Jumps and Monodromy of Abelian Varieties

Lars Halvard Halle and Johannes Nicaise^{1}

Received: October 10, 2010 Revised: August 19, 2011

Communicated by Peter Schneider

Abstract. We prove a strong form of the motivic monodromy conjecture for abelian varieties, by showing that the order of the unique pole of the motivic zeta function is equal to the maximal rank of a Jordan block of the corresponding monodromy eigenvalue. Moreover, we give a Hodge-theoretic interpretation of the fundamental invariants appearing in the proof.

2010 Mathematics Subject Classification: MSC2000: 11G10, 14D05, 14D07

1. Introduction

Let K be a henselian discretely valued field with algebraically closed residue field k, and let A be a tamely ramified abelian K-variety of dimension g. In [14], we introduced the motivic zeta function Z_{A}(T) of A. It is a formal power series over the localized Grothendieck ring of k-varieties M_{k}, and it measures the behaviour of the Néron model of A under tame base change. We showed that Z_{A}(𝕃^{−s}) has a unique pole, which coincides with Chai’s base change conductor c(A) of A, and that the order of this pole equals 1 + t_{pot}(A), where t_{pot}(A) denotes the potential toric rank of A. Moreover, we proved that for every embedding of ℚ_{ℓ} in ℂ, the value exp(2πc(A)i) is an eigenvalue of the tame monodromy transformation on the ℓ-adic cohomology of A in degree g. The main ingredient of the proof is Edixhoven’s filtration on the special fiber of the Néron model of A [12]. As we’ve explained in [14], this result is a global version of Denef and Loeser’s motivic monodromy conjecture for hypersurface singularities in characteristic zero. Denef and Loeser’s conjecture relates the poles of the motivic zeta function of the singularity to monodromy eigenvalues on the nearby cohomology. It is a motivic generalization of a conjecture of Igusa’s for the p-adic zeta function, which relates certain arithmetic properties of polynomials f in ℤ[x_{1}, …, x_{n}] (namely, the asymptotic behaviour of the number of solutions of the congruence f ≡ 0 modulo powers of a prime p) to the structure of the singularities of the complex hypersurface defined by the equation f = 0. The conjectures of Igusa and Denef-Loeser have been solved, for instance, in the case n = 2 [16][23], but the general case remains wide open. We refer to [20] for a survey. There also exists a stronger form of Igusa’s conjecture, which says that the real parts of the poles of the p-adic zeta function of f are roots of the Bernstein polynomial b_{f}(s) of f, and that the order of each pole is at most the multiplicity of the corresponding root of b_{f}(s). This stronger conjecture also has a motivic generalization, replacing the p-adic zeta function by the motivic zeta function, and taking for f any polynomial with coefficients in a field of characteristic zero (or, more generally, any regular function on a smooth algebraic variety over a field of characteristic zero). It is well-known that, for every complex polynomial f and every root α of b_{f}(s), the value α′ = exp(2πiα) is a monodromy eigenvalue on the nearby cohomology Rψ_{f}(ℂ)_{x} of f at some closed point x of the zero locus H_{f} of f [15][17]. Moreover, if H_{f} has an isolated singularity at x, then the multiplicity m_{α} of α as a root of the local Bernstein polynomial b_{f,x}(s) of f at x is closely related to the maximal size m_{α′} of the Jordan blocks with eigenvalue α′ of the monodromy transformation on R^{n−1}ψ_{f}(ℂ)_{x}. In particular, m_{α} ≤ m_{α′} if α ∉ ℤ [17, 7.1]. The aim of the present paper is twofold. First, we prove a strong form of the motivic monodromy conjecture for abelian varieties. There is no good notion of Bernstein polynomial in our setting, but we can look at the size of the Jordan blocks. We show that the order 1 + t_{pot}(A) of the unique pole s = c(A) of the motivic zeta function Z_{A}(𝕃^{−s}) is equal to the maximal rank of a Jordan block of the corresponding monodromy eigenvalue on the degree g cohomology of A (Theorem 5.9). Next, we use the theory of Néron models of variations of Hodge structures to give a Hodge-theoretic interpretation of the jumps in Edixhoven’s filtration. This is done in Theorems 6.2 and 6.3. In [14, 2.7], we speculated on a generalization of the monodromy conjecture to Calabi-Yau varieties over ℂ((t)) (i.e., smooth, proper, geometrically connected ℂ((t))-varieties with trivial canonical sheaf); we hope that the translation of Edixhoven’s invariants to Hodge theory will help to extend the proof of the monodromy conjecture to that setting. In order to obtain these results, we divide Edixhoven’s jumps into three types: toric, abelian, and dual abelian. The basic properties of these types are discussed in Section 3. Not all of these results are used in the proofs of the main results of the paper. We include them for the sake of completeness and because we believe that they are of independent interest. The reader who is only interested in Theorems 4.4, 5.9, 6.2 and 6.3 may skip Lemma 3.2, Proposition 3.4 and all the results in Section 3 after Proposition 3.5.

1The first author was partially supported by the Fund for Scientific Research-Flanders (G.0318.06) and by DFG under grant Hu 337/6-1. The second author was partially supported by the Fund for Scientific Research - Flanders (G.0415.10) and ANR-06-BLAN-0183.

The different types of jumps are related to the monodromy transformation on the Tate module of A in Section 4. Toric jumps correspond to monodromy eigenvalues with Jordan block of size two, and the abelian and dual abelian jumps to monodromy eigenvalues with Jordan block of size one (Theorem 4.4). If K = ℂ((t)), then the abelian and dual abelian jumps can be distinguished by looking at the Hodge type in the limit mixed Hodge structure (Theorem 6.3).

Acknowledgements. We are grateful to the referee for carefully reading the manuscript and suggesting several improvements to the paper. The second author is indebted to Antoine Chambert-Loir for helpful suggestions concerning Section 5.1.

2. Preliminaries and notation

We denote by R a Henselian discrete valuation ring, with quotient field K and algebraically closed residue field k. We denote by K^{a} an algebraic closure of K, by K^{s} the separable closure of K in K^{a}, and by K^{t} the tame closure of K in K^{s}. We fix a topological generator σ of the tame monodromy group G(K^{t}/K). We denote by p the characteristic exponent of k, and by ℕ′ the set of integers d > 0 prime to p. We denote by

(·)_{s} : (R − Schemes) → (k − Schemes) : X ↦ X_{s} = X ×_{R} k

the special fiber functor. For every abelian variety B over a field F, we denote its dual abelian variety by B^{∨}. For every abelian K-variety A with Néron model A, we denote by t(A), u(A) and a(A) the reductive, resp. unipotent, resp. abelian rank of A_{s}^{o}. We call these values the toric, resp. unipotent, resp. abelian rank of A. Obviously, their sum equals the dimension of A. By Grothendieck’s semi-stable reduction theorem, there exists a finite extension K′ of K in K^{s} such that A ×_{K} K′ has semi-abelian reduction [2, IX.3.6]. This means that the identity component of the special fiber of its Néron model is a semi-abelian k-variety; equivalently, u(A ×_{K} K′) = 0. The value t_{pot}(A) = t(A ×_{K} K′) is called the potential toric rank of A, and the value a_{pot}(A) = a(A ×_{K} K′) the potential abelian rank. It follows from [2, IX.3.9] that these values are independent of the choice of K′. We say that A is tamely ramified if we can take for K′ a tame finite extension of K (since k is algebraically closed, this means that the degree [K′ : K] is prime to p) . For every scheme S, every S-group scheme G and every integer n > 0, we denote by n_{G} : G → G the multiplication by n, and by _{n}G its kernel. If S is a set, and g : S → ℝ a function with finite support, we set

∥g∥ = ∑_{s∈S} g(s).

We denote the support of g by Supp(g).

Definition 2.1. For every function

f : ℚ/ℤ → ℝ

we define its reflection

f^{∗} : ℚ/ℤ → ℝ

by

f^{∗}(x) = f(−x).

We call f complete if for every x ∈ ℚ/ℤ, the value f(x) only depends on the order of x in the group ℚ/ℤ. We say that f is semi-complete if f + f^{∗} is complete.

Consider a function

f : ℚ/ℤ → ℕ

and assume that there exists an element e of ℤ_{>0} such that Supp(f) is contained in ((1/e)ℤ)/ℤ. Let F be any algebraically closed field such that e is prime to the characteristic exponent p′ of F. For each generator ζ of μ_{e}(F), we put

P_{f,ζ}(t) = ∏_{i∈((1/e)ℤ)/ℤ}(t − ζ^{i·e})^{f(i)}

in F [t]. For each integer d > 0, we denote by Φ_{d}(t) the cyclotomic polynomial whose roots are the primitive d-th roots of unity. We say that Φ_{d}(t) is F -tame if d is prime to p′.

Lemma 2.2. The function f is complete if and only if for some generator ζ of μ_{e}(F), the polynomial P_{f,ζ}(t) is the image of a product Q_{f}(t) of F -tame cyclotomic polynomials under the unique ring morphism

ρ : ℤ[t] → F [t]

mapping t to t. If f is complete, then P_{f,ζ}(t) is independent of ζ and e, and Q_{f}(t) is unique. In that case, if we choose a primitive e-th root of unity ξ in an algebraic closure ℚ^{a} of ℚ, then

Q_{f}(t) = ∏_{i∈((1/e)ℤ)/ℤ}(t − ξ^{i·e})^{f(i)}.

Proof. First, assume that f is complete, and put

Q_{f}(t) = ∏_{i∈((1/e)ℤ)/ℤ}(t − ξ^{i·e})^{f(i)}

for some primitive e-th root of unity ξ in ℚ^{a}. Then Q_{f}(t) is a product of F -tame cyclotomic polynomials, because f is complete and e is prime to p′. There is a unique ring morphism

ρ̃: ℤ[ξ][t] → F [t]

that maps ξ to ζ and t to t. We clearly have ρ̃(Q_{f}(t)) = P_{f,ζ}(T). Since Q_{f}(t) belongs to ℤ[t], it follows that ρ(Q_{f}(t)) = P_{f,ζ}(T) so that P_{f,ζ}(t) does not depend on ζ. Uniqueness of Q_{f}(t) follows from [14, 5.10].

Conversely, if P_{f,ζ}(t) is the image under ρ of a product Q(t) of F -tame cyclotomic polynomials, then it is easily seen that f is complete. □

3. Toric and abelian multiplicity

3.1. Galois action on Néron models. Let A be a tamely ramified abelian K-variety of dimension g, and let K′ be a finite extension of K in K^{t} such that A′ = A ×_{K} K′ has semi-abelian reduction. We denote by R′ the integral closure of R in K′, and by 𝔪′ the maximal ideal of R′. We put d = [K′ : K]. We denote by μ the Galois group G(K′/K), and we let μ act on K′ from the left. The action of ζ ∈ μ on 𝔪′/(𝔪′)^{2} is multiplication by ι(ζ), for some element ι(ζ) in the group μ_{d}(k) of d-th roots of unity in k, and the map

μ → μ_{d}(k) : ζ ↦ ι(ζ)

is an isomorphism. We denote by A and A′ the Néron models of A, resp. A′. By the universal property of the Néron model, there exists a unique morphism of R′-group schemes

h : A ×_{R} R′ → A′

that extends the canonical isomorphism between the generic fibers. It induces an injective morphism of free rank g R′-modules

Lie(h) : Lie(A ×_{R} R′) → Lie(A′).

Definition 3.1 (Chai [5]). The base change conductor c(A) of A is [K′ : K]^{−1} times the length of the cokernel of Lie(h).

The definition does not require that A is tamely ramified. The base change conductor is a positive rational number, independent of the choice of K′. It vanishes if and only if A has semi-abelian reduction. The right μ-action on A′ extends uniquely to a right μ-action on A′ such that the structural morphism

A′ → Spec R′

is μ-equivariant. We denote by

(3.1)  0 → T → (A′_{s})^{o} → B → 0

the Chevalley decomposition of (A′_{s})^{o}, with T a k-torus and B an abelian kvariety. There exist unique right μ-actions on T, resp. B, such that (3.1) is μ-equivariant. The right μ-action on B induces a left μ-action on the dual abelian variety B^{∨}.

Lemma 3.2. (1) The complex

(3.2)  0 → T^{μ} → ((A′_{s})^{o})^{μ} → B^{μ},

obtained from (3.1) by taking μ-invariants, is an exact complex of smooth group schemes over k. Taking identity components, we get a complex

(3.3)  0 → (T^{μ})^{o} → ((A′_{s})^{μ})^{o} → (B^{μ})^{o} → 0

of smooth group schemes over k that is exact at the left and at the right. The quotient

B′ = ((A′_{s})^{μ})^{o}/(T^{μ})^{o}

is an abelian k-variety, and the natural morphism f : B′ → (B^{μ})^{o} is a separable isogeny.

(2) If we denote by h the unique morphism

h : A ×_{R} R′ → A′

extending the natural isomorphism between the generic fibers, then the kmorphism h_{s} : A_{s} → A′_{s} factors through a morphism

g : A_{s} → (A′_{s})^{μ}.

The morphism g is smooth and surjective, and its kernel is a connected unipotent smooth algebraic k-group. The identity component ((A′_{s})^{μ})^{o} is semiabelian.

Proof. (1) It follows from [12, 3.4] that the group schemes in (3.2) are smooth over k. Exactness of (3.2) is clear. The morphism

α : (A′_{s})^{o} → B

is smooth, since T is smooth over k [1, VI_{B}.9.2]. It follows from [12, 3.5] that

α^{μ} : ((A′_{s})^{o})^{μ} → B^{μ}

is smooth, as well. Taking identity components in (3.2), we get a complex

(T^{μ})^{o} ⟶^{β} ((A′_{s})^{μ})^{o} ⟶^{γ=(α^{μ})^{o}} (B^{μ})^{o}

of smooth group schemes over k. It is obvious that β is a monomorphism, and thus a closed immersion [1, VI_{B}.1.4.2]. Surjectivity of γ follows from [1, VI_{B}.3.11], since γ is smooth, and thus flat. We put

B′ = ((A′_{s})^{μ})^{o}/(T^{μ})^{o}

This is a connected smooth algebraic k-group, by [1, VI_{A}.3.2 and VI_{B}.9.2]. Consider the natural morphism f : B′ → (B^{μ})^{o}. It is surjective, because γ is surjective. The dimension of B′ is equal to

dim (A′_{s})^{μ} − dim T^{μ},

which is at most dim B^{μ} by exactness of (3.2). Surjectivity of f then implies that B′ and B^{μ} must have the same dimension, so that f has finite kernel. Thus f is an isogeny and B′ is an abelian variety. The kernel of f is canonically isomorphic to

ker(γ)/(T^{μ})^{o}.

Since γ is smooth, we know that ker(γ) is smooth over k, so that ker(f) is smooth over k, by [1, VI_{B}.9.2]. Hence, f is a separable isogeny.

(2) Since h is μ-equivariant, and μ acts trivially on the special fiber A_{s} of A ×_{R} R′, the morphism h_{s} factors through a morphism g : A_{s} → (A′_{s})^{μ}. By [12, 5.3], the morphism g is smooth and surjective, and its kernel is a connected unipotent smooth algebraic k-group. By [12, 3.4], ((A′_{s})^{μ})^{o} is a connected smooth closed k-subgroup scheme of the semi-abelian k-group scheme (A′_{s})^{o}, so that ((A′_{s})^{μ})^{o} is semi-abelian by [14, 5.2]. □

3.2. Multiplicity functions. Fix an element e ∈ ℕ′. For every finite dimensional k-vector space V with a right μ_{e}(k)-action

∗ : V × μ_{e}(k) → V : (v, ζ) ↦ v ∗ ζ

and for every integer i in {0, …, e−1}, we denote by V [i] the maximal subspace of V such that

v ∗ ζ = ζ^{i} · v

for all ζ ∈ μ_{e}(k) and all v ∈ V [i]. We define the multiplicity function

m_{V,μ_{e}(k)} : ℚ/ℤ → ℕ

by

\{ \begin{matrix}m_{V,μ_{e}(k)}(i/e) & = & dim(V [i]) & for i ∈ \{0, …, e − 1\} \\ m_{V,μ_{e}(k)}(x) & = & 0 & if x ∉ ((1/e)ℤ)/ℤ \end{matrix}

Note that m_{V,μ_{e}(k)} determines the k[μ_{e}(k)]-module V up to isomorphism, since the order of μ_{e}(k) is invertible in k. In an analogous way, we define the multiplicity function m_{μ_{e}(k),W} for a finite dimensional k-vector space W with left μ_{e}(k)-action. The inverse of the right μ_{e}(k)-action on V is the left action

μ_{e}(k) × V → V : (ζ, v) ↦ v ∗ ζ^{−1}.

Its multiplicity function m_{μ_{e}(k),V} is equal to the reflection (m_{V,μ_{e}(k)})^{∗} of the multiplicity function m_{V,μ_{e}(k)}. Let A be a tamely ramified abelian K-variety. We adopt the notations of Section 3.1. In the set-up of (3.1), the group μ ≅ μ_{d}(k) acts on the k-vector spaces Lie(T), Lie(A′_{s}) and Lie(B) from the right, and on Lie(B^{∨}) from the left (via the dual action of μ on B^{∨}). Hence, we can state the following definitions.

Definition 3.3. We define the toric multiplicity function m_{A}^{tor} of A by

m_{A}^{tor} = m_{Lie(T),μ}.

We define the abelian multiplicity function m_{A}^{ab} of A by

m_{A}^{ab} = m_{Lie(B),μ}.

We define the dual abelian multiplicity function m̆_{A}^{ab} of A by

m̆_{A}^{ab} = m_{μ,Lie(B^{∨})}.

Finally, we define the multiplicity function m_{A} of A by

m_{A} = m_{A}^{tor} + m_{A}^{ab} = m_{Lie(A′_{s}),μ}.

Using [2, IX.3.9], it is easily checked that these definitions only depend on A, and not on the choice of K′.

Proposition 3.4. For every tamely ramified abelian K-variety A, we have

m̆_{A}^{ab} = (m_{A^{∨}}^{ab})^{∗}.

Proof. We adopt the notations of Section 3.1. We set (A^{∨})′ = A^{∨} ×_{K} K′ and we denote its Néron model by (A^{∨})′. The identity component of (A^{∨})′_{s} is a semi-abelian k-variety [2, IX.2.2.7]. We denote by C its abelian part. As explained in Section 3.1, the left Galois action of μ on K′ induces a right action of μ on C. The canonical divisorial correspondence on A ×_{K} A^{∨} induces a divisorial correspondence on B ×_{k} C that identifies C with the dual abelian variety of B [2, IX.5.4]. It suffices to show that the right μ-action on C is the inverse of the dual of the right μ-action on B. To this end, we take a closer look at the construction of the divisorial correspondence on B ×_{k} C. Here we need the language of biextensions [2, VII and VIII]. We note that the following proof does not use the assumption that A is tamely ramified and that K′ is a tame extension of K. The canonical divisorial correspondence on A ×_{K} A^{∨} can be interpreted as a Poincaré biextension 𝒫 of (A, A^{∨}) by 𝔾_{m,K} [2, VII.2.9.5], which is defined up to isomorphism. It induces a biextension 𝒫′ of (A′, (A^{∨})′) by 𝔾_{m,K′} by base change. By [2, VIII.7.1], the biextension 𝒫′ extends uniquely to a biextension of ((A′)^{o}, ((A^{∨})′)^{o}) by 𝔾_{m,R′}, which restricts to a biextension 𝒫′_{s} of ((A′)_{s}^{o}, ((A^{∨})′)_{s}^{o}) by 𝔾_{m,k}. By [2, VIII.4.8], 𝒫′_{s} induces a biextension 𝒬 of (B, C) by 𝔾_{m,k} that is characterized (up to isomorphism) by the fact that its pullback to ((A′)_{s}^{o}, ((A^{∨})′)_{s}^{o}) is isomorphic to 𝒫′_{s}. The theorem in [2, IX.5.4] asserts that 𝒬 is a Poincaré biextension. For every element ζ of μ, we denote by r_{ζ} the right multiplication by ζ on B and C. Since 𝒫′ is obtained from the biextension 𝒫 over K by base change to K′, it follows easily from the construction that the pullback of the biextension 𝒬 through the k-morphisms

r_{ζ} : B → B
r_{ζ} : C → C

is isomorphic to 𝒬. Interpreting 𝒬 as an isomorphism

i : B → C^{∨}

in the way of [2, VIII.3.2.2], this means that the diagram

B ⟶^{i} C^{∨}
r_{ζ} ↓  ↑^{(r_{ζ})^{∨}}
B ⟶^{i} C^{∨}

commutes, which is what we wanted to show. □

In the following proposition, we see how the multiplicity functions of a tamely ramified abelian K-variety A are related to Edixhoven’s jumps and Chai’s elementary divisors of A. These jumps and elementary divisors are rational numbers in [0, 1[ that measure the behaviour of the Néron model of A under tame ramification of the base field K. For the definition of Edixhoven’s jumps, we refer to [12, 5.4.5]. The terminology we use is the one from [14, 4.12]; in particular, the multiplicity of a jump is defined there. For Chai’s elementary divisors, we refer to [5, 2.4]. By definition, the base change conductor c(A) of A is equal to the sum of the elementary divisors.

Proposition 3.5. Let A be a tamely ramified abelian K-variety. The functions m_{A}, m_{A}^{tor}, m_{A}^{ab} and m̆_{A}^{ab} are supported on

((1/e)ℤ)/ℤ,

with e the degree of the minimal extension of K where A acquires semi-abelian reduction. If we identify [0, 1[ ∩ℚ with ℚ/ℤ via the bijection

[0, 1[ ∩ℚ → ℚ/ℤ : x ↦ x mod ℤ

then for every x ∈ [0, 1[ ∩ℚ, the value m_{A}(x) is equal to the multiplicity of x as a jump in Edixhoven’s filtration for A. In particular, the support of m_{A} is the set of jumps in Edixhoven’s filtration. The value m_{A}(x) is also equal to the number of Chai’s elementary divisors of A that are equal to x, and the base change conductor c(A) of A is given by

c(A) = ∑_{x∈[0,1[ ∩ℚ}(m_{A}(x) · x).

Proof. See [12, 5.3 and 5.4.5] and [14, 4.8 and 4.13 and 4.18]. □

Proposition 3.6. We have the following equalities:

∥m_{A}∥ = dim(A),  ∥m_{A}^{ab}∥ = ∥m̆_{A}^{ab}∥ = a_{pot}(A),
∥m_{A}^{tor}∥ = t_{pot}(A),  m_{A}^{ab}(0) = m̆_{A}^{ab}(0) = a(A),
m_{A}^{tor}(0) = t(A).

Moreover, we have

∑_{x∈(ℚ/ℤ)\{0}} m_{A}(x) = u(A).

Proof. We adopt the notations of Section 3.1. It follows immediately from the definitions that

∥m_{A}∥ = dim(Lie(A′_{s})) = dim(A),
∥m_{A}^{tor}∥ = dim(Lie(T)) = t_{pot}(A),
∥m̆_{A}^{ab}∥ = ∥m_{A}^{ab}∥ = dim(Lie(B)) = a_{pot}(A).

By Lemma 3.2, the abelian, resp., reductive rank of A_{s}^{o} is equal to the abelian, resp., reductive rank of the semi-abelian k-variety ((A′_{s})^{μ})^{o}. In the notation of Lemma 3.2, the Chevalley decomposition of ((A′_{s})^{μ})^{o} is given by

0 → (T^{μ})^{o} → ((A′_{s})^{μ})^{o} → B′ → 0

and there exists a separable isogeny f : B′ → (B^{μ})^{o}. By [12, 3.2], the natural morphisms

Lie(T^{μ}) → Lie(T)^{μ} = Lie(T)[0]
Lie(B^{μ}) → Lie(B)^{μ} = Lie(B)[0]

are isomorphisms. Since Lie(f) is also an isomorphism, we find

m_{A}^{tor}(0) = t(A),
m_{A}^{ab}(0) = a(A).

It follows that

∑_{x∈(ℚ/ℤ)\{0}} m_{A}(x) = ∥m_{A}∥ − m_{A}^{tor}(0) − m_{A}^{ab}(0)
= dim(A) − t(A) − a(A)
= u(A).

By Proposition 3.4, we have

m̆_{A}^{ab}(0) = m_{A^{∨}}^{ab}(0)

and we’ve just seen that this value is equal to the abelian rank a(A^{∨}) of A^{∨}. But the abelian ranks of A and A^{∨} coincide, by [2, 2.2.7], so that

m̆_{A}^{ab}(0) = a(A).
□

Lemma 3.7. If A_{1} and A_{2} are tamely ramified abelian K-varieties, then

m_{A_{1}×_{K}A_{2}}^{tor} = m_{A_{1}}^{tor} + m_{A_{2}}^{tor},
m_{A_{1}×_{K}A_{2}}^{ab} = m_{A_{1}}^{ab} + m_{A_{2}}^{ab},
m̆_{A_{1}×_{K}A_{2}}^{ab} = m̆_{A_{1}}^{ab} + m̆_{A_{2}}^{ab}.

Proof. If we denote by A_{1} and A_{2} the Néron models of A_{1}, resp. A_{2}, then it follows immediately from the universal property of the Néron model that A_{1} ×_{R} A_{2} is a Néron model for A_{1} ×_{K} A_{2}. Since the Chevalley decomposition of a connected smooth algebraic k-group commutes with finite fibered products over k and Lie(G_{1} ×_{k} G_{2}) is canonically isomorphic to Lie(G_{1}) ⊕ Lie(G_{2}) for any pair of algebraic k-groups G_{1}, G_{2}, the result follows. □

Proposition 3.8. Let A be a tamely ramified abelian K-variety. Let L be a finite tame extension of K of degree e, and put A_{L} = A ×_{K} L. Then for each x ∈ ℚ/ℤ, we have

m_{A_{L}}^{tor}(x) = ∑_{y∈ℚ/ℤ, e·y=x} m_{A}^{tor}(y)
m_{A_{L}}^{ab}(x) = ∑_{y∈ℚ/ℤ, e·y=x} m_{A}^{ab}(y)
m̆_{A_{L}}^{ab}(x) = ∑_{y∈ℚ/ℤ, e·y=x} m̆_{A}^{ab}(y)

Proof. We adopt the notations of Section 3.1. Since the multiplicity functions do not depend on the choice of the field K′ where A acquires semi-abelian reduction, we may assume that L is contained in K′. If ζ is a generator of μ = G(K′/K), then the Galois group G(K′/L) is generated by ζ^{e}. Now the result easily follows from the definition of the multiplicity functions. □

Proposition 3.9. If f : A_{1} → A_{2} is an isogeny of tamely ramified abelian K-varieties and if the degree deg(f) of f is prime to p, then

m_{A_{1}}^{ab} = m_{A_{2}}^{ab} and m̆_{A_{1}}^{ab} = m̆_{A_{2}}^{ab}.

Proof. We put n = deg(f). Since n is invertible in K, the morphism f is separable, so that ker(f) is étale over k. Thus ker(f) is a finite étale K-group scheme of rank n. Every finite group scheme over a field is killed by its rank. (See [1, VII_{A}.8.5]; in our case, the result is elementary, because ker(f) is étale, so that we can reduce to the case of a constant group by base change to an algebraic closure of K. As an aside, we recall that Deligne has shown that every commutative finite group scheme over an arbitrary base scheme is killed by its rank [28, p.4].) It follows that ker(f) is contained in _{n}(A_{1}). Hence, there exists an isogeny g : A_{2} → A_{1} such that g ∘ f = n_{A_{1}}. Let K′ be a tame finite extension of K such that A_{1} and A_{2} acquire semiabelian reduction over K′, and denote by R′ the integral closure of R in K′. We denote the Néron model of (A_{i}) ×_{K} K′ by A′_{i}, for i = 1, 2. The morphisms f ×_{K} K′ and g ×_{K} K′ extend uniquely to morphisms of R′-group schemes

\begin{matrix}f′ : A′_{1} → A′_{2} \\ g′ : A′_{2} → A′_{1}. \end{matrix}

For i = 1, 2, we denote by B_{i} the abelian part of the semi-abelian k-variety (A′_{i})_{s}^{o}. By functoriality of the Chevalley decomposition, f′_{s} induces a morphism of k-group schemes f′_{B} : B_{1} → B_{2}. Likewise, g′_{s} induces a morphism of k-group schemes g′_{B} : B_{2} → B_{1}. Since g′ ∘ f′ is multiplication by n, the same holds for g′_{B} ∘ f′_{B}. In particular, f′_{B} is an isogeny of degree prime to p. Thus f′_{B} is a μ-equivariant separable isogeny, so that Lie(f′_{B}) : Lie(B_{1}) → Lie(B_{2}) is a μ-equivariant isomorphism, and m_{A_{1}}^{ab} = m_{A_{2}}^{ab}. By [19, p. 143], the dual morphism (f′_{B})^{∨} is again an isogeny, and its kernel is the Cartier dual of the kernel of f′_{B}. In particular, f′_{B} and (f′_{B})^{∨} have the same degree, so that (f′_{B})^{∨} is separable. Since it is also equivariant for the left μ-action on B^{∨}, we find that m̆_{A_{1}}^{ab} = m̆_{A_{2}}^{ab}. □

Remark 3.10. The same proof shows that m_{A}^{tor} is invariant under isogenies of degree prime to p. We’ll see in Corollary 4.5 that, more generally, the functions m_{A}^{tor} and m_{A}^{ab} + m̆_{A}^{ab} are invariant under all isogenies.

Corollary 3.11. Let A be a tamely ramified abelian K-variety. If k has characteristic zero, or A is principally polarized, then

m_{A}^{ab} = m_{A^{∨}}^{ab}

and

m̆_{A}^{ab} = (m_{A}^{ab})^{∗}.

Proof. The first equality follows from Proposition 3.9. Together with Proposition 3.4, it implies the second equality. □

We will see in Theorem 6.3 that, when R is the ring of germs of holomorphic functions at the origin of ℂ, the equality

m̆_{A}^{ab} = (m_{A}^{ab})^{∗}

expresses that the monodromy eigenvalues on the (−1, 0)-component of a certain limit mixed Hodge structure associated to A are the complex conjugates of the monodromy eigenvalues on the (0, −1)-component. Corollary 3.11 generalizes this Hodge symmetry.

Question 3.12. Is it true that

m̆_{A}^{ab} = (m_{A}^{ab})^{∗}

for every tamely ramified abelian K-variety A?

4. Jumps and monodromy

Proposition 4.1. Let B be an abelian k-variety, and let T be an algebraic k-torus. Fix an element e ∈ ℕ′, and assume that μ_{e}(k) acts on B, resp. T from the right. We consider the dual left action of μ_{e}(k) on B^{∨}. The functions m_{1} := m_{Lie(T),μ_{e}(k)} and

m_{2} := m_{Lie(B),μ_{e}(k)} + m_{μ_{e}(k),Lie(B^{∨})}

are complete. Moreover, for every prime ℓ invertible in k and for each generator ζ of μ_{e}(k), the characteristic polynomial P_{1}(t) of ζ on the ℓ-adic Tate module

𝒱_{ℓ}T ≅ 𝒯_{ℓ}T ⊗_{ℤ_{ℓ}} ℚ_{ℓ}

is equal to Q_{m_{1}}(t) (in the notation of Lemma 2.2). Likewise, the characteristic polynomial P_{2}(t) of ζ on 𝒱_{ℓ}B is equal to Q_{m_{2}}(t).

Proof. We denote by

ρ : ℤ[t] → k[t]

the unique ring morphism that maps t to t. It is well-known that the characteristic polynomials P_{1}(t) and P_{2}(t) belong to ℤ[t]. For P_{1}(t), this follows from the canonical isomorphism

(4.1)  𝒱_{ℓ}T ≅ Hom_{ℤ}(X(T), ℚ_{ℓ}(1)),

where X(T) denotes the character module of T. For P_{2}(t), it follows from [19, § 19, Thm.4]. Since e is invertible in k, P_{1}(t) and P_{2}(t) are products of k-tame cyclotomic polynomials. Thus, by Lemma 2.2 (and using the notation introduced there), we only have to show the following claims.

Claim 1: The image of P_{1}(t) under ρ equals P_{m_{1},ζ}(t). Note that, by definition of the function m_{1}, the polynomial P_{m_{1},ζ}(t) is the characteristic polynomial of the automorphism induced by ζ on Lie(T). Thus Claim 1 is an immediate consequence of (4.1) and the canonical isomorphism

Lie(T) ≅ Hom_{ℤ}(X(T), k).

Claim 2: The image of P_{2}(t) under ρ equals P_{m_{2},ζ}(t). By definition of the function m_{2}, the polynomial P_{m_{2},ζ}(t) is the product of the characteristic polynomials of the automorphism induced by ζ on Lie(B) and the dual automorphism on Lie(B^{∨}). By [21, 5.1], the Hodge-de Rham spectral sequence of B degenerates at E_{1}. This yields a natural short exact sequence

0 → H^{0}(B, Ω_{B}^{1}) → H_{dR}^{1}(B) → H^{1}(B, O_{B}) → 0

where H_{dR}^{1}(B) is the degree one de Rham cohomology of B. We have natural isomorphisms

H^{0}(B, Ω_{B}^{1}) ≅ Lie(B)^{∨},
H^{1}(B, O_{B}) ≅ Lie(B^{∨}).

(The first isomorphism follows from [4, 4.2.2]; the second isomorphism from [19, § 13, Cor.3]). Thus it suffices to show that the image of P_{2}(t) under ρ is equal to the characteristic polynomial of ζ on H_{dR}^{1}(B). As explained in the proof of [14, 5.12], this can be deduced from the fact that étale cohomology is a Weil cohomology, as well as de Rham cohomology (if k has characteristic zero) and crystalline cohomology (if k has characteristic p > 0), so that the characteristic polynomials of ζ on the respective cohomology spaces must coincide. □

For every n ∈ ℤ_{>0} and every a ∈ ℂ, we denote by Diag_{n}(a) the rank n diagonal matrix with diagonal (a, …, a), and by Jord_{n}(a) the rank n Jordan matrix with diagonal (a, …, a) and subdiagonal (1, …, 1). For any two complex square matrices M and N, of rank m, resp. n, we denote by M ⊕ N the rank m + n square matrix

M ⊕ N = ( \begin{matrix}M & 0 \\ 0 & N \end{matrix} ) .

For every integer q > 0, we put

⊕^{q}M = \underbrace{M ⊕ ⋯ ⊕ M}_{q times} .

Definition 4.2. For i = 1, 2, let

m_{i} : ℚ/ℤ → ℕ

be a function with finite support. The Jordan matrix Jord(m_{1}, m_{2}) associated to the couple (m_{1}, m_{2}) is the complex square matrix of rank ∥m_{1}∥ + 2 · ∥m_{2}∥ given by

Jord(m_{1}, m_{2}) = ⨁_{x∈Supp(m_{1})} (Diag_{m_{1}(x)}(exp(2πix)))

⊕ ⨁_{y∈Supp(m_{2})} (⊕^{m_{2}(y)}Jord_{2}(exp(2πiy)))

where we ordered the set ℚ/ℤ using the bijection ℚ∩[0, 1[→ ℚ/ℤ and the usual ordering on [0, 1[.

Lemma 4.3. Let V be a finite dimensional vector space over an algebraically closed field F of characteristic zero, and let M be an endomorphism of V. Let d > 0 be an integer such that M^{d} is unipotent. Set

W = {v ∈ V | M^{d}(v) = v}

and assume that M^{d} acts trivially on V/W and that there exists an Mequivariant isomorphism between (V/W)^{∨} and an M-stable subspace W′ of W. Then the endomorphism M of V has the following Jordan form: for every eigenvalue α of M on W′ (counted with multiplicities), there is a Jordan block of size two with eigenvalue α, and for every eigenvalue β of M on W/W′ (counted with multiplicities), there is a Jordan block of size one with eigenvalue β.

Proof. Since M^{d} acts trivially on W and V/W, we know that (M^{d} − Id)^{2} = 0 on V, so that the minimal polynomial of M divides (t^{d} − 1)^{2} and the Jordan blocks of M have size at most two. The subspace W of V is generated by all the eigenvectors of M. Thus the dimension of V/W is equal to the number of Jordan blocks of M of size two, and the eigenvalues of these Jordan blocks are precisely the eigenvalues of M on V/W, or, equivalently, W′. It follows that the eigenvalues of M on V corresponding to a Jordan block of size one are the eigenvalues of M on W/W′. □

Theorem 4.4. We fix an embedding ℚ_{ℓ} ↪ ℂ. If A is a tamely ramified abelian K-variety, then the monodromy action of σ on H^{1}(A ×_{K} K^{t}, ℚ_{ℓ}) has Jordan form

Jord(m_{A}^{ab} + m̆_{A}^{ab}, m_{A}^{tor}).

Moreover, the functions m_{A}^{tor} and m_{A}^{ab} + m̆_{A}^{ab} are complete.

Proof. We adopt the notations of Section 3.1. We denote by 𝒯_{ℓ}A the ℓ-adic Tate module of A. We put I = G(K^{s}/K) and I′ = G(K^{s}/K′). Recall that there exists a canonical I-equivariant isomorphism

(4.2)  H^{1}(A ×_{K} K^{s}, ℚ_{ℓ}) ≅ Hom_{ℤ_{ℓ}}(𝒯_{ℓ}A, ℚ_{ℓ})

(see [18, 15.1]). Since A is tamely ramified, the wild inertia subgroup P ⊂ I acts trivially on H^{1}(A ×_{K} K^{s}, ℚ_{ℓ}) and 𝒯_{ℓ}A, so that the I-action on these modules factors through an action of I/P = G(K^{t}/K).

Since P is a p-group and p is prime to ℓ, there exists for every K-variety X and every integer i ≥ 0 a canonical G(K^{t}/K)-equivariant isomorphism

H^{i}(X ×_{K} K^{t}, ℚ_{ℓ}) ≅ H^{i}(X ×_{K} K^{s}, ℚ_{ℓ})^{P}

(see [2, I.2.7.1]). In our case, this yields a canonical G(K^{t}/K)-equivariant isomorphism

(4.3)  H^{1}(A ×_{K} K^{s}, ℚ_{ℓ}) = H^{1}(A ×_{K} K^{s}, ℚ_{ℓ})^{P} ≅ H^{1}(A ×_{K} K^{t}, ℚ_{ℓ}).

By (4.2) and (4.3), it suffices to show that the action of σ on

𝒱_{ℓ}A = 𝒯_{ℓ}A ⊗_{ℤ_{ℓ}} ℚ_{ℓ}

has Jordan form

Jord(m_{A}^{ab} + m̆_{A}^{ab}, m_{A}^{tor})

and that m_{A}^{ab} + m̆_{A}^{ab} and m_{A}^{tor} are complete. Consider the filtration

(4.4)  (𝒯_{ℓ}A)^{et} ⊂ (𝒯_{ℓ}A)^{ef} ⊂ 𝒯_{ℓ}A

from [2, IX.4.1.1], with (𝒯_{ℓ}A)^{ef} the essentially fixed part of the Tate module 𝒯_{ℓ}A, and (𝒯_{ℓ}A)^{et} the essentially toric part. By definition,

(𝒯_{ℓ}A)^{ef} = (𝒯_{ℓ}A)^{I′}

and (𝒯_{ℓ}A)^{et} is stable under the action of I on 𝒯_{ℓ}A. We denote by

(4.5)  (𝒱_{ℓ}A)^{et} ⊂ (𝒱_{ℓ}A)^{ef} = (𝒱_{ℓ}A)^{I′} ⊂ 𝒱_{ℓ}A

the filtration obtained from (4.4) by tensoring with ℚ_{ℓ}. By [2, IX.4.1.2] there exists an I-equivariant isomorphism

(4.6)  𝒱_{ℓ}A/(𝒱_{ℓ}A)^{ef} ≅ ((𝒱_{ℓ}A)^{et})^{∨}.

In particular, I′ acts trivially on 𝒱_{ℓ}A/(𝒱_{ℓ}A)^{ef}. It follows that the I′-action on 𝒱_{ℓ}A is unipotent of level ≤ 2 (this means that for every element γ of I′, the endomorphism (γ − Id)^{2} of 𝒱_{ℓ}A is zero), and that the I-action on (𝒱_{ℓ}A)^{et} and 𝒱_{ℓ}A/(𝒱_{ℓ}A)^{ef} factors through an action of I/I′ ≅ μ = μ_{d}(k), where d = [K′ : K]. We denote by \overline{σ} the image of σ under the projection G(K^{t}/K) → μ. The element σ^{d} is a topological generator of I′/P. Combining (4.5) and (4.6) and applying Lemma 4.3 to the σ-action on 𝒱_{ℓ}A, we see that it suffices to prove the following claims:

(1) the \overline{σ}-action on (𝒱_{ℓ}A)^{et} has Jordan form Jord(m_{A}^{tor}, 0), and m_{A}^{tor} is complete,

(2) the \overline{σ}-action on (𝒱_{ℓ}A)^{ef}/(𝒱_{ℓ}A)^{et} has Jordan form Jord(m_{A}^{ab} + m̆_{A}^{ab}, 0),

and m_{A}^{ab} + m̆_{A}^{ab} is complete. Since \overline{σ}^{d} is the identity, the Jordan forms of the \overline{σ}-actions in (1) and (2) are diagonal matrices. By [2, IX.4.2.7 and IX.4.2.9] there exist μ-equivariant isomorphisms

(𝒱_{ℓ}A)^{et} ≅ 𝒱_{ℓ}T
(𝒱_{ℓ}A)^{ef}/(𝒱_{ℓ}A)^{et} ≅ 𝒱_{ℓ}B

so that claims (1) and (2) follow from Proposition 4.1. □

Corollary 4.5. The functions m_{A}^{ab} + m̆_{A}^{ab} and m_{A}^{tor} are invariant under isogeny. In particular, m_{A}^{tor} = m_{A^{∨}}^{tor}, and

m_{A}^{ab} + m̆_{A}^{ab} = m_{A^{∨}}^{ab} + m̆_{A^{∨}}^{ab}.

The multiplicity function m_{A}, the jumps of A (with their multiplicities), the elementary divisors of A and the base change conductor c(A) are invariant under isogenies of degree prime to p.

Proof. This follows from Propositions 3.5 and 3.9, and Theorem 4.4. □

Remark 4.6. Beware that the base change conductor, and thus the function m_{A}^{ab}, of a tamely ramified abelian K-variety A can change under isogenies of degree p, if p > 0; see [5, 6.10.2] for an example.

Corollary 4.7. Using the notations of Proposition 3.5, the base change conductor of a tamely ramified abelian K-variety A is given by

c(A) = \frac{1}{2}(t_{pot}(A) − t(A)) + ∑_{x∈]0,1[∩ℚ} m_{A}^{ab}(x)x.

In particular, if A has potential purely multiplicative reduction (which means that a_{pot}(A) = 0), then

c(A) = \frac{u(A)}{2}.

Proof. By Proposition 3.5, we know that

c(A) = ∑_{x∈]0,1[∩ℚ} m_{A}^{tor}(x)x + ∑_{x∈]0,1[∩ℚ} m_{A}^{ab}(x)x.

Since m_{A}^{tor} is complete, we have that

∑_{x∈]0,1[∩ℚ} m_{A}^{tor}(x)x = \frac{1}{2} ( ∑_{x∈]0,1[∩ℚ} m_{A}^{tor}(x)x + ∑_{x∈]0,1[∩ℚ} m_{A}^{tor}(x)(1 − x))

= \frac{1}{2} ( ∑_{x∈]0,1[∩ℚ} m_{A}^{tor}(x))

= \frac{1}{2}( ∥m_{A}^{tor}∥ − m_{A}^{tor}(0))

= \frac{1}{2}(t_{pot}(A) − t(A))

where the last equality follows from Proposition 3.6. If a_{pot}(A) = 0, then it follows from Proposition 3.6 that m_{A}^{ab} = 0 and a(A) = 0, so that

c(A) = \frac{1}{2}(t_{pot}(A) − t(A)) = \frac{1}{2}(dim(A) − t(A)) = \frac{u(A)}{2}.

□

Remark 4.8. If A has potential purely multiplicative reduction, then Corollary 4.7 can be viewed as a special case of Chai’s result that for every abelian Kvariety A (not necessarily tamely ramified) with potential purely multiplicative reduction, the base change conductor c(A) equals one fourth of the Artin conductor of 𝒱_{ℓ}A [5, 5.2]. If A is tamely ramified, then this Artin conductor is simply the dimension of 𝒱_{ℓ}A/((𝒱_{ℓ}A)^{ss})^{I}, where I = G(K^{s}/K) and (𝒱_{ℓ}A)^{ss} is the semi-simplification of the ℓ-adic I-representation 𝒱_{ℓ}A. This value is precisely the number of eigenvalues of σ (counted with multiplicities) that are different from one. Combining Proposition 3.6 with Theorem 4.4, we find that the Artin conductor of 𝒱_{ℓ}A equals

2 dim(A) − m_{A}^{ab}(0) − m̆_{A}^{ab}(0) − 2m_{A}^{tor}(0) = 2u(A).

Corollary 4.9. Let A be a tamely ramified abelian K-variety, and let e be the degree of the minimal extension of K where A acquires semi-abelian reduction. Fix a primitive e-th root of unity ξ in an algebraic closure ℚ^{a} of ℚ. The characteristic polynomial

P_{σ}(t) = det(t · Id − σ | H^{1}(A ×_{K} K^{t}, ℚ_{ℓ}))

of σ on H^{1}(A ×_{K} K^{t}, ℚ_{ℓ}) is given by

P_{σ}(t) = ∏_{i∈((1/e)ℤ)/ℤ}(t − ξ^{e·i})^{m_{A}^{ab}(i)+m̆_{A}^{ab}(i)+2m_{A}^{tor}(i)} ∈ ℤ[t].

Proof. This is an immediate consequence of Theorem 4.4. □

Corollary 4.10. Let A be a tamely ramified abelian K-variety. Assume either that k has characteristic zero, or that A is principally polarized. Then m_{A}^{ab} and m̆_{A}^{ab} are semi-complete, and the monodromy action of σ on H^{1}(A ×_{K} K^{t}, ℚ_{ℓ}) has Jordan form

Jord(m_{A}^{ab} + (m_{A}^{ab})^{∗}, m_{A}^{tor}).

In the notation of Corollary 4.9, we have

P_{σ}(t) = ∏_{i∈((1/e)ℤ)/ℤ}(t − ξ^{e·i})^{m_{A}^{ab}(i)+m_{A}^{ab}(−i)+2m_{A}^{tor}(i)} ∈ ℤ[t].

Proof. Semi-completeness of m_{A}^{ab} and m̆_{A}^{ab} follows from Corollary 3.11 and Theorem 4.4. The remainder of the statement is a combination of Corollaries 3.11 and 4.9. □

5. Potential toric rank and Jordan blocks

5.1. The weight filtration associated to a nilpotent operator. Throughout this section, we fix a field F of characteristic zero and a finite dimensional vector space V over F. For every endomorphism M on V, we consider its Jordan-Chevalley decomposition

M = M_{s} + M_{n}

with M_{s} the semi-simple part of M and M_{n} its nilpotent part. We recall the following well-known property.

Proposition 5.1. Let N be a nilpotent endomorphism of V. Let w be an integer. There exists a unique finite ascending filtration W = (W_{i}, i ∈ ℤ) on V such that

(1) NW_{i} ⊂ W_{i−2} for all i in ℤ,

(2) the morphism of vector spaces

Gr_{w+α}^{W}V → Gr_{w−α}^{W}V

induced by N^{α} is an isomorphism for every integer α > 0. The filtration W is called the weight filtration centered at w associated to the nilpotent operator N.

Proof. See, for instance, [10, 1.6.1]. □

It is clear from the definition that the weight filtration centered at another integer w′ is the shifted filtration W′_{•} = W_{•−w′+w}. We define the amplitude of the filtration W in Proposition 5.1 as the smallest integer n ≥ 0 such that W_{w+n} = V. This value does not depend on the choice of the central weight w. The amplitude is related to sizes of Jordan blocks in the following way.

Proposition 5.2. Let M be an endomorphism of V. We denote by a the amplitude of the weight filtration W associated to M_{n} (centered at any weight w ∈ ℤ). Then a + 1 is the maximum of the ranks of the Jordan blocks of M.

Proof. We may assume that w = 0 and that M = M_{n}. Then the proposition is an immediate consequence of the explicit description of the weight filtration in [10, 1.6.7]. □

Proposition 5.3. Let N be a nilpotent endomorphism of V, and denote by W the associated weight filtration centered at w ∈ ℤ.

(1) If N′ is another nilpotent operator on V such that

(N − N′)W_{i} ⊂ W_{i−3}

for all i ∈ ℤ, then the weight filtration associated to N′ centered at w coincides with W.

(2) The weight filtration W does not change if we multiply N with an automorphism S of V that commutes with N.

Proof. (1) This follows immediately from the definition of the weight filtration in Proposition 5.1.

(2) Clearly, the filtration (S(W_{i}), i ∈ ℤ) on V also satisfies properties (1) and (2) in Proposition 5.1, so that S(W_{i}) = W_{i} for all i in ℤ by uniqueness of the weight filtration. This implies at once that W coincides with the weight filtration associated to NS centered at w. □

Definition 5.4. Let W = (W_{i}, i ∈ ℤ) be an ascending filtration on V. The dual filtration W^{∨} on V^{∨} is the ascending filtration defined by

(W^{∨})_{i} = (W_{−i−1})^{⊥}

for all i in ℤ. For every integer j ≥ 0, the degree j exterior power of W is the ascending filtration ∧^{j}W on ∧^{j}V given by

(∧^{j}W)_{i} = ∑_{i_{1}+...+i_{j}=i} W_{i_{1}} ∧ ⋯ ∧ W_{i_{j}}.

If V′ is another finite dimensional vector space over F, endowed with an ascending filtration W′, then the tensor product of W and W′ is the ascending filtration W ⊗ W′ on V ⊗ V′ given by

(W ⊗ W′)_{i} = ∑_{i_{1}+i_{2}=i} W_{i_{1}} ⊗ W′_{i_{2}}.

Proposition 5.5. Let V and V′ be finite dimensional vector spaces over F, endowed with nilpotent operators N and N′, respectively. We denote by W and W′ the associated weight filtrations on V and V′, centered at integers w and w′.

(1) The weight filtration on V^{∨} centered at −w associated to the nilpotent operator −N^{∨} is the dual W^{∨} of the weight filtration W.

(2) The weight filtration on V ⊗ V′ centered at w + w′ associated to the nilpotent operator N ⊗ Id + Id ⊗ N′ is the tensor product W ⊗ W′ of the weight filtrations W and W′.

(3) For every integer j > 0, the weight filtration on ∧^{j}V centered at w · j associated to the nilpotent operator

N^{(∧j)} = N ∧ Id ∧ ⋯ ∧ Id + Id ∧ N ∧ Id ∧ ⋯ ∧ Id + … + Id ∧ ⋯ ∧ Id ∧ N

is the exterior power ∧^{j}W of the weight filtration W.

Proof. Point (1) and (2) are proven in [10, 1.6.9], using the theorem of Jacobson-Morosov. Point (3) can be proven in exactly the same way, since the morphism of linear groups

h : GL(V) → GL(∧^{j}V) : M ↦ ∧^{j}M

induces a morphism of Lie algebras

Lie(h) : End(V) → End(∧^{j}V)

that sends N to N^{(∧j)}. □

Corollary 5.6. Let M be an automorphism of V, and consider an integer w and an integer j > 0. We denote by W the weight filtration centered at w associated to the nilpotent operator M_{n} on V.

(1) Let V′ be another finite dimensional vector space, endowed with an automorphism M′. If W′ is the weight filtration associated to M′_{n} centered at w′ ∈ ℤ, then W ⊗ W′ is the weight filtration centered at w + w′ associated to the nilpotent operator (M ⊗ M′)_{n} on V ⊗ V′.

(2) The exterior power filtration ∧^{j}W is the weight filtration centered at w · j associated to the nilpotent operator (∧^{j}M)_{n} on ∧^{j}V.

Proof. (1) By Proposition 5.3(2), we may assume that M and M′ are unipotent, since multiplying these operators with M_{s}^{−1} and (M′_{s})^{−1}, respectively, has no influence on the weight filtrations that we want to compare. Then M = Id+M_{n} and M′ = Id + M′_{n}, so that

(M ⊗ M′)_{n} − Id ⊗ M′_{n} − M_{n} ⊗ Id = M_{n} ⊗ M′_{n}.

It follows that

((M ⊗ M′)_{n} − Id ⊗ M′_{n} − M_{n} ⊗ Id)(W ⊗ W′)_{i} ⊂ (W ⊗ W′)_{i−4}

for all i ∈ ℤ. The result now follows from Propositions 5.3(1) and 5.5(2).

(2) The proof is completely similar to the proof of (1): one reduces to the case where M is unipotent, and one shows by direct computation that

(∧^{j}M)_{n} − (M_{n})^{(∧j)}

shifts weights by at least −4. □

The following lemma and proposition will allow us to compute the maximal size of certain Jordan blocks of monodromy on the cohomology of a tamely ramified abelian K-variety (Theorem 5.9).

Lemma 5.7. Let F be an algebraically closed field of characteristic zero, and let V ≠ {0} be a finite dimensional vector space over F. Let M be an automorphism of V, with Jordan form

Jord_{m}(ξ)

where m ∈ ℤ_{>0} and ξ ∈ F^{×}. Then for every integer j in [1, m] and every integer w, the weight filtration centered at w associated to the nilpotent operator (∧^{j}M)_{n} on ∧^{j}V has amplitude m(m − j).

Proof. We may assume that w = 0. Denote by W the weight filtration associated to M_{n} centered at 0. By Corollary 5.6, the weight filtration associated to (∧^{j}M)_{n} centered at 0 coincides with the exterior power filtration ∧^{j}W. By the explicit description of the weight filtration in [10, 1.6.7], the dimension of Gr_{α}^{W} V is one if α is an integer in [1 − m, m − 1] such that α − m is odd, and zero in all other cases. This easily implies that ∧^{j}W has amplitude

(m − 1) + (m − 3) + … + (m − 2j + 1) = m(m − j).
□

Proposition 5.8. Let F be an algebraically closed field of characteristic zero, and let V ≠ {0} be a finite dimensional vector space over F. Let M be an automorphism of V, with Jordan form

Jord_{m_{1}}(ξ_{1}) ⊕ ⋯ ⊕ Jord_{m_{q}}(ξ_{q})

where q ∈ ℤ_{>0}, m ∈ (ℤ_{>0})^{q} and ξ_{i} ∈ F^{×} for i = 1, …, q. We fix an integer j > 0. For every element ζ of F, we denote by max_{ζ} the maximum of the ranks of the Jordan blocks of ∧^{j}M on ∧^{j}V with eigenvalue ζ. If we denote by 𝒮 the set of tuples s ∈ ℕ^{q} such that ∥s∥ = j and s_{i} ≤ m_{i} for each i ∈ {1, …, q}, then

max_{ζ} = max{1 + ∑_{i=1}^{q} s_{i}(m_{i} − s_{i}) | s ∈ 𝒮 , ∏_{i=1}^{q}(ξ_{i})^{s_{i}} = ζ}

for every ζ ∈ F, with the convention that max ∅ = 0.

Proof. We can write

V = V_{1} ⊕ ⋯ ⊕ V_{q}

such that M(V_{i}) ⊂ V_{i} for each i and such that the restriction M_{i} of M to V_{i} has Jordan form Jord_{m_{i}}(ξ_{i}). If we put

V_{s} = (∧^{s_{1}}V_{1}) ⊗ ⋯ ⊗ (∧^{s_{q}}V_{q})

for each s ∈ 𝒮, then we have a canonical isomorphism

∧^{j}V ≅ ⨁_{s∈𝒮} V_{s}

such that every summand V_{s} is stable under ∧^{j}M and the restriction of ∧^{j}M to V_{s} equals

(∧^{s_{1}}M_{1}) ⊗ ⋯ ⊗ (∧^{s_{q}}M_{q}).

The automorphism ∧^{j}M has a unique eigenvalue on V_{s}, which is equal to

∏_{i=1}^{q}(ξ_{i})^{s_{i}}.

By Proposition 5.2, Corollary 5.6(1) and Lemma 5.7, the maximal rank of a Jordan block of ∧^{j}M on V_{s} equals

1 + ∑_{i=1}^{q} s_{i}(m_{i} − s_{i}).

This yields the desired formula for max_{ζ}. □

5.2. The strong form of the monodromy conjecture.

Theorem 5.9. Let A be a tamely ramified abelian K-variety of dimension g. For every embedding of ℚ_{ℓ} in ℂ, the value α = exp(2πic(A)) is an eigenvalue of σ on H^{g}(A ×_{K} K^{t}, ℚ_{ℓ}). Each Jordan block of σ on H^{g}(A ×_{K} K^{t}, ℚ_{ℓ}) has rank at most t_{pot}(A) + 1, and σ has a Jordan block with eigenvalue α on H^{g}(A ×_{K} K^{t}, ℚ_{ℓ}) with rank t_{pot}(A) + 1.

Proof. Since A is tamely ramified, we have a canonical G(K^{t}/K)-equivariant isomorphism of ℚ_{ℓ}-vector spaces

H^{g}(A ×_{K} K^{t}, ℚ_{ℓ}) ≅ ⋀^{g} H^{1}(A ×_{K} K^{t}, ℚ_{ℓ}).

By Theorem 4.4, the monodromy operator σ has precisely ∥m_{A}^{tor}∥ Jordan blocks of size 2 on H^{1}(A ×_{K} K^{t}, ℚ_{ℓ}), and no larger Jordan blocks. It follows from Proposition 5.8 that the size of the Jordan blocks of σ on H^{g}(A ×_{K} K^{t}, ℚ_{ℓ}) is bounded by 1 + ∥m_{A}^{tor}∥. By Proposition 3.6, we know that ∥m_{A}^{tor}∥ = t_{pot}(A). By Proposition 3.5, the image in ℚ/ℤ of the base change conductor c(A) equals

∑_{x∈ℚ/ℤ}((m_{A}^{tor}(x) + m_{A}^{ab}(x)) · x)

and by Proposition 3.6, we have

∑_{x∈ℚ/ℤ}(m_{A}^{tor}(x) + m_{A}^{ab}(x)) = g.

Hence, by Theorem 4.4 and Proposition 5.8, σ has a Jordan block of size 1 + t_{pot}(A) with eigenvalue α on H^{g}(A ×_{K} K^{t}, ℚ_{ℓ}). □

6. Limit Mixed Hodge structure

Let A be a tamely ramified abelian K-variety of dimension g. Theorem 4.4 shows that the couple of functions (m_{A}^{tor}, m_{A}^{ab} + m̆_{A}^{ab}) and the Jordan form of σ on the tame degree one cohomology of A determine each other. It does not tell us how to recover m_{A}^{ab} and m̆_{A}^{ab} individually from the cohomology of A. In this section, we assume that A is obtained by base change from a family of abelian varieties over a smooth complex curve. We will give an interpretation of the multiplicity functions m_{A}^{ab}, m̆_{A}^{ab} and m_{A}^{tor} in terms of the limit mixed Hodge structure of the family.

6.1. Limit mixed Hodge structure of a family of abelian varieties. Let \overline{S} be a connected smooth complex algebraic curve, let s be a closed point on \overline{S}, and choose a local parameter t on \overline{S} at s. We put K = ℂ((t)), R = ℂ[[t]] and S = \overline{S} \ \{s\}. Let

f : X → S

be a smooth projective family of abelian varieties over S, of relative dimension g, and put

A = X ×_{S} Spec K.

We choose an extension of f to a flat projective morphism

\overline{f} : \overline{X} → \overline{S},

and we denote by \overline{X}_{s} the fiber of \overline{f} over the point s. We denote by (·)^{an} the complex analytic GAGA functor, but we will usually omit it from the notation if no confusion can occur. For instance, when we speak of the sheaf R^{i}f_{∗}(ℤ), it should be clear that we mean R^{i}f_{∗}^{an}(ℤ). For every i ∈ ℕ, we consider the degree i limit cohomology, resp. homology,

H^{i}(X_{∞}, ℤ) := ℍ^{i}(\overline{X}_{s}(ℂ), Rψ\overline{f}(ℤ))
H_{i}(X_{∞}, ℤ) := ℍ^{2g−i}(\overline{X}_{s}(ℂ), Rψ\overline{f}(ℤ))(g) = H^{2g−i}(X_{∞}, ℤ)(g)

of \overline{f} at s. Here

Rψ\overline{f}(ℤ) ∈ D_{c}^{b}(\overline{X}_{s}(ℂ), ℤ)

denotes the complex of nearby cycles associated to \overline{f}^{an}. For every i in ℕ, the ℤ-module H^{i}(X_{∞}, ℤ) is non-canonically isomorphic to H^{i}(X_{z}(ℂ), ℤ), where z is any point of S(ℂ) and X_{z} is the fiber of f over z [3, XIV.1.3.3.2]. Likewise, by Poincaré duality, H_{i}(X_{∞}, ℤ) is non-canonically isomorphic to H_{i}(X_{z}(ℂ), ℤ). The limit cohomology and homology are independent of the chosen compactification \overline{f}, as can be deduced from [11, 4.2.11] by dominating two compactifications by a third one. We set

H^{i}(X_{∞}, ℚ) := H^{i}(X_{∞}, ℤ) ⊗_{ℤ} ℚ
H^{i}(X_{∞}, ℂ) := H^{i}(X_{∞}, ℤ) ⊗_{ℤ} ℂ

and we use similar notations for the limit homology H_{i}(X_{∞}, ∗). For all i ∈ ℕ, we denote by M the monodromy operators on H_{i}(X_{∞}, ℤ) and H^{i}(X_{∞}, ℤ), and by N the logarithm of the unipotent part M_{u} of M. The ℤ-modules H^{i}(X_{∞}, ℤ) and H_{i}(X_{∞}, ℤ) carry natural mixed Hodge structures, which are the limits at s of the variations of Hodge structures

R^{i}f_{∗}(ℤ), resp. R^{2g−i}f_{∗}(ℤ)(g),

on S. The existence of these limit mixed Hodge structures was conjectured by Deligne, and they were constructed by Schmid [25] and Steenbrink [26]. The weight filtrations on H^{i}(X_{∞}, ℚ) and H_{i}(X_{∞}, ℚ) are the weight filtrations centered at i, resp. −i, associated to the nilpotent operator N. We will briefly recall Schmid’s construction of the limit Hodge filtration below. The action of the semi-simple part M_{s} of M on H_{i}(X_{∞}, ℚ) and H^{i}(X_{∞}, ℚ) is a morphism of rational mixed Hodge structures, by [27, 2.13]. For every i ∈ ℕ, there exists an isomorphism of ℚ_{ℓ}-vector spaces

(6.1)  H^{i}(X_{∞}, ℤ) ⊗_{ℤ} ℚ_{ℓ} ≅ H^{i}(A ×_{K} K^{a}, ℚ_{ℓ})

such that the monodromy action on the left hand side corresponds to the action of the canonical topological generator of G(K^{a}/K) ≅ ℤ̂(1)(ℂ) on the right hand side. This follows from Deligne’s comparison theorem for ℓ-adic versus complex analytic nearby cycles [3, XIV.2.8]. Thus, if K′ is a finite extension of K such that A ×_{K} K′ has semi-abelian reduction and if we set d = [K′ : K], then (M_{s})^{d} is the identity on H^{i}(X_{∞}, ℚ) and H_{i}(X_{∞}, ℚ) for all i ≥ 0. Identifying M_{s} with the canonical generator exp(2πi/d) of μ_{d}(ℂ), we obtain an action of μ_{d}(ℂ) on the rational mixed Hodge structures H^{i}(X_{∞}, ℚ) and H_{i}(X_{∞}, ℚ), for all i ≥ 0. For the definition of the dual and the exterior powers of a mixed Hodge structure, we refer to [22, 3.2].

Proposition 6.1. (1) For every i ∈ ℕ, there exists a natural isomorphism of mixed Hodge structures

∧_{ℤ}^{i}H^{1}(X_{∞}, ℤ) → H^{i}(X_{∞}, ℤ)

that is compatible with the action of M on the underlying ℤ-modules.

(2) For every i ∈ ℕ, there exists a natural isomorphism of mixed Hodge

structures

H^{i}(X_{∞}, ℤ)^{∨} → H_{i}(X_{∞}, ℤ)

that is compatible with the action of M on the underlying ℤ-modules.

Proof. (1) The cup product defines a morphism

(6.2)  ∧_{ℤ}^{i}R^{1}f_{∗}(ℤ) → R^{i}f_{∗}(ℤ)

of sheaves on S. This is an isomorphism on every fiber, by the proper base change theorem and [19, p. 3], and thus an isomorphism of sheaves. Moreover, it is an isomorphism of variations of Hodge structures, because the cup product defines a morphism of pure Hodge structures on the cohomology of every fiber of f [22, 5.45]. Looking at Schmid’s construction of the limit mixed Hodge structure in [25], one checks in a straightforward way that taking the limit of a variation of Hodge structures commutes with taking exterior powers. Compatibility of the Hodge filtrations is easy, since the exterior power defines a holomorphic map between the relevant classifying spaces. The compatibility of the weight filtrations follows from Corollary 5.6. Thus, taking the limit at s of the isomorphism (6.2), we obtain an isomorphism of mixed Hodge structures

∧_{ℤ}^{i}H^{1}(X_{∞}, ℤ) → H^{i}(X_{∞}, ℤ)

that is compatible with the action of M.

(2) For every i ≥ 0, Poincaré duality yields a natural isomorphism of sheaves of ℤ-modules

(6.3)  α : R^{i}f_{∗}(ℤ)^{∨} → R^{2g−i}f_{∗}(ℤ)(g)

(note that Poincaré duality holds with coefficients in ℤ because R^{i}f_{∗}(ℤ) is a locally free sheaf of ℤ-modules for all i ≥ 0). By [22, 6.19], this isomorphism respects the Hodge structure on every fiber of R^{i}f_{∗}(ℤ)^{∨} and R^{2g−i}f_{∗}(ℤ)(g). Thus (6.3) is an isomorphism of variations of Hodge structures on S. As in (1), one checks in a straightforward way that the limit of R^{i}f_{∗}(ℤ)^{∨} at s is the dual of the limit of R^{i}f_{∗}(ℤ), using Proposition 5.5 to verify the compatibility of the weight filtrations. Thus the isomorphism (6.3) induces an isomorphism of mixed Hodge structures

H^{i}(X_{∞}, ℤ)^{∨} → H_{i}(X_{∞}, ℤ).
□

By Proposition 6.1, in order to describe the limit mixed Hodge structure on H_{i}(X_{∞}, ℚ) and H^{i}(X_{∞}, ℤ) for all i ≥ 0, it suffices to determine the limit mixed Hodge structure on H_{1}(X_{∞}, ℤ).

6.2. Description of the mixed Hodge structure on H_{1}(X_{∞}, ℤ). We denote by

𝒱 → S^{an}

the polarized variation of Hodge structures

R^{2g−1}f_{∗}(ℤ)(g)

of type {(0, −1), (−1, 0)} [8, 4.4.3]. We denote by 𝒱_{ℤ}, 𝒱_{ℚ} and 𝒱_{ℂ} the integer, resp. rational, resp. complex component of 𝒱 . The sheaf 𝒱_{ℤ} is a locally free sheaf of ℤ-modules on S^{an} of rank 2g. The fiber of 𝒱 over a point z of S^{an} is canonically isomorphic to the weight −1 Hodge structure

H^{2g−1}(X_{z}(ℂ), ℤ)(g),

where X_{z} denotes the fiber of f over z. By Poincaré duality, there is a canonical isomorphism of ℤ-modules

H^{2g−1}(X_{z}(ℂ), ℤ) ≅ H_{1}(X_{z}(ℂ), ℤ).

The limit of 𝒱 at the point s is a mixed Hodge structure that was constructed by Schmid [25]. In our notation, this limit is precisely the mixed Hodge structure H_{1}(X_{∞}, ℤ). For a quick introduction to limit mixed Hodge structures, we refer to [13] and [22, §10 and §11]. Here we will only briefly sketch the main ingredients of the construction. We consider a small disc Δ around s in S^{an} and we denote by Δ^{∗} the punctured disc Δ \ {s}. It follows from the definition of the nearby cycles functor that H_{1}(X_{∞}, ℤ) is the ℤ-module of global sections of the pullback of 𝒱_{ℤ} to a universal cover of Δ^{∗} [3, XIV.1.3.3]. By the comparison isomorphism (6.1) and [2, IX.3.5], the action of the monodromy operator M^{d} on H_{1}(X_{∞}, ℚ) is unipotent of level ≤ 2 (meaning that (M^{d} − Id)^{2} = 0). The level of unipotency can also be deduced from the Monodromy Theorem [25, 6.1] and the fact that the fibers of 𝒱 are of type {(−1, 0), (0, −1)}. The weight filtration W on H_{1}(X_{∞}, ℚ) is the weight filtration with center −1 associated to the nilpotent operator N (recall that N is the logarithm of the unipotent part M_{u} of the monodromy operator M). Since M_{u}^{d} = M^{d}, we have

dN = log(M_{u}^{d}) = M^{d} − Id

so that N^{2} = 0 and the weight filtration is of the form

{0} ⊂ W_{−2} ⊂ W_{−1} ⊂ W_{0} = H_{1}(X_{∞}, ℚ).

Explicitly, we have W_{−1} = ker(N) and W_{−2} = im(N). This filtration induces a weight filtration on the lattice H_{1}(X_{∞}, ℤ) in H_{1}(X_{∞}, ℚ). Note that

W_{−1}H_{1}(X_{∞}, ℤ) := W_{−1}H_{1}(X_{∞}, ℚ) ∩ H_{1}(X_{∞}, ℤ)

is stable under the action of M_{s}, since H_{1}(X_{∞}, ℤ) is stable under M and M is semi-simple on W_{−1}H_{1}(X_{∞}, ℚ) = ker(M^{d} − Id). Consider the finite covering

Δ̃ → Δ

obtained by taking a d-th root t′ of the coordinate t on Δ. This covering is totally ramified over the origin s of Δ. With a slight abuse of notation, we denote again by s the unique point of the open disc Δ̃ that lies over s ∈ Δ. We denote by Δ̃^{∗} the punctured disc Δ̃ \ {s}. Pulling back 𝒱 to a variation of Hodge structures 𝒱 ′ on Δ̃^{∗} has the effect of raising the monodromy operator M to the power d. This has no influence on the associated weight filtration on H_{1}(X_{∞}, ℚ), since the logarithm of (M_{u})^{d} = M^{d} equals dN. Pulling back 𝒱 to Δ̃^{∗} is the first step in the construction of the limit Hodge filtration F on H_{1}(X_{∞}, ℚ). The important point is that the monodromy M^{d} of the variation 𝒱 ′ is unipotent. Schmid considers a complex manifold Ď that parameterizes descending filtrations

F^{1} = H_{1}(X_{∞}, ℂ) ⊃ F^{0} ⊃ {0}

on H_{1}(X_{∞}, ℂ) that satisfy a certain compatibility relation with the polarization on H_{1}(X_{∞}, ℂ) and such that F^{0} has dimension g. “Untwisting” the monodromy action on the fibers of 𝒱 ′, he constructs a map

Ψ̃ : Δ̃^{∗} → Ď.

The Nilpotent Orbit Theorem (for one variable) in [25, 4.9] guarantees that Ψ̃ extends to a holomorphic map

Ψ : Δ̃ → Ď.

The point Ψ(s) of Ď corresponds to a descending filtration

F^{−1} = H_{1}(X_{∞}, ℂ) ⊃ F^{0} ⊃ {0}

that is called the limit Hodge filtration on H_{1}(X_{∞}, ℂ). Schmid’s fundamental result [25, 6.16] states that the weight filtration W and the limit Hodge filtration F define a polarized mixed Hodge structure on H_{1}(X_{∞}, ℤ), which is called the limit mixed Hodge structure of the variation of Hodge structures 𝒱 ′. We see from the shape of the weight filtration and the limit Hodge filtration that the limit mixed Hodge structure on H_{1}(X_{∞}, ℤ) is of type

{(0, 0), (−1, 0), (0, −1), (−1, −1)}.

Moreover, since

Gr_{−1}^{W}H_{1}(X_{∞}, ℤ)

is polarizable, the mixed Hodge structure

(H_{1}(X_{∞}, ℤ), W, F)

is a mixed Hodge 1-motive in the sense of [9, § 10]. The construction of the limit Hodge filtration can be reformulated in terms of the Deligne extension or canonical extension. Consider the holomorphic vector bundle

(𝒱 ′)^{h} = 𝒱′_{ℤ} ⊗_{ℤ} OΔ̃^{∗}

on the punctured disc Δ̃^{∗}. The locally constant subsheaf 𝒱′_{ℂ} of this vector bundle defines a connection ∇ on (𝒱 ′)^{h}, called the Gauss-Manin connection. The vector bundle (𝒱 ′)^{h} extends in a unique way to a vector bundle 𝒱̂ ′ on Δ̃ such that ∇ extends to a logarithmic connection on 𝒱̂ ′ whose residue at s is nilpotent [7, 5.2]. We call 𝒱̂ ′ the Deligne extension of the variation of Hodge structures 𝒱 ′. The fiber of the Deligne extension over the origin s of Δ̃ can be identified with H_{1}(X_{∞}, ℂ), by [22, XI-8] (this identification depends on the choice of a local coordinate on Δ̃; we take the coordinate t′). The Hodge flags F^{0} on the fibers of 𝒱 ′ glue to a holomorphic subbundle F^{0}(𝒱 ′)^{h} of (𝒱 ′)^{h}, which extends uniquely to a holomorphic subbundle F^{0}𝒱̂ ′ of 𝒱̂ ′ [22, 11.10]. Taking fibers at s, we obtain a descending filtration

(𝒱̂ ′)_{s} = H_{1}(X_{∞}, ℂ) ⊃ (F^{0}𝒱̂ ′)_{s} ⊃ {0}

and this is precisely the limit Hodge filtration on H_{1}(X_{∞}, ℂ) [22, 11.10]. Schmid’s construction of the limit mixed Hodge structure works for abstract variations of Hodge structures, which need not necessarily come from geometry. If the variation of Hodge structures comes from the cohomology of a proper and smooth family over S (such as in the case that we are considering), the Deligne extension and the extension of the Hodge bundels can be constructed explicitly using a relative logarithmic de Rham complex associated to a suitable compactification \overline{f}. This is Steenbrink’s construction [26]. We will not need this approach in this paper.

Theorem 6.2. We apply the terminology of Section 3.1 to the abelian K-variety A and define in this way the degree d extension K′ of K, as well as the torus T and the abelian variety B over ℂ, endowed with a right action of μ ≅ μ_{d}(ℂ). There exist canonical μ-equivariant isomorphisms of pure Hodge structures

Gr_{0}^{W} (H_{1}(X_{∞}, ℚ)) ≅ H_{1}(T (ℂ), ℚ)(−1)
Gr_{−1}^{W}(H_{1}(X_{∞}, ℤ)) ≅ H_{1}(B(ℂ), ℤ)
Gr_{−2}^{W}(H_{1}(X_{∞}, ℤ)) ≅ H_{1}(T (ℂ), ℤ).

Proof. We denote by ℂ(S′) the algebraic closure of the function field ℂ(S) in K′, and we consider the normalization

\overline{S}′ → \overline{S}

of \overline{S} in ℂ(S′). This is a ramified Galois covering, obtained by taking a d-th root of the local parameter t. Its Galois group is canonically isomorphic to μ.

With abuse of notation, we denote again by s the unique point of the inverse image of s in \overline{S}′, and we put S′ = \overline{S}′ \ \{s\}. Then

f′ : X′ = X ×_{S} S′ → S′

is a smooth projective family of abelian varieties, and we have a canonical isomorphism

A′ = A ×_{K} K′ ≅ X′ ×_{S′} Spec K′.

As we’ve argued above, the fact that A′ has semi-abelian reduction implies that the variation of Hodge structures

𝒱 ′ = 𝒱 ×_{S} S′ ≅ R^{2g−1}f′_{∗}(ℤ)(g)

has unipotent monodromy around s. We denote by X′ the Néron model of X′ over \overline{S}′, and by A′ the Néron model of A′ over R′, where R′ denotes the integral closure of R in K′. Note that there is a canonical isomorphism of R′-schemes

A′ ≅ X′ ×\overline{S}′ Spec R′.

The analytic family of abelian varieties

(f′)^{an} : (X′)^{an} → (S′)^{an}

is canonically isomorphic to the Jacobian

J(𝒱 ′) → (S′)^{an}

of the variation of Hodge structures 𝒱 ′ [24, 2.10.1]. We will now explain the relation between the complex semi-abelian variety (A′)_{s}^{o} and the limit mixed Hodge structure H_{1}(X_{∞}, ℤ) of 𝒱 ′ at the point s. To simplify notation, we put H_{C} = H_{1}(X_{∞}, C) for C = ℤ, ℚ, ℂ, and we denote by H the mixed Hodge structure

(H_{ℤ}, W_{•}H_{ℚ}, F^{•}H_{ℂ}).

By [24, 4.5(i)], (X′)^{an} is canonically isomorphic to Clemens’ Néron model of 𝒱 ′ over \overline{S}′; see [6] and [24, 2.5] for a definition. In [24, 2.5], Clemens’ Néron model is constructed by gluing copies of the Zucker extension J_{\overline{S}′}^{Z}(𝒱 ′) of 𝒱 ′, which is defined in [29] and [24, 2.1]. It follows immediately from this construction that the identity component

((X′)^{o})^{an}

of Clemens’ Néron model is canonically isomorphic to the Zucker extension J_{\overline{S}′}^{Z}(𝒱 ′). The Zucker extension is given explicitly by

J_{\overline{S}′}^{Z}(𝒱 ′) = j_{∗}𝒱′_{ℤ}\𝒱̂ ′/F^{0}𝒱̂ ′

where 𝒱̂ ′ is the Deligne extension of 𝒱 ′ to \overline{S}′, j is the open immersion of S′ into \overline{S}′, and F^{0}𝒱̂ ′ is the unique extension of the holomorphic vector bundle

F^{0}(𝒱′_{ℤ} ⊗_{ℤ} O_{(S′)^{an}})

to a holomorphic subbundle of 𝒱̂ ′. We can describe the fiber

J_{\overline{S}′}^{Z}(𝒱 ′)_{s} ≅ ((X′)_{s}^{o})^{an} ≅ ((A′)_{s}^{o})^{an}

of J_{\overline{S}′}^{Z}(𝒱 ′) at s in terms of the mixed Hodge structure H, as follows.

As we’ve explained above, the fiber of 𝒱̂ ′ over s is isomorphic to H_{ℂ}, and F^{0}(𝒱̂ ′)_{s} coincides with the degree zero part of the Hodge filtration on H_{ℂ}. Moreover, the fiber of j_{∗}𝒱′_{ℤ} at s is the ℤ-module of elements in H_{ℤ} that are invariant under the monodromy action of M^{d}. By definition, the weight filtration on H_{ℚ} is the filtration centered at −1 defined by the logarithm N of the unipotent part M_{u} of M, or, equivalently, the logarithm N′ = dN of M^{d}. Since (M^{d} − Id)^{2} = 0, we have N′ = M^{d} − Id and (N′)^{2} = 0, and we see that

(j_{∗}𝒱′_{ℤ})_{s} = ker(N′) = W_{−1}H_{ℤ}.

Thus, we find canonical isomorphisms

((A′)_{s}^{o})^{an} ≅ J_{\overline{S}′}^{Z}(𝒱 ′)_{s}
≅ W_{−1}H_{ℤ}\H_{ℂ}/F^{0}H_{ℂ}
≅ W_{−1}H_{ℤ}\W_{−1}H_{ℂ}/(F^{0}H_{ℂ} ∩ W_{−1}H_{ℂ}).

The last isomorphism is obtained as follows: since Gr_{0}^{W} H is purely of type (0, 0), we have

F^{0}Gr_{0}^{W} H_{ℂ} = Gr_{0}^{W} H_{ℂ} = H_{ℂ}/W_{−1}H_{ℂ},

so that the morphism

W_{−1}H_{ℂ} → H_{ℂ}/F^{0}H_{ℂ}

induced by the inclusion of W_{−1}H_{ℂ} in H_{ℂ} is surjective. Its kernel is precisely F^{0}H_{ℂ} ∩ W_{−1}H_{ℂ}. By [9, 10.1], we have an extension

(6.4)  0 → J(Gr_{−2}^{W}H) → ((A′)_{s}^{o})^{an} → J(Gr_{−1}^{W}H) → 0

where

J(Gr_{−2}^{W}H) = Gr_{−2}^{W}H_{ℂ}/Gr_{−2}^{W}H_{ℤ}

is a torus, and

J(Gr_{−1}^{W}H) = H_{ℤ}\Gr_{−1}^{W}H_{ℂ}/F^{0}Gr_{−1}^{W}H_{ℂ}

an abelian variety, because the Hodge structure Gr_{−1}^{W}H is polarizable. By [9, 10.1.3.3], the extension (6.4) must be the analytification of the Chevalley decomposition

0 → T → (A′)_{s}^{o} → B → 0.

Hence, there exist canonical isomorphisms of pure Hodge structures

(6.5)  Gr_{−1}^{W}(H) ≅ H_{1}(B(ℂ), ℤ),
(6.6)  Gr_{−2}^{W}(H) ≅ H_{1}(T (ℂ), ℤ).

Moreover, by definition of the weight filtration on H_{ℚ}, the operator N defines a μ-equivariant isomorphism of ℚ-Hodge structures

Gr_{0}^{W} (H) ⊗_{ℤ} ℚ → Gr_{−2}^{W}(H)(−1) ⊗_{ℤ} ℚ.

It remains to show that the isomorphisms (6.5) and (6.6) are μ-equivariant. It is enough to prove that the Galois action of μ on

𝒱 ′ → S′

extends analytically to the Zucker extension

J_{\overline{S}′}^{Z}(𝒱 ′) → \overline{S}′

in such a way that the action of the canonical generator of μ = μ_{d}(ℂ) on

J_{\overline{S}′}^{Z}(𝒱 ′)_{s} = W_{−1}H_{ℤ}\H_{ℂ}/F^{0}H_{ℂ}

coincides with the semi-simple part M_{s} of the monodromy action. This follows easily from the constructions. □

6.3. Multiplicity functions and limit mixed Hodge structure.

Theorem 6.3. We keep the notations of Section 6.1.

(1) The potential toric rank t_{pot}(A) is equal to the largest integer α such that exp(2πc(A)i) is an eigenvalue of M_{s} on Gr_{g+α}^{W}H^{g}(X_{∞}, ℚ).

(2) The Jordan form of M_{s} on

(Gr_{−1}^{W}H_{1}(X_{∞}, ℚ))^{1,0} is Jord(m_{A}^{ab}, 0),
(Gr_{−1}^{W}H_{1}(X_{∞}, ℚ))^{0,1} is Jord(m̆_{A}^{ab}, 0),
Gr_{−2}^{W}H_{1}(X_{∞}, ℚ)  is Jord(m_{A}^{tor}, 0),
Gr_{0}^{W} H_{1}(X_{∞}, ℚ)  is Jord(m_{A}^{tor}, 0).

Proof. We denote by M_{u} the unipotent part of the monodromy operator M, and by N its logarithm. By definition, the weight filtration on H^{g}(X_{∞}, ℚ) is the filtration with center g associated to the nilpotent operator N. Thus (1) follows from Proposition 5.2, Theorem 5.9 and the isomorphism (6.1). Point (2) follows from Theorem 6.2 and the canonical μ-equivariant isomorphisms

H_{1}(B(ℂ), ℂ)^{1,0} ≅ Lie(B)
H_{1}(B(ℂ), ℂ)^{0,1} ≅ Lie(B^{∨})^{∨}
H_{1}(T (ℂ), ℂ)  ≅ Lie(T)

(see [19, pp. 4 and 86] for the dual isomorphisms on the level of cohomology).

□

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Lars Halvard Halle

Institut für

Algebraische Geometrie Gottfried-Wilhelm-Leibniz-Universität 30167 Hannover Germany and Matematisk Institutt Universitetet i Oslo Postboks 1053 Blindern 0316 Oslo Norway larshhal@math.uio.no Johannes Nicaise KULeuven Department of Mathematics Celestijnenlaan 200B 3001 Heverlee Belgium johannes.nicaise@ wis.kuleuven.be