On the semisimplicity of pure sheaves

The semisimplicity conjecture says that for any smooth projective scheme $X_0$ over a finite field $\mathbf {F}_q$, the Frobenius correspondence acts semisimply on $H^i(X\otimes _{\mathbf { F}_q} \mathbf { F}, \overline {\mathbf { Q}}_l)$, where $\mathbf { F}$ is an algebraic closure of $\mathbf { F}_q$. Based on the works of Deligne and Laumon, we reduce this conjecture to a problem about the Galois representations of function fields. This reduction was also achieved by Laumon a few years ago (unpublished).

This introductory chapter sets out the book's purpose, which is to study Weil's conjecture over function fields: that is, fields K which arise as rational functions on an algebraic curve X over a finite field F q. It reformulates Weil's conjecture as a mass formula, which counts the number of principal G-bundles over the algebraic curve X. An essential feature of the function field setting is that the objects that we want to count (in this case, principal G-bundles) admit a “geometric” parametrization: they can be identified with Fq-valued points of an algebraic stack BunG(X). This observation is used to reformulate Weil's conjecture yet again: it essentially reduces to a statement about the ℓ-adic cohomolog of BunG(X), reflecting the heuristic idea that it should admit a “continuous Künneth decomposition”.

The characteristic polynomials of abelian varieties of higher dimension over finite fields

§1. In this note, we indicate a few improvements to [3]. Let X be an irreducible, non-singular projective algebraic curve defined over a finite field Fq with q elements, of characteristic p. Let SL(n, d) be the coarse moduli scheme of isomorphism classes of stable vector bundles of rank n and determinant isomorphic to an lFq-rational line bundle L of degree d. [We will assume that (n, d) = 1.] By replacing lFq by a finite extension if necessary, we may assume that SL(n, d) is defined over IFq. As one might expect, it is then indeed true that the IFq-rational points of SL(n, d) are precisely the stable vector bundles on X defined over IFq (see [3]). By the Weil conjectures, it is easy to write down the Poincar6 polynomials of SL(n, d) once the number of its lFq-rational points is known. In order to compute the latter, one first notices that the fact that the Tamagawa number of SL(n) is 1 can be interpreted as follows.

A central concern of number theory is the study of local-to-global principles, which describe the behavior of a global field K in terms of the behavior of various completions of K. This book looks at a specific example of a local-to-global principle: Weil's conjecture on the Tamagawa number of a semisimple algebraic group G over K. In the case where K is the function field of an algebraic curve X, this conjecture counts the number of G-bundles on X (global information) in terms of the reduction of G at the points of X (local information). The goal of this book is to give a conceptual proof of Weil's conjecture, based on the geometry of the moduli stack of G-bundles. Inspired by ideas from algebraic topology, it introduces a theory of factorization homology in the setting ℓ-adic sheaves. Using this theory, the authors articulate a different local-to-global principle: a product formula that expresses the cohomology of the moduli stack of G-bundles (a global object) as a tensor product of local factors. Using a version of the Grothendieck–Lefschetz trace formula, the book shows that this product formula implies Weil's conjecture. The proof of the product formula will appear in a sequel volume.

A central concern of number theory is the study of local-to-global principles, which describe the behavior of a global field K in terms of the behavior of various completions of K. This book looks at a specific example of a local-to-global principle: Weil's conjecture on the Tamagawa number of a semisimple algebraic group G over K. In the case where K is the function field of an algebraic curve X, this conjecture counts the number of G-bundles on X (global information) in terms of the reduction of G at the points of X (local information). The goal of this book is to give a conceptual proof of Weil's conjecture, based on the geometry of the moduli stack of G-bundles. Inspired by ideas from algebraic topology, it introduces a theory of factorization homology in the setting ℓ-adic sheaves. Using this theory, the authors articulate a different local-to-global principle: a product formula that expresses the cohomology of the moduli stack of G-bundles (a global object) as a tensor product of local factors. Using a version of the Grothendieck–Lefschetz trace formula, the book shows that this product formula implies Weil's conjecture. The proof of the product formula will appear in a sequel volume.

Number of Points on the Projective Curves aYl=bXl+cZl and aY2l=bX2l+cZ2l Defined over Finite Fields, l an Odd Prime

We give an algorithm to compute representatives of the conjugacy classes of semisimple square integral matrices with given minimal and characteristic polynomials. We also give an algorithm to compute the Fq-isomorphism classes of abelian varieties over a finite field Fq which belong to an isogeny class determined by a characteristic polynomial h of Frobenius when h is ordinary, or q is prime and h has no real roots.

On the cyclicity of the rational points group of abelian varieties over finite fields

Any projection of the visual field is a manipulation. It is therefore essential to select a projection which gives the most useful representation of the field and its clinical defects. It should also provide a uniform numerical assessment of the sensitivity of vision throughout the field and incorporate a percentage functional field estimate along the lines proposed by Dr Esterman.To study these requirements a mathematical system of representation of the visual field which progressively augments it parabolically towards the centre is proposed. It is applicable to all types of perimetry but the Octopus perimeter has been used as a test-bed employing the Sargon programme with both the traditional and the sine-bell stimulus. This work and the case for the adoption of the system are briefly presented.

Stable reduction of modular curves.- On p-adic families of automorphic forms.- ?-curves and abelian varieties of GL2-type from dihedral genus 2 curves.- The old subvariety of J0(NM).- Irreducibility of Galois actions on level 1 Siegel cusp forms.- On elliptic K-curves.- ?-curves and Galois representations.- On the local behaviour of ordinary modular Galois representations.- Arithmetic of ?-curves.- Serre's conjecture for mod 7 Galois representations.- Pairings in the arithmetic of elliptic curves.- Explicit parametrizations of ordinary and supersingular regions of X0(Pn).- Elliptic ?-curves with complex multiplication.- Abelian varieties over ? with large endomorphism algebras and their simple components over ?.- Abelian varieties over ? and modular forms.- Shimura curves embedded in Igusa's threefold.- Shafarevich-Tate groups of nonsquare order.

In this thesis we study Galois representations corresponding to abelian varieties with certain reduction conditions. We show that these conditions force the image of the representations to be big, so that the Mumford-Tate conjecture (:= MT) holds. We also prove that the set of abelian varieties satisfying these conditions is dense in a corresponding moduli space. The main results of the thesis are the following two theorems. Theorem A: Let A be an absolutely simple abelian variety, End° (A) = k : imaginary quadratic field, g = dim(A). Assume either dim(A) ≤ 4, or A has bad reduction at some prime ϕ, with the dimension of the toric part of the reduction equal to 2r, and gcd(r,g) = 1, and (r,g) ≠ (15,56) or (m -1, m(m+1)/2). Then MT holds. Theorem B: Let M be the moduli space of abelian varieties with fixed polarization, level structure and a k-action. It is defined over a number field F. The subset of M(Q) corresponding to absolutely simple abelian varieties with a prescribed stable reduction at a large enough prime ϕ of F is dense in M(C) in the complex topology. In particular, the set of simple abelian varieties having bad reductions with fixed dimension of the toric parts is dense. Besides this we also established the following results: (1) MT holds for some other classes of abelian varieties with similar reduction conditions. For example, if A is an abelian variety with End° (A) = Q and the dimension of the toric part of its reduction is prime to dim( A), then MT holds. (2) MT holds for Ribet-type abelian varieties. (3) The Hodge and the Tate conjectures are equivalent for abelian 4-folds. (4) MT holds for abelian 4-folds of type II, III, IV (Theorem 5.0(2)) and some 4-folds of type I. (5) For some abelian varieties either MT or the Hodge conjecture holds.

Introduction This paper surveys some aspects of the theory of abelian varieties relevant to the Pila–Zannier proof of the Manin–Mumford conjecture and to the Andre– Oort conjecture. An abelian variety is a complete algebraic variety with a group law. The geometry of abelian varieties is tightly constrained and well-behaved, and they are important tools in algebraic geometry. Abelian varieties defined over number fields pose interesting arithmetic problems, for example concerning their rational points and associated Galois representations. The paper is in three parts: (1) an introduction to abelian varieties; (2) an outline of moduli spaces of principally polarised abelian varieties, which are the fundamental examples of Shimura varieties; (3) a detailed proof of the Ax–Lindemann–Weierstrass theorem for abelian varieties, following amethod using o-minimal geometry due to Pila, Ullmo and Yafaev. The first part assumes only an elementary knowledge of algebraic varieties and complex analytic geometry. The second part makes heavier use of algebraic geometry, but still at the level of varieties, and a little algebraic number theory. Like the first part, the algebraic geometry in the third part is elementary; the third part also assumes familiarity with the concept of semialgebraic sets, and uses cell decomposition for semialgebraic sets and the Pila–Wilkie theorem as black boxes. The second and third parts are independent of each other, so the reader interested primarily in the Ax–Lindemann–Weierstrass theorem may skip the second part (sections 4 to 6). In the first part of the paper (sections 2 and 3), we introduce abelian varieties over fields of characteristic zero, and especially over the complex numbers. The theory of abelian varieties over fields of positive characteristic introduces additional complications which we will not discuss. Our choice of topics is driven by Pila and Zannier's proof of the Manin–Mumford conjecture using o-minimal geometry. We will not discuss the Manin–Mumford conjecture or its proofs directly in this paper; aspects of the proof, and its generalisation to Shimura varieties, are discussed in other papers in this volume.

In w 1 we consider the situation: L/K is a finite separable field extension, A is an abelian variety over L, and A, is the abelian variety over K obtained from A by restriction of scalars. We study the arithmetic properties of A, relative to those of A, and in particular show that the conjectures of Birch and Swinnerton-Dyer hold for A if and only if they hold for A, . In w 2 we study certain twisted products of abelian varieties and use our results to show that the conjectures of Birch and Swinnerton-Dyer are true for a large class of twisted constant elliptic curves over function fields. In w we develop a method of handling abelian varieties over a number field K which are of CM-type but which do not have all their complex multiplications defined over K. In particular we compute under quite general conditions the conductors and zeta functions of such abelian varieties and so verify Serre's conjecture [12] on the form of the functional equation. Similar, but less complete, results have been obtained by Deuring [1] for elliptic curves and Shimura [15] for abelian varieties.

The generic fibre of a fibre system of polarized abelian varieties with level structure, and with endomorphism structure coming from a CMfield, is defined over the function field of the moduli space for the abelian varieties.We prove that the points on this generic abelian variety which are defined over the function field of the moduli space form a finite group.The methods of proof generalize those of Mordell-Weil groups o generic abelian varieties, Invent.Math.81 (1985), 71-106, to which this paper is a sequel. Introduction.In this paper we will consider fibre systems of polarized abelian varieties over C characterized by having a fixed CM-field embedded in their endomorphism algebras.If V is the moduli space for such abelian varieties (with level structure), and W is the fibre variety constructed in [4], then the fibre over the generic point of V is an abelian variety defined over the function field of V. We consider the points of this abelian variety which are defined over C(V).Our result is THEOREM.If dimV > 0, then the Mordell-Weil group of the generic fibre is finite.Here, V is isomorphic to a product of domains of the form {rxs complex matrices Z\l -ZlZ > 0} modulo the action of a discrete subgroup of a unitary group.The setting for which the theorem holds is described more precisely in 1.The analogous theorem was proved for V a noncompact quotient of the complex upper half plane by Shioda (Theorem 5.1 of [5]).When F is a quotient of a Hilbert-Siegel space HTS and V is the moduli space for abelian varieties whose endomorphism algebras contain a fixed totally real field, CM-field, totally indefinite quaterion algebra over a totally real field, or quaternion algebra over a CM-field, the analogous theorem was proved in [6] (see also [7]).The present paper makes use of new methods to remove the restriction on the base variety V in the CM-field case.I owe many thanks to Goro Shimura for suggesting the problem and giving guidance and advice, and to Robert Indik for the proof of Theorem 6.NOTATION.We will write diag(j4i,..., As) for the matrix with blocks Ai,...,As on the diagonal, I(r) for the identity matrix in GLr(Z), and '7 for the transpose of the matrix 7.