Let S be a scheme, and letGbe a ¢nite, £at, commutative group scheme over S. In this paper we show that (subject to a mild technical assumption) every primitive class in Pic(G) is realisable. This gives an af¢rmative answer to a question of Waterhouse. We also discuss applications to locally free classgroups and to Selmer groups of Abelian varieties. Mathematics Subject Classi¢cations (2000). 1...

In this paper, we consider the Λ-adic deformations of Galois representations associated to elliptic curves. We prove that the Pontryagin dual of the Selmer group of a Λ-adic deformation over certain p-adic Lie extensions of a number field, that are not necessarily commutative, has no non-zero pseudo-null submodule. We also study the structure of various arithmetic Iwasawa modules associated to ...

The purpose of these notes is to describe the notion of an Euler system, a collection of compatible cohomology classes arising from a tower of fields that can be used to bound the size of Selmer groups. There are applications to the study of the ideal class group, Iwasawa’s main conjecture, Mordell-Weil group of an elliptic curve, X (the Safarevich-Tate group), Birch-Swinnerton-Dyer conjecture,...

This is an introduction to classical descent theory, in the context of abelian varieties over number fields.

Let $A$ be an abelian variety defined over a number field $F$ with supersingular reduction at all primes of above $p$. We establish equivalence between the weak Leopoldt conjecture and expected value corank classical Selmer group $p$-adic Lie extension (not neccesasily containing cyclotomic $\Zp$-extension). As application, we obtain exactness defining sequence group. In event that is one-dimen...

In this paper we extend the niteness result on the p-primary torsion subgroup in the Chow group of zero cycles on the selfproduct of a semistable elliptic curve obtained in joint work with S. Saito to primes p dividing the conductor. On the way we show the niteness of the Selmer group associated to the symmetric square of the elliptic curve for those primes. The proof uses p-adic techniques, in...

For a square-free number n = p1p2 . . . pk Feng and Xiong in [FX] give a way to construct a corresponding graph on k-vertices and then give necessary and sufficient conditions on these graphs for the integers n to determine when the elliptic curve En : y = x3−n2x has trivial 2-Selmer groups. These conditions involve understanding when a graph is even. In this note we give a substantial understa...

We prove that when all hyperelliptic curves of genus n ≥ 1 having a rational Weierstrass point are ordered by height, the average size of the 2-Selmer group of their Jacobians is equal to 3. It follows that (the limsup of) the average rank of the Mordell-Weil group of their Jacobians is at most 3/2. The method of Chabauty can then be used to obtain an effective bound on the number of rational p...