2}})/(1 + pO_{K_{2}})

is an isomorphism.

Remark 6.10. This implies the existence of a weight 2 newform on X_{0}(p^{3}) whose corresponding representation is wildly ramified at p, which has been independently verified by Jared Weinstein.

Let D_{x} be the wide open residue class above x in SD_{A} (recall that x is a fixed point of \overline{w}_{ρ}) and D̃_{x} the residue class above D_{x} in Z_{A}. Let s: D_{x} → X_{1}(p) be a section of X_{1}(p) → X_{0}(p) on the image of D_{x} as in Lemma 8.6 of [CMc]. For ζ ∈ μ_{p} we defined an automorphism S̃_{s,ζ} of D̃_{x}. For b ∈ O_{K}^{∗}, let ν(b) = 0 if b is a square and 1 otherwise.

Corollary 6.11. Suppose x̃ = (E, C). Then there exists an n_{x} ∈ {0, 1} such that for b ∈ O_{K}^{∗}

w(Art_{K}(b))|_{C_{x}} = \overline{S̃_{s,e_{1}(P,Q)}}τ_{x}^{ν(b)n_{x}},

where s(E, C) = (E, P), Q ∈ E[p]\C, e_{1}( , ) is the Weil pairing on E[p] and (P + Q) = N_{F} (b^{−1})(Q).

This follows from the theorem and the following lemma whose proof we leave as an exercise for the reader.

Lemma 6.12. The automorphism group of the affine curve y^{2} = x^{p}−x has order 2(p − 1)p and is generated by α : (x, y) ↦ (x + 1, y) and β_{b} : (x, y) ↦ (ax, by), where a ∈ 𝔽_{p}^{∗} and b^{2} = a. These satisfy α^{p} = β_{b}^{2(p−1)} = 1 and β_{b}αβ_{b}^{−1} = α^{a}. In particular, there is only one p-Sylow subgroup, and its centralizer is Abelian and generated by α and β_{−1}.

We will show that n_{x} = 1. Suppose σ ∈ I and σ(\sqrt{p}) = −\sqrt{p}. Then w(σ) on the bridging component above A, which has the equation

X^{(p+1)/i(A)} + X^{−(p+1)/i(A)} = Z^{p},

is X → (−1)^{i(A)}X. When j(A) ≠ 0 or 1728, this follows from Equation (4) of [CMc, §8]. The general case then follows from Lemma 6.1. As in the proof of Proposition 8.3 of [CMc], the involutions \overline{w̃}_{ρ} are

(X, Z) → (ζ/X, Z),

where ζ runs over the (p + 1)/i(A)-th roots of unity (X^{−p} may be identified with a parameter U on SD_{A} so that the involutions \overline{w}_{ρ} are U → ζ/U). The fixed points of \overline{w}_{ρ} on \overline{SD}_{A} are the solutions x of U(x)^{2} ≡ ζ. We now label the fixed points of the involutions \overline{w}_{ρ} by the 2(p + 1)/i(A)-th roots of unity. So for each such root of unity ξ, there is a component C_{ξ} of \overline{X_{0}(p^{3})}. It follows from the above that w(σ) restricts to an isomorphism from C_{ξ} to C_{(−1)^{i(A)}ξ}.

The group B^{∗} acts on the part of stable model over A. If α ∈ B^{∗}, let h(α) be the corresponding automorphism of that part of the reduction. If σ ∈ I, because the action of B^{∗} is defined over ℚ_{p}^{nr} (and by Lemma 6.1), we have

w(σ) ∘ h(α) = h(α) ∘ w(σ).  (2)

Using Lemma 3.6, we see that if α ∈ B′ and α^{2} ∈ ℤ_{p}^{∗}, then

0 = α^{2} − (α′)^{2} = (α − α′)(α + α′),

and thus α = ±α′. In particular, we have αx̃ = α′x̃. So

w_{ρ}(αx̃) = ρw(α′x̃) = ραw(x̃),

which equals αx̃ if αρ = ρα and doesn’t equal x̃ if α ∉ Aut(A)ℤ_{p}^{∗}(1 + jB). (So for example, if ρ ∈ ℤ_{p}[i] and α = i and j(A) ≠ 1728.) Suppose this to be the case (i.e. α satisfies these conditions). Then if x̃ ∼ (F, ι, C), the other fixed point of w_{ρ} is αx̃ ∼ (F, αι, C). (This implies Corollary 3.10 in this case.) Let x′ = \overline{αx}. Then h(α) takes C_{x} to C_{x′} and vice-versa.

Let X_{ξ} = C_{ξ} ∐ C_{−ξ}. For τ ∈ I, identifying C_{ξ} with C_{−ξ} via h(α) and using (2), we can write

w(τ)|_{X_{ξ}} = (a(τ), s(τ)) ∈ Aut(C_{ξ}) × {±1}

where s(τ) = τ(\sqrt{p})/\sqrt{p}. If τ = Art_{K}(v), it follows that s(τ) = 1. Suppose s(σ) = −1. Then on X_{ξ}, on one hand we have

w(σ)w(τ)w(σ^{−1}) = w(στσ^{−1}) = (a(τ^{−1}), 1),

and on the other we have

w(σ)w(τ)w(σ^{−1}) = (a(σ)a(τ)a(σ^{−1}), 1).

It follows that a(σ) is not in the commutative subgroup (α, β_{−1}) of Aut(y^{2} = x^{p} − x). This implies a(σ)^{2} ∉ (α) so n_{x} = 1. Thus, in particular, there exists σ such that s(σ) = −1, a(σ) = β_{ϵ}, where ϵ^{2} = −1.

Suppose now that j(A) = 1728. Let ξ be a (p + 1)/2-th root of unity and X = C_{ξ}. It follows that if σ ∈ I, w(σ)(X) = X. Let L = End F ⊗ℚ_{p}. We know X is a double cover of P^{1} and its branch points correspond to non-canonical subgroups of F [p]. Moreover, if Q is such a subgroup and b ∈ O_{L}^{∗}, we have

w(Art_{L}(b))Q = b^{−1}Q.

In particular, w(Art_{L}(ℤ_{p}^{∗}(1+pO_{L}))) is the identity. Now suppose that σ|_{L} ≠ id. Then

σArt_{L}(b) = Art_{L}(b̄)σ.

So if τ = Art_{L}(b), then

w(σ)w(τ)w(σ)^{−1} = w(τ)^{−1}.

It follows from the theorem and Lemma 6.12 that if the order of w(σ) is prime to p, w(σ^{2}) = w(σ)^{2} = τ_{x} and thus n_{x} = 1.

7 Stable Model of X_{0}(p^{3}) when p < 13

Recall from Section 2 that the argument which we used in [CMc] to compute the stable model of X_{0}(p^{3}) does not only apply when p < 13. The reason for this is that in these cases there is no supersingular region where one can apply analysis of de Shalit, as restated in Theorem 2.2, which approximates the forgetful map from X_{0}(p) to X(1). In this section, we recall in greater detail how the theorem is used to construct components in the supersingular regions of X_{0}(p^{2}) and X_{0}(p^{3}) when p ≥ 13. We then use explicit equations for X_{0}(p) when p = 5, 7, and 11, to derive formulas analogous to Theorem 2.2, and subsequently construct the analogous stable reduction components. This should serve not only to extend the result of [CMc] to p > 3, but also to make the construction more understandable and concrete.

7.1 Explicit Analysis of a “Good” Supersingular Region

Suppose that p > 13, and hence by the result of Howe that there is a supersingular A/𝔽_{p} with j(A) ≠ 0, 1728. All of the information from Theorem 2.2 which we need to do the explicit analysis of [CMc] can be summarized as follows. First of all, we have parameters, t and s, on W_{A}(p) and W_{A}(1) (respectively), which identify these regions with the annulus, 0 < v(t) < 1, and the disk, v(s) > 0. Moreover, in terms of these parameters, the maps π_{f} and w_{1} satisfy

w_{1}(t) = \frac{κ}{t}  and  s = π_{f}(t) ≡ t + (\frac{κ}{t})^{p} (mod p),

for some κ ∈ W (𝔽_{p^{2}}) with v(κ) = 1. Finally, the three special circles inside W_{A}(p), namely TS_{A}, SD_{A}, and C_{A}, are described by v(t) = \frac{p}{p+1}, v(t) = \frac{1}{2}, and v(t) = 1 − \frac{1}{2p} (respectively).

Using the above information, we now recall briefly how to explicitly calculate the reduction of the affinoid, Y_{A} := π_{ν}^{−1}(TS_{A}) ⊆ W_{A}(p^{2}). First of all, we show in [CMc, Lemma 5.1] that Y_{A} is isomorphic to the rigid space:

T_{A} := { (x, y) ∈ TS_{A} × TS_{A} | x ≠ y, π_{f}(x) = π_{f}(y) }.

Then we take u, v = α^{p}/t as parameters on two copies of TS_{A}, for any α with v(α) = \frac{1}{p+1}. With these parameters, the condition that π_{f}(x) = π_{f}(y) leads to the following congruence.

u^{−1} − v^{−1} ≡ (v^{p} − u^{p})(κ/α^{p+1})^{p} (mod α).

By making the change of variables s = 1/(v − u) and r = (v + u)/(v − u), we disregard the u = v component and arrive at an isomorphism between \overline{Y}_{A} and an affine curve of the form s^{p+1} = c(r^{2} − 1). Note that this equation also ends up describing the reductions of the affinoids, E_{1 A}, E_{2 A} ⊆ W_{A}(p^{3}), which lie over Y_{A} via π_{f} and π_{ν} (see [CMc, Remark 9.3]).

Similarly, we compute the reduction of Z_{A} := π_{1 1}^{−1}(SD_{A}) ⊆ W_{A}(p^{3}) by first identifying it with the rigid space:

S_{A} := { (x, y) ∈ C_{A} × C_{A} | τ_{f}(x) = w_{1} ∘ τ_{f}(y) }

(this is done in [CMc, Prop 7.1]). Recall that τ_{f} : C_{A} → SD_{A} is the map which takes (E, C) to (E, H_{1}(E)), and hence that π_{f} ∘ τ_{f} = π_{f}. Therefore, in terms of the parameter t from above, it is easy to show that

τ_{f}(t) ≡ t + (κ/t)^{p} (mod p).

Now we may proceed as before, taking U = t/\sqrt{κ} as a parameter on SD_{A} (note that w_{1}(U) = 1/U), and taking X, Y = t/α as parameters on two copies of C_{A}, for any α with v(α) = 1 − \frac{1}{2p}. With a careful choice of α (see [CMc, Prop 8.2]), the definition of S_{A} translates into the following congruence.

(X^{−p} + αX/\sqrt{κ})(Y^{−p} + αY/\sqrt{κ}) ≡ 1 (mod \sqrt{p})

Finally, after a second change of variables of the form, Z = c(XY − 1), we obtain the following equation for \overline{Z}_{A}.

X^{p+1} + X^{−(p+1)} = Z^{p}

Once again, when p < 13, the preceding calculations do not apply because there is no such A. So for the specific primes, p = 5, 7, and 11, we will now derive a formula which is analogous to Theorem 2.2 for each supersingular region, and then use it to compute the reductions of Y_{A} and Z_{A} (as above). In each case, we do arrive at equations which are consistent with Theorem 2.5. Our claim is that the rest of the argument of [CMc] (as summarized in Section 2) is completely analogous for these primes, and need not be repeated. As a final note, we do not address X_{0}(8) and X_{0}(27) primarily because both have good reduction. Additionally, complications arise from the fact that j = 0 and j = 1728 lie in the same residue disk of X(1) in both cases.

7.2 X_{0}(5^{3})

Borrowing directly from [M1, §4], we can choose a parameter on the genus 0 curve, X_{0}(5), by taking t = η_{1}^{6}/η_{5}^{6}. The only supersingular j-invariant is j = 0, and the corresponding annulus is described by 0 < v(t) < 3. Furthermore, from [M1, Table 3], the formulas for the forgetful map and Atkin-Lehner involution are then given by

π_{f}^{∗}j = \frac{(t^{2} + 2 · 5^{3}t + 5^{5})^{3}}{t^{5}}  and  w_{1}^{∗}t = 125/t.

The circles SD and C are described by v(t) = 1.5 and v(t) = 2.7 (respectively), and the too-supersingular circle, TS, is where v(t) = 2.5.

First we compute the reduction of the affinoid, Y := π_{ν}^{−1}(TS) ⊆ X_{0}(25), by embedding it into TS × TS as above. For parameters on two copies of TS, we may choose X, Y = t/(25\sqrt{5}). Then plugging these parameters into the equation, π_{f}(x) = π_{f}(y), we quickly arrive at the congruence:

(X^{2} + 1)^{3}/X^{5} ≡ (Y^{2} + 1)^{3}/Y^{5} (mod \sqrt{5}).

From the definition of T_{A}, we are interested only in the component where X ≠ Y, and by letting X = s/(r − 1)^{3} and Y = s/(r + 1)^{3} we see that this is isomorphic to the affine curve:

s^{2} = r^{2} − 1.  (3)

Now we compute the reduction of Z := π_{1 1}^{−1}(SD) ⊆ X_{0}(125) by means of the embedding into C × C. We need an approximation for τ_{f} : C → SD, and from the above formula for π_{f}, it is easy to show that

τ_{f}(t) ≡ \frac{5^{15}}{t^{5}} + \frac{3 · 5^{10}}{t^{3}} (mod 5^{2}).

As in [CMc, §8.1], we need to work over a larger extension of ℤ_{5} to define and compute the reduction of Z. In particular, it suffices to work over R := ℤ_{5}[β, \sqrt{5}], where β ∈ ℂ_{5} is chosen so that β^{25} ≡ 5 (mod 5\sqrt{5}). For parameters on SD and (both copies of) C, we then choose the functions U = t/(5\sqrt{5}) and X, Y = t/(25β^{5}\sqrt{5}). Using these parameters and our formula for τ_{f}, the equation τ_{f}(x) = w_{1} ∘ τ_{f}(y) then leads to the following congruence.

(\frac{1}{X^{5}} + \frac{3β^{10}}{X^{3}} ) (\frac{1}{Y^{5}} + \frac{3β^{10}}{Y^{3}} ) ≡ 1 (mod \sqrt{5})

After making the substitution, Z = (XY − 1)/(3β^{2}), we arrive at our final equation for \overline{Z}.

X^{2} + X^{−2} = Z^{5}  (4)

Remark 7.1. Note that Equations (3) and (4), describing \overline{Y} and \overline{Z}, are consistent with Theorem 2.5, since p = 5 and i(A) = 3 in this case.

7.3 X_{0}(7^{3})

This is very similar to the previous example, as X_{0}(7) also has genus 0 and only one supersingular annulus corresponding to j = 1728. If we take t = η_{1}^{4}/η_{7}^{4} as a parameter (see [M1, §2]), the supersingular annulus is the region described by 0 < v(t) < 2, and the formulas for the forgetful map and Atkin-Lehner involution are as follows.

π_{f}^{∗}(j − 1728) = \frac{(t^{4} − 10 · 7^{2}t^{3} − 9 · 7^{4}t^{2} − 2 · 7^{6}t − 7^{7})^{2}}{t^{7}} w_{1}^{∗}t = \frac{49}{t}

The circles, SD and C, are now given by v(t) = 1 and v(t) = 13/7 (respectively), and TS is the circle where v(t) = 7/4.

We begin by embedding Y := π_{ν}^{−1}(TS) ⊆ X_{0}(49) into TS × TS, taking X, Y = αt/49 as parameters on both copies of TS, where α^{4} = −7. With these parameters, the equation π_{f}(x) = π_{f}(y) yields the congruence:

(X^{4} + 1)^{2}/X^{7} ≡ (Y^{4} + 1)^{2}/Y^{7} (mod α).

Then the substitution, X = s/(r − 1)^{2} and Y = s/(r + 1)^{2}, defines an isomorphism between the component with X ≠ Y and the genus 1 affine curve:

s^{4} = r^{2} − 1.  (5)

Likewise we compute the reduction of Z by embedding it into C×C. This time the approximation formula for τ_{f} : C → SD, which we derive from the formula for π_{f}, is as follows.

τ_{f}(t) ≡ \frac{7^{14}}{t^{7}} − \frac{2 · 7^{7}}{t^{3}} (mod 7\sqrt{7})

Working over the extension given by R = ℤ_{7}[β] with β^{49} ≡ 7 (mod 7\sqrt{7}), we take X, Y = β^{7}t/49 as parameters on two copies of C. We also take U = t/7 as our parameter on SD. Then the equation τ_{f}(x) = w_{1} ∘ τ_{f}(y) gives us the congruence:

(\frac{1}{X^{7}} − \frac{2β^{21}}{X^{3}} ) (\frac{1}{Y^{7}} − \frac{2β^{21}}{Y^{3}} ) ≡ 1 (mod \sqrt{7}).

To complete the calculation, we make the substitution, Z = (1 − XY)/(2β^{3}), which results in our final equation for \overline{Z}.

X^{4} + X^{−4} = Z^{7}  (6)

Remark 7.2. Once again, note that Equations (5) and (6) are consistent with Theorem 2.5, as p = 7 and i(A) is now 2.

7.4 X_{0}(11^{3})

When p = 11, X_{0}(p) has genus 1 and two supersingular annuli corresponding to j = 0 and j = 1728. In order to work out formulas for the forgetful map on these supersingular regions, we must choose an appropriate model for the overall curve. One convenient model relates the following two functions.

t = (\frac{η_{1}}{η_{11}} )^{12} x = \frac{dt/t}{(η_{1}η_{11})^{2}}

The first function has divisor 5(0)−5(∞), and the second is a degree 2 function with a simple pole at each cusp. Since w_{1} interchanges the two cusps, it follows that x is actually a parameter on the genus 0 quotient, X_{0}(11)^{+} := X_{0}(11)/w_{1}. Therefore, t must be quadratic over x, and indeed by comparing q-expansions we have the equation:

t^{2} + \frac{1}{5^{5}} (x^{5} + 170x^{4} + 9345x^{3} + 167320x^{2} − 7903458)t + 11^{6} = 0.

This model is singular, but we can normalize by adjoining a square root of the discriminant. In particular, the following gives a nonsingular model for X_{0}(11).

y = \frac{2 · 5^{5}t + (x^{5} + 170x^{4} + 9345x^{3} + 167320x^{2} − 7903458)}{(x + 47)(x^{2} + 89x + 1424)}

y^{2} = f(x) = (x − 8)(x^{3} + 76x^{2} − 8x + 188)

Remark 7.3. Since x is invariant under w_{1}, the formulas for the Atkin-Lehner involution are given by w_{1}^{∗}t = 11^{6}/t and w_{1}^{∗}y = −y.

Note that there are two pairs of branch points in the degree two extension from X_{0}(11)^{+} up to X_{0}(11), and they lie in the two residue disks where v(x − 2) > 0 and v(x + 3) > 0. It follows that the regions lying over these residue disks are annuli. In fact, from [DR, §VI 6.16] (or the explicit calculations which follow) these are precisely the two supersingular annuli.

7.4.1 j = 0

Let r_{1}, r_{2} be the two roots of f(x) close to x = −3, and let s_{1}, s_{2} be those close to x = 2. Then the following map defines an isomorphism (over ℚ_{11}) between the annulus, 0 < v(z) < 3, and the subspace of X_{0}(11) which lies over the residue disk, v(x − 2) > 0.

x = z + \frac{(s_{1} + s_{2})}{2} + \frac{(s_{1} − s_{2})^{2}}{16z} ≡ z + 310 − 212 ( \frac{11^{3}}{z} ) (mod 11^{3})
y = (−z + \frac{(s_{1} − s_{2})^{2}}{16z} ) \sqrt{(x(z) − r_{1})(x(z) − r_{2})}

≈ (−z − 212 ( \frac{11^{3}}{z} )) ( ⋯ + 484 ( \frac{11^{3}}{z} )^{3} + 363 ( \frac{11^{3}}{z} )^{2} +

393 ( \frac{11^{3}}{z} ) + 775 + 243z + 484z^{2} + 968z^{3} + ⋯)

To obtain an approximation formula for π_{f} over the annulus, we begin with the following explicit formula (easily verified by q-expansions).

π_{f}^{∗}j = \frac{(60y + 61x^{2} + 864x − 2016)^{3}}{5^{6}t}

Over the whole annulus, 60y + 61x^{2} + 864x − 2016 is well approximated by 5z − 11^{11}/z^{3}, in the sense that the error is always strictly smaller than the larger of these two terms. Similarly, t is always close to −4z^{2}. From this we may conclude that v(j) > 0 over the whole annulus (as claimed).

Now we are able to embed Z ⊆ X_{0}(11^{3}) into C×C and explicitly compute its reduction. We know that the circles, SD and C, are described by v(z) = 3/2 and v(z) = 63/22 (respectively). So our formula for π_{f} leads to the following approximation for τ_{f} : C → SD in terms of z.

τ_{f}(z) ≡ \frac{−3 · 11^{33}}{z^{11}} + \frac{11^{22}}{z^{7}} (mod 11^{2}).

Also, from the fact that x(z) must be fixed by w_{1}, we see that

w_{1}(z) = \frac{(s_{1} − s_{2})^{2}}{16z} ≡ \frac{−3 · 11^{3}}{z} .

At this point our analysis closely parallels that of the two previous examples. In particular, let R = ℤ_{11}[\sqrt{−11}, β] where β ∈ ℂ_{11} satisfies β^{121} ≡ −11 (mod 11\sqrt{−11}), and choose the following functions as parameters on SD and (two copies of) C.

U = \frac{2z}{11\sqrt{−11}} X, Y = \frac{−2z}{121β^{44}\sqrt{−11}}

In terms of these new parameters, the relation τ_{f}(x) = w_{1} ∘ τ_{f}(y) leads to the following congruence.

(\frac{1}{X^{11}} + \frac{3β^{55}}{X^{7}} ) (\frac{1}{Y^{11}} + \frac{3β^{55}}{Y^{7}} ) ≡ 1 (mod \sqrt{−11})

Finally, with the substitution, Z = (XY − 1)/(3β^{5}), we obtain the desired equation for the reduction of Z over R.

X^{4} + X^{−4} = Z^{11}

Similarly, the equation for the (j = 0) supersingular component of X_{0}(11^{2}) follows directly from the embedding of Y into TS × TS (where TS is now the circle, v(z) = 11/4). The approximating formula for π_{f} on that circle is

j = π_{f}(z) ≡ \frac{(5z − 11^{11}/z^{3})^{3}}{5^{6}(−4z^{2})} (mod 11^{3}).

So now let α^{2} = \sqrt{−11} and take as parameters on both copies of TS the functions X, Y = αz/(5 · 11^{3}). Then the relation π_{f}(x) = π_{f}(y) reduces to

(X^{4} + 1)^{3}/X^{11} ≡ (Y^{4} + 1)^{3}/Y^{11} (mod α).

The irreducible component of this curve where X ≠ Y is then isomorphic to the genus 1 curve, s^{4} = r^{2} − 1, by the following map.

X = s/(r − 1)^{3} Y = s/(r + 1)^{3}

Remark 7.4. These equations match those of Theorem 2.5 when p = 11 and i(A) = 3.

7.4.2 j = 1728

The supersingular annulus of X_{0}(11) corresponding to j = 1728 is the region which lies over the residue disk, v(x + 3) > 0. To see this, we first parameterize the region using the annulus, 0 < v(z) < 2, and the following map.

x = z + \frac{(r_{1} + r_{2})}{2} + \frac{(r_{1} − r_{2})^{2}}{16z} ≡ z + 2318 + 12356 ( \frac{11^{2}}{z} ) (mod 11^{4})
y = (−z + \frac{(r1−r2)2}{16z} ) \sqrt{(x(z) − s_{1})(x(z) − s_{2})}

≈ (−z + 12356 ( \frac{11^{2}}{z} )) ( ⋯ + 5324 ( \frac{11^{2}}{z} )^{3} + 3993 ( \frac{11^{2}}{z} )^{2} +

4370 ( \frac{11^{2}}{z} ) + 6001 + 11980z + 5324z^{2} + 3993z^{3} + ⋯)

Then we choose an explicit formula for the forgetful map which is convenient for analysis near j = 1728, in particular

π_{f}^{∗}(j − 1728) = \frac{(665x^{3} + 666xy + 22680x^{2} + 2592y − 120960x + 22680)^{2}}{5^{6}t} .

Over the entire annulus, the cubic function in the numerator is well approximated (in the above sense) by 5z^{2} − 3 · 11^{11}/z^{4}, and t is well approximated by −3z^{3}. Therefore by counting valuations it follows that v(j − 1728) > 0 over the entire annulus, as claimed.

As in the previous examples we now compute the equations for the bridging component of X_{0}(11^{3}) and supersingular component of X_{0}(11^{2}) corresponding to j = 1728. By approximating π_{f} on C (where v(z) = 21/11) and SD (where v(z) = 1), we find the following formula for τ_{f} in terms of z.

τ_{f}(z) ≡ 3 (\frac{11^{2}}{z} )^{11} + 11 (\frac{11^{2}}{z} )^{5} (mod 11\sqrt{11})

After an appropriate choice of parameters on C and SD, this leads to the equation,

X^{6} + X^{−6} = Z^{11},

which describes the bridging component. Then on the too-supersingular circle, where v(z) = 11/6, we approximate π_{f} by

j − 1728 = π_{f}(z) ≡ \frac{(5z^{2} − 3 · 11^{11}/z^{4})^{2}}{5^{6}(−3z^{3})} (mod 11^{2}).

After making an appropriate change of variables, this leads to the equation which one should expect for \overline{Y}:

s^{6} = r^{2} − 1.

8 Stable Model of X_{0}(Np^{3})

Intuitively, one might expect the stable model of X_{0}(Np^{3}) (when (N, p) = 1) to follow fairly directly from the stable model of X_{0}(p^{3}). Indeed, X_{0}(Np^{3}) is birational to X_{0}(p^{3}) ×_{X(1)} X_{0}(N), and X_{0}(N) has good reduction. Using semi-stable maps (as in [C2]) to make this line of reasoning precise, we are able to show the following (compare with Theorem 2.5).

Theorem 8.1. The stable reduction of X_{0}(Np^{3}) has six ordinary components: two isomorphic to X_{0}(N) and four isomorphic to (the normalization of) Ig(p)× X_{0}(N). Also, for each supersingular point P of X_{0}(N), there is a “necklace” of components whose graph is given below in Figure 2. Set i(P) = 2 or 3 if P is elliptic and j(P) = 1728 or 0 (respectively). Set i(P) = 1 otherwise. Then E_{2,P} and E_{1,P} are isomorphic to y^{2} = x^{(p+1)/i(P)} + 1, while Z_{P} is crossed by 2(p + 1)/i(P) components isomorphic to y^{2} = x^{p} − x.

[figure: Z_{P}; ordinary; E_{1,P}; .; .; .; .; .; .; .; .; .; .; .; .; .; ordinary; E_{2,P}]

Figure 2: Partial Graph of the Stable Reduction of X_{0}(Np^{3})

Corollary 8.2. The stable reduction of the p-new part of the Jacobian of X_{0}(Np^{3}) has c_{N}(p^{2} − 1)/6 copies of the Jacobian of y^{2} = x^{p} − x, where c_{N} = [Γ : Γ_{0}(N)].

8.1 Semi-stable Maps

We begin by giving the definition of semi-stable map, and by proving the lemma which will form the blueprint for our overall construction.

Definition 8.3. Let K ⊆ ℂ_{p} be a complete subfield with ring of integers R. Then f : X → Y is a semi-stable map over R, if X/R and Y/R are semi-stable (as in [CMc, Definition 2.6]) and f is finite. In this case we say that f extends the restriction map, f_{K} : X_{K} → Y_{K}, on generic fibers.

Lemma 8.4. Let f : X → Z and g : Y → Z be semi-stable maps over R. Suppose that the following conditions hold.

(i) X̄ and Ȳ have (only) smooth components. (ii) f and g take singular points to singular points (and vice-versa). (iii) For each pair, (x_{i}, y_{i}), of singular points in X̄ and Ȳ with f(x_{i}) = g(y_{i}), A_{x_{i}} × A_{y_{i}} is the disjoint union of annuli (over K), where A_{P} = red^{−1}(P).

(iv) For each pair, (X_{i}, Y_{i}), of components of X̄ and Ȳ with f(X_{i}) = g(Y_{i}), X_{i} × Y_{i} is irreducible, and smooth away from all the points from (iii). Then f × g : X × Y → Z is a semi-stable map (over R), and in particular X × Y is semi-stable.

Proof. Choose any pair (X_{i}, Y_{i}) of irreducible components of X̄ and Ȳ with f(X_{i}) = g(Y_{i}). Let W_{i} = red^{−1}(X_{i}) × red^{−1}(Y_{i}) and A_{i} = red^{−1}(X_{i}^{ns}) × red^{−1}(Y_{i}^{ns}). Then (i)-(iv) guarantee that each W_{i} is a basic wide open (as in [CMc, §2]. Furthermore, the W_{i}’s forms a semi-stable covering of X_{K} × Y_{K}, and then it follows from [CMc, Prop 2.7] that X × Y is semi-stable. Finiteness of f × g is immediate.

Remark 8.5. We will apply Lemma 8.4 to semi-stable extensions of the forgetful maps from X_{0}(p^{3}) and X_{0}(N) to X(1), but with one caveat. Technically, condition (iv) will fail at all points of the form (P, Q) ∈ X_{0}(p^{3}) × X_{0}(N) where P and Q lie over j = 0 (or j = 1728) and both ramify. This issue can basically be ignored, however, as these singularities are resolved in X_{0}(Np^{3}).

8.2 Semi-stable Extensions of the Forgetful Map

We begin by constructing a semi-stable map which extends π_{f} : X_{0}(p^{3}) → X(1) (as in the main theorem of [C2]). This can be done by starting with the stable models for X_{0}(p^{3}) and X(1) (say, Spec(ℤ_{p}[j])) and performing a series of blow-ups. At each step, we choose a component of X_{0}(p^{3}) which has finite image in X(1). There is a unique minimal way to blow-up our models for X(1) and X_{0}(p^{3}) so that this component no longer has finite image and so that π_{f} still extends. After finitely many steps, the process terminates and we have our semi-stable map. A partial picture of this map (showing one supersingular region only) is given below in Figure 3, and the components in the final models for X_{0}(p^{3}) and X(1) can be described in words as follows.

First of all, the ordinary regions of X_{0}(p^{3}) and X(1) are unchanged. In other words, the final model for X_{0}(p^{3}) still has six ordinary components corresponding to the six ordinary affinoids, X_{a b}^{±} (defined in §2), and these all map onto the same component of X(1). For each supersingular elliptic curve, A, the special fiber of X(1) also contains a “necklace” of trivial components. More specifically, each necklace contains a chain of four components which correspond to the circles (and disk) where h(E) = \frac{1}{p(p+1)}, h(E) = \frac{1}{2p}, h(E) = \frac{1}{p+1}, and h(E) ≥ \frac{p}{p+1} (as in §2.1). Intersecting the second of these we also have 2(p + 1)/i(A) components which correspond to residue classes within that circle. The components in our final model for X_{0}(p^{3}) can be given similar descriptions, i.e. we can describe the components within a fixed supersingular region by describing points of the corresponding affinoids in moduli-theoretic terms. Remember that these affinoids should map onto the ones which were just described for X(1) via the forgetful map.

E_{2,A} = \{ (E, C) | h(E) = \frac{1}{p(p+1)}, pC = K_{2}(E) \}
Z_{A} = \{ (E, C) | h(E) = \frac{1}{2p}, pC = K_{2}(E) \}

(with its 2(p + 1)/i(A) nontrivial residue classes)

E_{1,A} = \{ (E, C) | h(E) = \frac{1}{p+1}, p^{2}C = K_{1}(E) \}
W_{1,2}^{±} ⊇ \{ (E, C) | h(E) = \frac{1}{2p}, |C ∩ K_{2}(E)| = p \}

(also blow-up 2(p + 1)/i(A) residue classes)

⊇ \{ (E, C) | h(E) = \frac{1}{p(p+1)}, |C ∩ K_{2}(E)| = p \}

W_{0,3} ⊇ \{ (E, C) | h(E) = \frac{p}{p+1} \} (so E is too-ss)

⊇ \{ (E, C) | h(E) = \frac{1}{p+1}, |C ∩ K_{1}(E)| = 1 \}

⊇ \{ (E, C) | h(E) = \frac{1}{2p}, |C ∩ K_{2}(E)| = 1 \}

(also blow-up 2(p + 1)/i(A) residue classes)

⊇ \{ (E, C) | h(E) = \frac{1}{p(p+1)}, |C ∩ K_{2}(E)| = 1 \}

Remark 8.6. Recall that W_{a b}^{±} is a wide open neighborhood of the ordinary affinoid, X_{a b}^{±}, which extends into the supersingular locus (see [CMc, §3.2]).

In order to apply Lemma 8.4, we also need to construct a semi-stable map extending π_{f} : X_{0}(N) → X(1) (involving the same model for X(1)). Basically, we start with the good reduction model for π_{f} : X_{0}(N) → X(1). Then every time we blow-up X(1) (as above), this forces a blow-up of X_{0}(N) so that π_{f} still extends. Again the ordinary locus of X_{0}(N) is unchanged. To understand the supersingular regions, consider π_{f} : X_{0}(N) → X(1) first as a map of smooth curves over \overline{𝔽}_{p}, and let P be a point of X_{0}(N) such that π_{f}(P) is supersingular. If P does not ramify, π_{f} must restrict to an isomorphism on the corresponding residue class of X_{0}(N). The only other option is that either e(P) = 3 and j(π_{f}(P)) = 0, or e(P) = 2 and j(π_{f}(P)) = 1728. There are two key points to make in either case. First of all, the corresponding residue class of X_{0}(N) (over ℂ_{p}) is an extension of a disk which is ramified (totally) at exactly one point (degree 2 if j = 1728, degree 3 if j = 0). Hence, the extension can be generated analytically by adjoining either \sqrt{j − 1728} or \sqrt[3]{j}. Secondly, when j = 0 or 1728 is supersingular, it is necessarily too-supersingular. Hence it reduces to a smooth point on the innermost component of its residue class in our final model for X(1). Therefore, like its image in X(1), the residue class of X_{0}(N) corresponding to such a P contains a chain of 4 components on which π_{f} is given locally by t → t^{2} or t → t^{3}. The “bridging component” then intersects 2(p + 1) copies of P^{1}, which map 2 : 1 or 3 : 1 onto analogous components of X(1).

To summarize the semi-stable extensions of both forgetful maps to X(1), a picture of the special fibers is now given below in Figure 3. Once again, the graph shows the entire ordinary locus, but only one supersingular region for each curve.

[figure: π_{f}; >; ∨; π_{f}; .; .; .; .; .; .; .; .; X(1); .; .; .; .; .; .; E_{2,A}; X_{2,1}^{−}; X_{2,1}^{+}; X_{3,0}; Z_{A}; X_{0}(p^{3}); .; .; .; .; .; .; .; .; .; X_{0,3}; X_{1,2}^{−}; X_{1,2}^{+}; .; .; .; .; .; .; E_{1,A}; .; .; .; .; .; .; X_{0}(N); .; .; .; .; .; .; .; .]

Figure 3: Partial Graph of Semi-Stable Maps from X_{0}(p^{3}) and X_{0}(N) to X(1)

8.3 Crossing the Semi-Stable Maps

At this point the proof comes down to verifying the hypotheses of Lemma 8.4 and computing the products of irreducible components with common image. The first two hypotheses follow immediately from the construction. To verify condition (iii), we first observe that π_{f} : X_{0}(N) → X(1) can only ramify over j = 0, 1728, or ∞, all of which have smooth reduction on our model for X(1). Furthermore, each supersingular residue class of X_{0}(N) maps with total degree at most 3. Therefore, for any pair of double points, (x, y) (as in the lemma), the corresponding product of annuli, A_{x} × A_{y}, is an unramified extension of some annulus of X_{0}(p^{3}) with degree less than p. Hence it can only be the disjoint union of annuli by [CMc, Lemma 3.3].

Now we compute the products of the irreducible components, starting with the ordinary locus. When we cross X_{0}(N) with the reduction of X_{3,0}, we are essentially crossing with P^{1} trivially (as π_{f} has degree 1 on X_{3,0}). So we simply get a copy of X_{0}(N). When we cross X_{0}(N) with the reduction of X_{2,1}^{±}, we get the curve Ig(p) × X_{0}(N) which is at least irreducible from [E1, Thm 2.1.2]. Recall that Ig(p)/X(1) is a degree (p − 1)/2 extension which is totally ramified over supersingular points, ramified with index 3 or 2 when j = 0 or 1728 is ordinary, and unramified elsewhere. Hence the only singular points of Ig(p) × X_{0}(N) can be ignored as a result of Remark 8.5. The remaining ordinary components can be dealt with by applying an appropriate Atkin-Lehner involution.

Next we consider a fixed supersingular region corresponding to a point P of X_{0}(N) (as above). If j(P) ≠ 0, 1728, or if P is an elliptic point, there’s nothing to do, since π_{f} : X_{0}(N) → X(1) must be an isomorphism on the residue class corresponding to P. But now suppose that j(P) = 0 or 1728, and P is not elliptic. By [E2, 2.3.1] we can choose parameters on E_{2,A} so that it has the equation,

y^{2} = x^{\frac{p+1}{i(A)}} + 1.

Furthermore, the two infinite points are where E_{2,A} meets X_{3,0} and Z_{A}, and (0, ±1) are the points where E_{2,A} meets X_{2,1}^{±}. The forgetful map induces a degree p map on E_{2,A} which has ramification indices of 1, (p − 1)/2, and p at the intersections with X_{3,0}, X_{2,1}^{±}, and Z_{A}. Therefore, if t is a parameter on the image of E_{2,A} in X(1), with t = 0 and ∞ at the double points, it follows that

π_{f}^{∗}t = \frac{cx^{\frac{p−1}{2}}}{(y − x^{\frac{p+1}{2i(A)}} )^{i(A)}} .

Now, we have already seen that the extension from X(1) up to X_{0}(N) is equivalent to adjoining an i(A)-th root of t in this case. Hence one can show that the extension of E_{2,A} can be obtained by adjoining an i(A)-th root of x. Subsequently, by a change of coordinates, the component lying over E_{2,A} in X_{0}(Np^{3}) will have the equation, y^{2} = x^{p+1} + 1. The argument for the remaining components is very similar. For example, on the bridging component, Z_{A}, we may choose a parameter x such that it meets E_{1,A} and E_{2,A} at 0 and ∞, and such that π_{f} is given by t = x^{p}. Adjoining an i(A)-th root of t then generates the same extension as adjoining an i(A)-th root of x. Thus we obtain a bridging component, Z_{P}, as in the statement of the theorem, which is crossed by 2(p+1) components that lie i(A) : 1 over their counterparts on Z_{A}. At this point the remaining components can be computed in a similar manner, or dealt with by applying an appropriate Atkin-Lehner involution. Thus all the supersingular components of X_{0}(Np^{3}) are as claimed, and the theorem is proved. One final remark is that when P ramifies over j = 0 or j = 1728 and is supersingular, we do technically get singularities in X_{0}(N) × X_{0}(p^{3}) which lie over the (smooth) reduction of j = 0 or 1728. These singularities can be ignored, however, by Remark 8.5.

8.4 Examples

It is now fairly straightforward to generate complete graphs with genera for the stable reduction of X_{0}(Np^{3}). First we determine the supersingular values mod p, and the ramification of π_{f} : X_{0}(N) → X(1) over j = 0 and j = 1728. The latter can be derived from [S, Prop 1.43], which gives both the degree and number of elliptic points of each type. The components in the supersingular region then follow directly from Theorem 8.1. The only things which remain to be computed are the genera of X_{0}(N) and Ig(p) × X_{0}(N).

The genus of X_{0}(N) can be computed with [S, Prop 1.40]. Then Riemann-Hurwitz can be applied to the forgetful map from X_{0}(N) × Ig(p) to X_{0}(N). By way of illustration, we now describe the stable reductions of X_{0}(Np^{3}) in two examples: X_{0}(3 · 11^{3}) and X_{0}(7 · 13^{3}).

Example 1: X_{0}(3 · 11^{3})

Only j = 0 and j = 1728 are supersingular mod 11. In the degree 4 extension from X_{0}(3) → X(1), j = 0 splits into two points with e = 1 and e = 3, while j = 1728 splits into two points with e = 2. So we have a total of four supersingular necklaces. For the one corresponding to the unique elliptic point, there are 8 genus 5 components along the bridging component, and two outer components which meet the ordinary locus and have genus 1. The other three supersingular regions have 24 genus 5 components along the bridging component, and two outer components which also have genus 5. Now we compute the genera of the ordinary components. X_{0}(3) has genus 0, and by Riemann-Hurwitz the genus of Ig(11) × X_{0}(3) is then 4. Indeed, it lies over X_{0}(3) with degree 5, and is totally ramified over 4 points and unramified elsewhere. This implies a total genus of:

2(0) + 4(4) + 1[2(1) + 8(5)] + 3[2(5) + 24(5)] + (4 − 1)(6 − 1) = 463,

which can easily be verified with [S, Prop 1.40].

Example 2: X_{0}(7 · 13^{3})

The unique supersingular j-invariant for p = 13 is j = 5. Since this is neither 0 nor 1728, we simply get 8 supersingular regions which are all isomorphic to the supersingular region of X_{0}(13^{3}). In particular, each necklace has 28 genus 6 components along the bridging component and then 2 more genus 6 components which meet the ordinary locus. Now we compute the genera of X_{0}(7) and X_{0}(7) × Ig(13). The first has genus 0, and for the second we again apply Riemann-Hurwitz. The degree is 6, and we have total ramification over the 8 supersingular points. There are also two elliptic points of X_{0}(7) lying over j = 0, each of which must split into two points with e = 3 in X_{0}(7) × Ig(13). So the genus of Ig(13) × X_{0}(7) is 19. That means if we add up the total genus of X_{0}(7 · 13^{3}) we get

2(0) + 4(19) + 8[2(6) + 28(6)] + (8 − 1)(6 − 1) = 1551,

which again can be easily verified with [S, Prop 1.40].

9 Index of Important Notation

K(E), canonical subgroup of E §2.1 H_{n}(E), canonical subgroup of E of order p^{n} X_{a b}^{±}, ordinary affinoids Ig(p^{n}), level p^{n} Igusa curve h(E), valuation of Hasse invariant of E W_{A}(p^{n}), wide open subspace of X_{0}(p^{n}) where Ē ≅ A i(A) := |Aut(A)|/2 TS_{A}, SD_{A}, too-supersingular and self-dual circles inside W_{A}(p) w_{n}, Atkin-Lehner involution on X_{0}(p^{n}) (F, A, α), Woods Hole representation of an elliptic curve π_{f}, forgetful map W (𝔽_{p^{n}}), Witt vectors of 𝔽_{p^{n}} B, quaternionic order over ℤ_{p} isomorphic to End(Â) B′, special subset of B^{∗} Φ, Gross-Hopkins period map w_{ρ}, generalized Atkin-Lehner involution of SD_{A} for ρ ∈ B′ Y_{A}, nontrivial affinoid in W_{A}(p^{2}) §2.2 π_{ν}, moduli-theoretic map taking (E, C) to (E/C[p], C/C[p]) E_{1,A}, E_{2,A}, two pullbacks of Y_{A} to X_{0}(p^{3}) π_{1 1} := π_{f} ∘ π_{ν} Z_{A}, affinoid in W_{A}(p^{3}) corresponding to “bridging component” C_{A}, τ_{f}, special circle of W_{A}(p) and map to SD_{A} w̃_{ρ}, generalized Atkin-Lehner involution of Z_{A} for ρ ∈ B′ R, maximal orders in the quadratic extensions of ℚ_{p} §3 α_{∗}, embedding of End(F) into B when (F, α) has fake CM §3.2 X_{0}(p^{n}), stable model of X_{0}(p^{n}) §4.1 P, pairing on K_{a}(E) onto μ_{p^{b}} which distinguishes X_{a b}^{±} M_{n}, field found by Krir over which J_{0}(p^{n}) has stable reduction §5 F_{n}, field over which our stable model for X_{0}(p^{n}) is defined (n ≤ 3) W_{a b}^{±}, wide open neighborhood of X_{a b}^{±} §5.2 V_{i}(A), U(A), wide open neighborhoods of E_{i,A} and Z(A) S(A), singular residue classes of Z_{A} Û(A), basic wide open refinement of U(A) I = I_{ℚ_{p}}, w = w_{X}, inertia group and inertia action on X = X_{0}(p^{n}) §6 m_{n}, minimal extension of ℚ_{p}^{nr} over which X_{0}(p^{n}) has stable reduction C_{x}, component of X_{0}(p^{3}) corresponding to a w_{ρ} fixed point §6.3 τ_{x}, F_{x}, hyper-elliptic involution on C_{x}, and its p fixed points D_{x}, D̃_{x}, residue classes of SD_{A} and Z_{A} S̃_{s,ζ}, order p automorphism of D̃_{x} References

[B] K. Buzzard, Analytic continuation of overconvergent eigenforms, J. Amer. Math. Soc. 16 (2003), no. 1, 29–55.

[BLR] S. Bosch, W. Lütkebohmert, M. Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 21. Springer-Verlag, Berlin, 1990.

[C1] R. Coleman, On the Components of X_{0}(p^{n}), J. Number Theory 110 (2005), no. 1, 3–21.

[C2] R. Coleman, Stable Maps of Curves, Kazuya Kato’s fiftieth birthday, Doc. Math. (extra volume), 217–225.

[CMc] R. Coleman, K. McMurdy, Stable Reduction of X_{0}(p^{3}), preprint available at http://math.berkeley.edu/∼coleman/X 0(pˆ 3)/Stable Xp3.pdf

[dS1] E. de Shalit, Kronecker’s polynomial, supersingular elliptic curves, and p-adic periods of modular curves, p-adic monodromy and the Birch and Swinnerton-Dyer conjecture, (Boston, 1991), 135–148, Contemp. Math. 165 (1994)

[dS2] E. de Shalit, Relative Lubin-Tate Groups, Proceedings of the AMS 95 (1985), no. 1,1–4.

[DR] P. Deligne, M. Rapoport, Schemas de modules de courbes elliptiques, Lecture Notes in Math. 349 (1973), 143–316.

[DM] , D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109.

[E1] B. Edixhoven, Minimal resolution and stable reduction of X_{0}(N), Ann. Inst. Fourier (Grenoble) 40 (1990), no. 1, 31–67.

[E2] , Stable models of modular curves and applicatons, Thèse de doctorat à l’université d’Utrecht, juin 1989. Available at http://www.math.leidenuniv.nl/∼edix/public html rennes/publications/ prschr.html

[G] B. Gross, On canonical and quasi-canonical liftings, Invent. Math. 84 (1986), no. 2, 321–326.

[GH] , M. Hopkins, Equivariant vector bundles on the Lubin-Tate moduli space, Topology and representation theory (Evanston, IL, 1992), 23–88, Contemp. Math., 158 (1994).

[Ig] J-i Igusa, On the algebraic theory of elliptic modular functions, J. Math. Soc. Japan 20 (1968), 96–106.

[Iw] K. Iwasawa, Local class field theory, Oxford Science Publications. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1986. viii+155 pp

[K] M. Krir, Degré d’une extension de Q_{p}^{nr} sur laquelle J_{0}(N) est semi-stable, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 2, 279–291.

[M1] K. McMurdy, Explicit parameterizations of ordinary and supersingular regions of X_{0}(p^{n}), Modular curves and abelian varieties (Barcelona, 2002), 165–179, Prog. Math. 224 (2004).

[M2] , Stable Reduction of X_{0}(81), preprint available.

[R] K. Ribet, On modular representations of Gal(\overline{Q}/Q) arising from modular forms, Invent. Math. 100 (1990), no. 2, 431–476.

[S] G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton Univ. Press, Princteon, N.J., 1971.

[T] J. Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966), 134–144.

[WH] J. Lubin, J.-P. Serre, J. Tate, Elliptic curves and formal groups, Lecture notes from the Summer Institute on Algebraic Geometry (Woods Hole, MA, 1964), available at http://www.ma.utexas.edu/users/voloch/lst.html

Ken McMurdy Department of Mathematics Rose-Hulman Institute of Technology USA mcmurdy@rose-hulman.edu Robert Coleman Department of Mathematics U. C. Berkeley USA coleman@math.berkeley.edu