the mordell-weil theorem states that the group of rational points‎ ‎on an elliptic curve over the rational numbers is a finitely‎ ‎generated abelian group‎. ‎in our previous paper, h‎. ‎daghigh‎, ‎and s‎. ‎didari‎, on the elliptic curves of the form $ y^2=x^3-3px$‎, ‎‎bull‎. ‎iranian math‎. ‎soc‎.‎‎ 40 (2014)‎, no‎. ‎5‎, ‎1119--1133‎.‎, ‎using selmer groups‎, ‎we have shown that for a prime $p$...

This article establishes new cases of the Birch and Swinnerton-Dyer conjecture in analytic rank 0, for elliptic curves over Q viewed over the fields cut out by certain self-dual Artin representations of dimension at most 4. When the associated L-function vanishes (to even order ≥ 2) at its central point, two canonical classes in the corresponding Selmer group are constructed and shown to be lin...

Let f1 (resp. f2) denote two (elliptic) newforms of prime level N , trivial character and weight 2 (resp. k + 2, where k ∈ {8, 12}). We provide evidence for the Bloch-Kato conjecture for the motive M = ρf1⊗ρf2 (−k/2−1) by proving that under some assumptions the p-valuation of the order of the Bloch-Kato Selmer group of M is bounded from below by the p-valuation of the relevant L-value (a specia...

Given a curve C of genus 2 defined over a field k of characteristic different from 2, with Jacobian variety J , we show that the two-coverings corresponding to elements of a large subgroup of H ` Gal(k/k), J [2](k) ́ (containing the Selmer group when k is a global field) can be embedded as intersection of 72 quadrics in P k , just as the Jacobian J itself. Moreover, we actually give explicit equ...

The following are extended notes of a lecture given by the author at the international colloquium on L-functions and Automorphic Representation held at TIFR in january 2012. This lecture reported on some joint work of Chris Skinner and the author on the link between central L-values and Selmer groups of elliptic curves. The detailed proofs of our results will appear in [SU13]. The author presen...

For a given Coleman family of modular forms, we construct formal model and prove the existence Galois representations associated to family. As an application, study variations Iwasawa $$\lambda $$ - $$\mu -invariants dual fine (strict) Selmer groups over cyclotomic $$\mathbb {Z}_p$$ -extension {Q}$$ in families forms. This generalizes earlier work Jha Sujatha for Hida families.

 The Mordell-Weil theorem states that the group of rational points‎ ‎on an elliptic curve over the rational numbers is a finitely‎ ‎generated abelian group‎. ‎In our previous paper, H‎. ‎Daghigh‎, ‎and S‎. ‎Didari‎, On the elliptic curves of the form $ y^2=x^3-3px$‎, ‎‎Bull‎. ‎Iranian Math‎. ‎Soc‎.‎‎ 40 (2014)‎, no‎. ‎5‎, ‎1119--1133‎.‎, ‎using Selmer groups‎, ‎we have shown that for a prime $p...

In this note, we consider an `-isogeny descent on a pair of elliptic curves over Q. We assume that ` > 3 is a prime. The main result expresses the relevant Selmer groups as kernels of simple explicit maps between finitedimensional F`-vector spaces defined in terms of the splitting fields of the kernels of the two isogenies. We give examples of proving the `-part of the Birch and Swinnerton-Dyer...

We give an overview over recent results concerning rational points on hyperelliptic curves. One result says that ‘most’ hyperelliptic curves of high genus have very few rational points. Another result gives a bound on the number of rational points in terms of the genus and the Mordell-Weil rank, provided the latter is sufficiently small. The first result relies on work by Bhargava and Gross on ...

We show, in the large $q$ limit, that average size of $n$-Selmer groups elliptic curves bounded height over $\mathbb F_q(t)$ is sum divisors $n$. As a corollary, again we deduce $100\%$ have rank $0$ or $1$.