Let F be a global field of characteristic p > 0, F/F a Galois extension with Gal(F/F ) ≃ Z l (for some prime l 6= p) and E/F a non-isotrivial elliptic curve. We study the behaviour of Selmer groups SelE(L)r (r any prime) as L varies through the subextensions of F via appropriate versions of Mazur’s Control Theorem. With mild hypotheses on SelE(F )r (essentially a consequence of the Birch and Sw...

In this discrete time process, guards are allowed to traverse edges of a network under the constraint that the neighbourhood of each vertex must be visited within a specified time interval t. Setting t = 0 the guards form a dominating set at all times. We show every rooted tree with relatively few leaves has a branch with a certain order and number of pendant vertices. This result is used to es...

In this short survey paper we present an outline for using the Saito-Kurokawa correspondence to provide evidence for the Bloch-Kato conjecture for modular forms. Specific results will be stated, but the aim is to provide the framework for such results with an aim towards future research. 1. The Bloch-Kato conjecture for modular forms In this section we will review the Bloch-Kato conjecture for ...

Heegner points play an outstanding role in the study of the Birch and Swinnerton-Dyer conjecture, providing canonical Mordell–Weil generators whose heights encode first derivatives of the associated Hasse–Weil L-series. Yet the fruitful connection between Heegner points and L-series also accounts for their main limitation, namely that they are torsion in (analytic) rank> 1. This partly exposito...

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...

Given fields k ? L , our results concern one parameter L-parametric polynomials over and their relation to generic polynomials. The former are P ( T Y ) ? [ ] of group G which parametrize all Galois extensions via specialization in the latter those -parametric for every field ? . We show, example, that being with taken be single C V U is fact sufficient a polynomial generic. As corollary, we ob...

A conjecture of Moore claims that if Γ is a group and H a finite index subgroup of Γ such that Γ −H has no elements of prime order (e.g. Γ is torsion free), then a Γ-module which is projective over H is projective over Γ. The conjecture is known for finite groups. In that case, it is a direct consequence of Chouinard’s theorem which is based on a fundamental result of Serre on the vanishing of ...