An example of non-cotorsion Selmer group

COTORSION DIMENSIONS OVER GROUP RINGS

Let $Gamma$ be a group, $Gamma'$ a subgroup of $Gamma$ with finite index and $M$ be a $Gamma$-module. We show that $M$ is cotorsion if and only if it is cotorsion as a $Gamma'$-module. Using this result, we prove that the global cotorsion dimensions of rings $ZGamma$ and $ZGamma'$ are equal.

Computing the p-Selmer Group of an Elliptic Curve

In this paper we explain how to bound the p-Selmer group of an elliptic curve over K, a number eld. Our method is an algorithm which is relatively simple to implement, although it requires data such as units and class groups from number elds of degree at most p 1. Our method is practical for p = 3 but for larger values of p becomes impractical with current computing power. In the examples we ha...

The Bloch–Kato Selmer Group

The “weak” BSD conjecture predicts that for an elliptic curve E over a number field K, we have rankE(K) = ords=1 L(E/K, s). The miracle of this formula is that it relates two quantities with very different origins: the left hand side is an algebraic object while the right hand side is an analytic object. Furthermore, the algebraic rank is “global” in nature, while the analytic rank can be defin...

Selmer Group Heuristics and Sieves

In the past decade or so, the most elementary of the sieve methods of analytic number theory has been adapted to a geometric setting. In this geometric setting, the primes are replaced by the closed points of a variety over a finite field or more generally of a scheme of finite type over Z. We will present the method and some of the results that have been proved using it. For instance, the prob...

Selmer Group Estimates Arising from the Existence of Canonical Subgroups