Generalizing the work of Lubin in the one dimensional case, conditions are found for the existence of canonical subgroups of finite height commutative formal groups of arbitrary dimension over local rings of mixed characteristic (0, p). These are p-torsion subgroups which are optimally close to being kernels of Frobenius homomorphisms. The Fp ranks of the first flat cohomology groups of these c...

We study the Iwasawa theory of fine Selmer group associated to certain Galois representations. The vanishing $\mu$-invariant is shown follow in some cases from a natural property satisfied by deformation rings. outline conditions under which $\mu=0$ conjecture hold for various representations interest.

This paper gives a fairly detailed description of an algorithm that computes (the size of) the 2-Selmer group of the Jacobian of a hyperellitptic curve over Q. The curve is assumed to have even genus or to possess a Q-rational Weierstraa point.

In this paper we develop a theory of class invariants associated to p-adic representations of absolute Galois groups of number fields. Our main tool for doing this involves a new way of describing certain Selmer groups attached to p-adic representations in terms of resolvends associated to torsors of finite group schemes.

Strikingly, this is not true when E has a single rational point of order two. In this case E has a degree two isogeny φ : E → E ′ and an associated Selmer group Selφ(E/Q). Work of Xiong shows that if E does not have a cyclic 4-isogeny de ned over Q(E[2]), then the distribution of the ranks of Selφ(E /Q) as d varies among the squarefree integers less than X tends to the distribution Max{0,N (0, ...

— We report on a large-scale project to investigate the ranks of elliptic curves in a quadratic twist family, focussing on the congruent number curve. Our methods to exclude candidate curves include 2-Selmer, 4-Selmer, and 8-Selmer tests, the use of the Guinand-Weil explicit formula, and even 3-descent in a couple of cases. We find that rank 6 quadratic twists are reasonably common (though stil...

Let p be a prime. Iwasawa’s famous conjecture relating Kubota-Leopoldt p-adic L-functions to the structure of certain Galois groups has been proven by Mazur and Wiles in [10]. Wiles later proved a far-reaching generalization involving p-adic L-functions for Hecke characters of finite order for a totally real number field in [14]. As we discussed in [5], an analogue of Iwasawa’s conjecture for p...

A classical twisting lemma says that given a finitely generated torsion module $M$ over the Iwasawa algebra $\mathbb{Z}_p[[\Gamma ]]$ with $\Gamma \cong \mathbb{Z}_p, \exists$ continuous character $\theta: \Gamma \rightarrow \mathbb{Z}_p^\times$ such that, $ \Gamma^{n}$-Euler characteristic of twist $M(\theta)$ is finite for every $n$. This has been generalized general compact $p$-adic Lie gro...

In this article, we propose a new probabilistic model for the distribution of ranks of elliptic curves in families of fixed Selmer rank, and compare the predictions with previous results, and with the databases of curves over the rationals that we have at our disposal. In addition, we document a phenomenon we refer to as Selmer bias that seems to play an important role in the data and in our mo...