Selmer Groups and Anticyclotomic Zp-extensions

On the Freeness of Anticyclotomic Selmer Groups

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...

Iwasawa’s Main Conjecture for elliptic curves over anticyclotomic Zp-extensions

Tate–Shafarevich groups in anticyclotomic p-extensions at supersingular primes

Let E/Q be an elliptic curve and p a prime of supersingular reduction for E. Denote by K∞ the anticyclotomic Zp-extension of an imaginary quadratic field K which satisfies the Heegner hypothesis. Assuming that p splits in K/Q, we prove that X(K∞, E)p∞ has trivial Λ-corank and, in the process, also show that HSel(K∞, Ep∞) and E(K∞)⊗Qp/Zp both have Λ-corank two.

Selmer Groups of Abelian Varieties in Extensions of Function Fields

Let k be a field of characteristic q, C a smooth connected curve defined over k with function field K := k(C). Let A/K be a non constant abelian variety defined over K of dimension d. We assume that q = 0 or > 2d + 1. Let p 6= q be a prime number and C → C a finite geometrically Galois and étale cover defined over k with function field K ′ := k(C). Let (τ , B) be the K /k-trace of A/K. We give ...

Congruences between Selmer groups ∗

The study of congruences between arithmetically interesting numbers has a long history and plays important roles in several areas of number theory. Examples of such congruences include the Kummer congruences between Bernoulli numbers and congruences between coefficients of modular forms. Many of these congruences could be interpreted as congruences between special values of L-functions of arith...