Related papers: Non-vanishing Theorems for Quadratic Twists of Ell…

The Birch and Swinnerton-Dyer conjecture has been numerically verified for the Jacobians of 32 modular hyperelliptic curves of genus 2 by Flynn, Lepr\'evost, Schaefer, Stein, Stoll and Wetherell, using modular methods. In the calculation of…

We develop the theory and algorithms necessary to be able to verify the strong Birch--Swinnerton-Dyer Conjecture for absolutely simple modular abelian varieties over $\mathbf{Q}$. We apply our methods to all 28 Atkin--Lehner quotients of…

Let $E$ be an elliptic curve over the rational field $\mathbf{Q}$. Let $K$ be a quadratic extension over $\mathbf{Q}$. Let $\mathrm{ST}(E/K)$ dente the Shafarevich-Tate group of $E$ over $K$. We show that (under mild conditions on $E$) for…

We give a criterion for the non-vanishing of some generalised modular symbols. The criterion is in terms of the non-vanishing of some classes on the compact dual of the locally symmetric manifold. As an application, we construct non-zero…

By introducing a new point of view in Algebraic Topology relating elliptic curves in $\mathbb{R}^2$ and suitable bordism groups, the congruent number problem is solved showing that the Tunnell's theorem is also sufficient. This could be…

We present theoretical and numerical evidence for a random matrix theoretic approach to a conjecture about vanishings of quadratic twists of certain L-functions

Let $E/\mathbb{Q}$ be an optimal elliptic curve of analytic rank zero. It follows from the Birch and Swinnerton-Dyer conjecture for elliptic curves of analytic rank zero that the order of the torsion subgroup of $E/\mathbb{Q}$ divides the…

Let $\mathsf{E}/\mathbb{Q}$ be an elliptic curve. By the modularity theorem, it admits a surjection from a modular curve $X_0(N) \to \mathsf{E}$, and the minimal degree among such maps is called the modular degree of $\mathsf{E}$. By the…

This is an investigation into the possible existence and consequences of a Birch-Swinnerton-Dyer-type formula for L-functions of elliptic curves twisted by Artin representations. We translate expected properties of L-functions into purely…

Let $E:y^2=x^3+ax+b$ be an elliptic curve defined over $\mathbb{Q}$. We compute certain twists of the classical modular curves $X(8)$. Searching for rational points on these twists enables us to find non-trivial pairs of $8$-congruent…

We extend work of Swinnerton-Dyer on the density of the number of twists of a given elliptic curve that have 2-Selmer group of a particular rank.

We investigate in this paper the vanishing at $s=1$ of the twisted $L$-functions of elliptic curves $E$ defined over the rational function field $\mathbb{F}_q(t)$ (where $\mathbb{F}_q$ is a finite field of $q$ elements and characteristic…

We consider heuristic predictions for small non-zero algebraic central values of twists of the $L$-function of an elliptic curve $E/\mathbb{Q}$ by Dirichlet characters. We provide computational evidence for these predictions and…

Let $E/\mathbb{Q}$ be a semistable elliptic curve such that $\mathrm{ord}_{s=1}L(E,s) = 1$. We prove the $p$-part of the Birch and Swinnerton-Dyer formula for $E/\mathbb{Q}$ for each prime $p \geq 5$ of good reduction such that $E[p]$ is…

We show that a positive proportion of Hecke $L$-functions attached to the quartic residue symbols $\big( \frac{\cdot}{q} \big)_4$ for squarefree $q \in \mathbb{Z}[i]$ do not vanish at the central point. Our method also extends to the Hecke…

Let $E$ be a $\mathbb Q$-curve without complex multiplication. We address the problem of deciding whether $E$ is geometrically isomorphic to a strongly modular $\mathbb Q$-curve. We show that the question has a positive answer if and only…

Suppose $E$ is an elliptic curve over $\mathbb{Q}$ and $\chi$ is a Dirichlet character. We use statistical properties of modular symbols to estimate heuristically the probability that $L(E,\chi,1) = 0$. Via the Birch and Swinnerton-Dyer…

We prove that for any reductive group $G$ of adjoint type cuspidal automorphic twisted D-modules have non-vanishing quantum Whittaker coefficients. The argument provides a microlocal interpretation of quantum Whittaker coefficients for any…

We study $N$-congruences between quadratic twists of elliptic curves. If $N$ has exactly two distinct prime factors we show that these are parametrised by double covers of certain modular curves. In many, but not all cases, the modular…

Given two elliptic curves defined over a number field K, not both with j-invariant zero, we show that there are infinitely many $D\in K^\times$ with pairwise distinct image in $ K^\times/{K^\times}^2 $, such that the quadratic twist of both…