Let E be an abelian variety deened over a number eld K. Let p be a prime number. Let X(K;E) p 1 be the p-Tate-Shafarevich group of E and S class E p 1 (K) the p 1-Selmer group of E. Thirty years ago, Tate proved a local duality theorem for E and used it to establish a global duality for E, later called Cassels-Tate pairing 3, 14]. It states that there is a pairing between X(K;E) p 1 and the p-T...

In this thesis, we study the Selmer group of the p-adic étale realization of certain motives using Kolyvagin’s method of Euler systems [34]. In Chapter 3, we use an Euler system of Heegner cycles to bound the Selmer group associated to a modular form of higher even weight twisted by a ring class character. This is an extension of Nekovář’s result [39] that uses Bertolini and Darmon’s refinement...

In [8], Swinnerton-Dyer considered the proportion of twists of an elliptic curve with full 2-torsion that have 2-Selmer group of a particular dimension. Swinnerton-Dyer obtained asymptotic results on the number of such twists using an unusual notion of asymptotic density. We build on this work to obtain similar results on the density of twists with particular rank of 2-Selmer group using the na...

We introduce a common generalization of essentially all known methods for explicit computation of Selmer groups, which are used to bound the ranks of abelian varieties over global fields. We also simplify and extend the proofs relating what is computed to the cohomologically-defined Selmer groups. Selmer group computations have been practical for many Jacobians of curves over Q of genus up to 2...

In this paper we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2-Selmer rank. As a consequence, under appropriate hypotheses we can find m...

For an elliptic curve over the rational number field and a prime number p, we study the structure of the classical Selmer group of p-power torsion points. In our previous paper [12], assuming the main conjecture and the non-degeneracy of the p-adic height pairing, we proved that the structure of the Selmer group with respect to p-power torsion points is determined by some analytic elements δ̃m d...

In this paper we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2-Selmer rank. As a consequence, under appropriate hypotheses we can find m...

This paper connects the vanishing at the central critical value of the Lfunctions of certain polarized regular motives with the positivity of the rank of the associated p-adic (Bloch-Kato) Selmer groups. For the motives studied it is shown that vanishing of the L-value implies positivity of the rank of the Selmer group. It is further shown that if the the order of vanishing is positive and even...

Let A be an abelian variety defined over a number field k and F a finite Galois extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A and F we investigate the explicit Galois structure of the p-primary Selmer group of A over F . We also use the results so obtained to derive new bounds on the growth of the Selmer rank of A over extensions of k.