Tate module

Abstract: Almost all of the general facts about abelian varieties which we use without comment or refer to as "well known" are due to WEIL, and the references for them are [12] and [3]. Let k be a field, k its algebraic closure, and A an abelian variety defined over k, of dimension g. For each integer m > 1, let A m denote the group of elements aeA(k) such that ma=O. Let l be a prime number different from the characteristic of k, and let T~(A) denote the projective limit of the groups A~ with respect to the maps A~n.l~Av, which are induced by multiplication by l. It is well known that Tt(A) is a free module of rank 2g over the ring Z l of l-adic integers. The group G=Gal(k./k) operates on Tt(A). Let A' and A" be abelian varieties defined over k. The group HOmk(A', A") of homomorphisms of A' into A" defined over k is Z-free, and the canonical map

Abstract: We study the relation between abelian varieties having good reduction at a discrete valuation and the action of the inertia group on torsion points of the abelian variety For onedimensional abelian varieties, ie, elliptic curves, such a connection was found by Ogg in 1967 and seemed to be known to Shafarevich around the same time [1] Ogg proved that an elliptic curve over a local field K with discrete valuation v has good reduction at v if and only if the inertia group of the absolute Galois group of K acts trivially on certain torsion points Then Serre and Tate generalized this result in 1968 to general abelian varieties over an arbitrary field K with a discrete valuation v in their article [2] They called the result the ‘criterion of Neron-Ogg-Shafarevich’ The aim of this thesis is to understand the proof of this criterion which is given in [2] Therefore, we first recall some general theory concerning abelian varieties over a field K The ’multiplication by n’, with n ∈ Z, endomorphisms are studied as well as their kernel, the ntorsion points Out of these torsion points we form the l-adic Tate module, where l is a prime unequal to the characteristic of K The l-adic Tate module comes equipped with a continuous action of the absolute Galois group of K The associated l-adic Galois representation ρl will be studied carefully as it will turn out to play a big role in Neron-Ogg-Shafarevich Furthermore, we address the theory of reducing schemes over a ring R modulo a prime p⊂ R and we discuss what it means to have good reduction Afterwards, we shortly review the theory of Neron models of abelian varieties They make up for a technical aspect in proving the main criterion After proving the criterion of Neron-Ogg-Shafarevich we study important corollaries Among others, these include a study of abelian varieties with potential good reduction at a discrete valuation Moreover, we examine the concept of abelian varieties with complex multiplica-

Abstract: Let k be a perfect field of characteristic p>0. When p>2, Fontaine and Laffaille have classified p-divisibles groups and finite flat p-groups over the Witt vectors W(k) in terms of filtered modules. Still assuming p>2, we extend these classifications over an arbitrary complete discrete valuation ring A with unequal characteristic (0,p) and residue field k by using "generalized" filtered modules. In particular, there is no restriction on the ramification index. In the case k is included in \bar{F}_p (and p>2), we then use this new classification to prove that any crystalline representation of the Galois group of Frac(A) with Hodge-Tate weights in {0,1} contains as a lattice the Tate module of a p-divisible group over A.

Abstract: For an abelian surface A over a number field k, we study the limiting distribution of the normalized Euler factors of the L-function of A. This distribution is expected to correspond to taking characteristic polynomials of a uniform random matrix in some closed subgroup of USp(4); this Sato-Tate group may be obtained from the Galois action on any Tate module of A. We show that the Sato-Tate group is limited to a particular list of 55 groups up to conjugacy. We then classify A according to the Galois module structure on the R-algebra generated by endomorphisms of A_Qbar (the Galois type), and establish a matching with the classification of Sato-Tate groups; this shows that there are at most 52 groups up to conjugacy which occur as Sato-Tate groups for suitable A and k, of which 34 can occur for k = Q. Finally, we exhibit examples of Jacobians of hyperelliptic curves exhibiting each Galois type (over Q whenever possible), and observe numerical agreement with the expected Sato-Tate distribution by comparing moment statistics.