The structure of analytic τ-sheaves

Let K be a finite extension of ℚ, A an abelian variety defined over K, π = Gal(K/K) the absolute Galois group of K, and l a prime number. Then π acts on the (so-called) Tate module $$ {T_l}(A) = \mathop{{\lim }}\limits_{{\mathop{ \leftarrow }\limits_n }} \,A[{l^n}](\overline K ) $$ The goal of this chapter is to give a proof of the following results: (a) The representation of π on \( {T_l}(A){ \otimes_{{{\mathbb{Z}_l}}}}{\mathbb{Q}_l} \) s is semisimple. (b) The map $$ {\text{En}}{{\text{d}}_K}(A){ \otimes_{\mathbb{Z}}}{\mathbb{Z}_l} \to {\text{En}}{{\text{d}}_{\pi }}({T_l}(A)) $$ is an isomorphism. (c) Let S be a finite set of places of K, and let d > 0. Then there are only finitely many isomorphism classes of abelian varieties over K with polarizations of degree d which have good reduction outside of S.

Potentially good reduction of Barsotti–Tate groups

LetK~$\tilde K$be a fixed algebraic closure of an infinite fieldK. We consider an absolutely integral curveΓinPKn$\mathbb{P}_{K}^{n}$withn≥ 2. The curveΓK~$\it\Gamma_{\tilde{K}}$should have only finitely many inflection points, finitely many double tangents, and there exists no point inPK~n$\mathbb{P}_{\tilde{K}}^{n}$through which infinitely many tangents toΓK~$\it\Gamma_{\tilde{K}}$go. In addition there exists a prime numberqsuch thatΓK~$\it\Gamma_{\tilde{K}}$has a cusp of multiplicityqand the multiplicities of all other points ofΓK~$\it\Gamma_{\tilde{K}}$are at mostq. Under these assumptions, we construct a non-empty Zariski-open subsetOofPK~n$\mathbb{P}_{\tilde{K}}^{n}$such that ifn≥ 3, the projection from each pointo∈O(K) birationally mapsΓonto an absolutely integral curveΓ′inPKn−1$\mathbb{P}_{K}^{n-1}$with the same properties asΓ(keepingqunchanged). Ifn= 2, then the projection from eacho∈O(K) mapsΓontoPK1$\mathbb{P}_{K}^{1}$and leads to astabilizingelementtof the function fieldFofΓoverK. The latter means thatF/K(t) is a finite separable extension whose Galois closureF^${\hat F}$is regular overK.

We investigate Néron models of Jacobians of singular curves over strictly Henselian discretely valued fields and their behavior under tame base change. For a semiabelian variety, this behavior is governed by a finite sequence of (a priori) real numbers between 0 and 1, called jumps. The jumps are conjectured to be rational, which is known in some cases. The purpose of this paper is to prove this conjecture in the case where the semiabelian variety is the Jacobian of a geometrically integral curve with a push-out singularity. Along the way, we prove the conjecture for algebraic tori which are induced along finite separable extensions and generalize Raynaud’s description of the identity component of the Néron model of the Jacobian of a smooth curve (in terms of the Picard functor of a proper, flat, and regular model) to our situation. The main technical result of this paper is that the exact sequence that decomposes the Jacobian of one of our singular curves into its toric and Abelian parts extends to an exact sequence of Néron models. Previously, only split semiabelian varieties were known to have this property.

Let k be a field of characteristic different from 2. Let \(E/k\) be a finite separable extension with a {\it k}-linear involution \(\sigma\). For every \(\sigma\)-symmetric element \(\mu\in E^*\), we define a hermitian scaled trace form by \(x\in E\mapsto\mathrm{Tr}_{E/k}(\mu x x^\sigma)\). If \(\mu=1\), it is called a hermitian trace form. In the following, we show that every even-dimensional quadratic form over a hilbertian field, which is not isomorphic to the hyperbolic plane, is isomorphic to a hermitian scaled trace form. Then we give a characterization of Witt classes of hermitian trace forms over some hilbertian fields.

The generalized Rédei-matrix for function fields

Compared with algebraic varieties the local monodromy of Drinfeld modules appears to be hopelessly complex: the image of the wild inertia subgroup under Tate module representations is infinite save for the case of potential good reduction. Nonetheless, we show that Tate modules of Drinfeld modules are ramified in a limited way: the image of a sufficiently deep ramification subgroup is trivial. This leads to a new invariant, the local conductor of a Drinfeld module. We establish an upper bound on the conductor in terms of the volume of the period lattice. As an intermediate step we develop a theory of normed lattices in function field arithmetic including the notion of volume. We relate normed lattices to vector bundles on projective curves. With the aid of Castelnuovo–Mumford regularity this implies a volume bound on norms of lattice generators, and the conductor inequality follows. Last but not least we describe the image of inertia for Drinfeld modules with period lattices of rank $1$ . Just as in the theory of local $\ell$ -adic Galois representations this image is commensurable with a commutative unipotent algebraic subgroup. However, in the case of Drinfeld modules such a subgroup can be a product of several copies of $\mathbf {G}_a$ .

This chapter develops tools for a more penetrating study of algebraic number theory than was possible in Chapter V and concludes by formulating two of the main three theorems of Chapter V in the modern setting of “adeles” and “ideles” commonly used in the subject.Sections 1–5 introduce discrete valuations, absolute values, and completions for fields, always paying attention to implications for number fields and for certain kinds of function fields. Section 1 contains a prototype for all these notions in the construction of the fieldQp of p-adic numbers formed out of the rationals. Discrete valuations in Section 2 are a generalization of the order-of-vanishing function about a point in the theory of one complex variable. Absolute values in Section 3 are real-valued multiplicative functions that give a metric on a field, and the pair consisting of a field and an absolute value is called a valued field. Inequivalent absolute values have a certain independence property that is captured by the Weak Approximation Theorem. Completions in Section 4 are functions mapping valued fields into their metric-space completions. Section 5 concerns Hensel’s Lemma, which in its simplest form allows one to lift roots of polynomials over finite prime fields Fp to roots of corresponding polynomials over p-adic fields Qp.Section 6 contains the main theorem for investigating the fundamental question of how prime ideals split in extensions. Let K be a finite separable extension of a field F, let R be a Dedekind domain with field of fractions F, and let T be the integral closure of R in K. The question concerns the factorization of an ideal pT in T when p is a nonzero prime ideal in R. If Fp denotes the completion of F with respect to p, the theorem explains how the tensor product K ⊗F Fp splits uniquely as a direct sum of completions of valued fields. The theorem in effect reduces the question of the splitting of pT in T to the splitting of Fp in a complete field in which only one of the prime factors of pT plays a role.Section 7 is a brief aside mentioning additional conclusions one can draw when the extension K/F is a Galois extension.Section 8 applies the main theorem of Section 6 to an analysis of the different of K/F and ultimately to the absolute discriminant of a number field. With the new sharp tools developed in the present chapter, including a Strong Approximation Theorem that is proved in Section 8, a complete proof is given for the Dedekind Discriminant Theorem; only a partial proof had been accessible in Chapter V.Sections 9–10 specialize to the case of number fields and to function fields that are finite separable extensions of Fq (X), where Fq is a finite field. The adele ring and the idele group are introduced for each of these kinds of fields, and it is shown how the original field embeds discretely in the adeles and how the multiplicative group embeds discretely in the ideles. The main theorems are compactness theorems about the quotient of the adeles by the embedded field and about the quotient of the normalized ideles by the embedded multiplicative group. Proofs are given only for number fields. In the first case the compactness encodes the Strong Approximation Theorem of Section 8 and the Artin product formula of Section 9. In the second case the compactness encodes both the finiteness of the class number and the Dirichlet Unit Theorem.KeywordsPrime IdealValuation RingBasic AlgebraDedekind DomainFractional IdealThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let R be a 2-dimensional normal excellent henselian local domain in which 2 is invertible and let L and k be its fraction field and residue field respectively. Let Ω R be the set of rank 1 discrete valuations of L corresponding to codimension 1 points of regular proper models of Spec R . We prove that a quadratic form q over L satisfies the local-global principle with respect to Ω R in the following two cases: (1) q has rank 3 or 4; (2) q has rank ≥ 5 and R = A [ [ y ] ] , where A is a complete discrete valuation ring with a not too restrictive condition on the residue field k , which is satisfied when k is C 1 .

Let G be a unitary group of rank one over a non-archimedean local field K (whose residue field has a characteristic ≠ 2). We consider the action of G on the projective plane. A G(K) equivariant map from the set of points in the projective plane that are semistable for every maximal K split torus in G to the set of convex subsets of the building of G(K) is constructed. This map gives rise to an equivariant map from the set of points that are stable for every maximal K split torus to the building. Using these maps one describes a G(K) invariant pure affinoid covering of the set of stable points. The reduction of the affinoid covering is given.

Given a field K equipped with a set of discrete valuations V , we develop a general theory to relate reduction properties of skew-hermitian forms over a quaternion K -algebra Q to quadratic forms over the function field K(Q) obtained via Morita equivalence. Using this we show that if (K,V) satisfies certain conditions, then the number of K -isomorphism classes of the universal coverings of the special unitary groups of quaternionic skew-hermitian forms that have good reduction at all valuations in V is finite and bounded by a value that depends on size of a quotient of the Picard group of V and the size of the kernel and cokernel of residue maps in Galois cohomology of K with finite coefficients. As a corollary we prove a conjecture of Chernousov, Rapinchuk, Rapinchuk for groups of this type.

On the torsion points of Drinfeld modules in abelian extensions

Let p be an odd prime number, W an absolutely unramified p-adically complete discrete valuation ring with algebraically closed residue field, and X a curve of genus at least two over the field of fractions K of W. In the present paper, we study, under the assumption that X has stable reduction over W, torsion points on X, i.e., torsion points of the Jacobian variety J of X which lie on the image of the Albanese embedding X ?→ J with respect to a K-rational point of X. A consequence of the main result of the present paper is that if, moreover, J has good reduction over W, then every torsion point on X is K-rational after multiplying p. This result is closely related to a conjecture of R. Coleman concerning the ramification of torsion points. For instance, this result leads us to a solution of the conjecture in the case where a given curve is hyperelliptic and of genus at least p.

The complexity of linear problems in fields

For any field K, a “K-curve” is an elliptic curve E defined over some finite separable extension of K such that every Galois conjugate of E is isogenous with E over a separable closure of K. A K-curve must either have complex multiplication (CM) or be isogenous to a collection of K-curves parametrized by a K-rational point on some modular curve X* (N). We give the first proof of this result, obtained in 1993 but not heretofore published. We also indicate some of the reasons for interest in K-curves, and in particular in Q-curves, and give some computational examples and open questions.