We study the average behaviour of Iwasawa invariants for Selmer groups elliptic curves, setting out new directions in arithmetic statistics and theory.

We prove an anticyclotomic Iwasawa main conjecture proposed by Perrin-Riou for Heegner points semi-stable elliptic curves E over a quadratic imaginary field K satisfying certain generalized hypothesis, at ordinary prime p. It states that the square of index family in equals characteristic ideal torsion part its Bloch-Kato Selmer group (see Theorem 1.3 precise statement). As byproduct we also eq...

In this article, we study the Beilinson–Bloch–Kato conjecture for motives associated to Rankin–Selberg products of conjugate self-dual automorphic representations, within framework Gan–Gross–Prasad conjecture. We show that if central critical value L-function does not vanish, then Bloch–Kato Selmer group with coefficients in a favorable field corresponding motive vanishes. also class constructe...

For an abelian number field of odd degree, we study the structure its 2 2 -Selmer group as a bilinear space and Galois module. We prove struct...

0 Introduction The central point in the Bloch-Kato conjectures is to establish formulas for the order of the Selmer groups attached to Galois representations in terms of the special values of their L-functions. In order to give upper bound, the main way is to construct Euler systems following Kolyvagin. Besides, lower bounds have been obtained by using congruences between automorphic forms. So,...

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...

We study the Selmer varieties of smooth projective curves genus at least two defined over $\mathbb{Q}$ which geometrically dominate a curve with CM Jacobian. extend result Coates and Kim to show that Kim's non-abelian Chabauty method applies such curve. By combining this results Bogomolov-Tschinkel Poonen on unramified correspondences, we deduce any cover $\mathbf{P}^1$ solvable Galois group, i...

We discuss approaches to computing in the Shafarevich-Tate group of Jacobians of higher genus curves, with an emphasis on the theory and practice of visualisation. Especially for hyperelliptic curves, this often enables the computation of ranks of Jacobians, even when the 2-Selmer bound does not bound the rank sharply. This was previously only possible for a few special cases. For curves of gen...

of the absolute Galois group of a number field F . Assume that ρ̄ is ordinary in the sense that the image of any decomposition group at a place v dividing p lies in some Borel subgroup Bv of G. Assume also that ρ̄ satisfies the conditions of [11, Section 7] which guarantee that it has a reasonable deformation theory; see Section 3.1 for details. In this paper we show that the Iwasawa invariants o...

Let $$p\ge 5$$ be a prime. We construct modular Galois representations for which the $$\mathbb {Z}_p$$ -corank of p-primary Selmer group (i.e., its $$\lambda $$ -invariant) over cyclotomic -extension is large. More precisely, any natural number n, one constructs representation such that associated -invariant $$\ge n$$ . The method based on study congruences between forms, and leverages results ...