Let K be a CM field and O be its ring of integers. Let p be an odd prime integer and p be a prime in K lying above p. Let F be a Galois extension of K unramified over p. For an Abelian variety A defined over F with complex multiplication by O, we study the variation of the p-ranks of the Selmer groups in pro-p algebraic extensions. We first study the Zpextension case. When K is quadratic imagin...

We take an approach toward counting the number of n for which the curve En : y = x3−n2x has 2-Selmer groups of a given size. This question was also discussed in a pair of papers by Roger Heath-Brown [6, 7]. We discuss the connection between computing the size of these Selmer groups and verifying cases of the Birch and Swinnerton-Dyer Conjecture. The key ingredient for the asymptotic formulae is...

Let $p$ be an odd prime. Associated to a pair $(E, \mathcal{F}_\infty)$ consisting of rational elliptic curve $E$ and $p$-adic Lie extension $\mathcal{F}_\infty$ $\mathbb{Q}$, is the $p$-primary Selmer group $Sel_{p^\infty}(E/\mathcal{F}_\infty)$ over $\mathcal{F}_\infty$. In this paper, we study arithmetic statistics for algebraic structure group. The results provide insights into asymptotics ...

In this talk K be a global field and GK := Gal(K/K) the absolute Galois group over K. We let V be a p-adic representation of GK (finite-dimensional over Qp), and Σ a finite set of places of K containing p and ∞, outside which V is unramified. So we can and will view V as a representation of GK,Σ for technical reasons, it is at times important to work with that rather than the full GK ! We are g...

The goal of this note is to prove, under some assumptions, a formula relating the Selmer groups of isogenous Galois representations. Local and global Euler-Poincaré characteristic formulas are key tools in the proof. With additional hypotheses, we use the isogeny formula to study how the formation of Selmer groups interacts with normalization of the coefficient ring and discuss how a main conje...

We establish the freeness of certain anticyclotomic Selmer groups of modular forms. The freeness of these Selmer groups plays a key role in the Euler system arguments introduced in [BD05]. In particular, our result fills some implicit gaps in [PW11] and [CH15] which in turn allows the results of these papers to hold for modular forms whose residual representations are not minimally ramified. Re...

In this article, we show that the Abel-Jacobi images of Heegner cycles over Shimura curves constructed by Nekovar, Besser and theta elements contructed Chida-Hsieh form a bipartite Euler system in sense Howard. As an application this, deduce converse to Gross-Zagier-Kolyvagin type theorem for higher weight modular forms generalizing works Wei Zhang Skinner two. That is, if rank certain residual...