Related papers: Patterns on elliptic curves beyond Bremner's conje…

Linnik proved in the late 1950's the equidistribution of integer points on large spheres under a congruence condition. The congruence condition was lifted in 1988 by Duke (building on a break-through by Iwaniec) using completely different…

Given a smooth projective curve C defined over a number field and given two elliptic surfaces E_1/C and E_2/C along with sections P_i and Q_i of E_i (for i = 1,2), we prove that if there exist infinitely many algebraic points t on C such…

Clemm and Trebat-Leder (2014) proved that the number of quadratic number fields with absolute discriminant bounded by $x$ over which there exist elliptic curves with good reduction everywhere and rational $j$-invariant is $\gg…

In a remark to Green's conjecture, Paranjape and Ramanan analyzed the vector bundle $E$ which is the pullback by the canonical map of the universal quotient bundle $T_{\Pp^{g-1}}(-1)$ on $\Pp^{g-1}$ and stated a more general conjecture and…

We prove that Zhang's dynamical Bogomolov conjecture holds uniformly along $1$-parameter families of rational split maps and curves. This provides dynamical analogues of recent results of Dimitrov-Gao-Habegger and K\"uhne. In fact, we prove…

Given an elliptic curve $E$ over $\mathbb{Q}$, a celebrated conjecture of Goldfeld asserts that a positive proportion of its quadratic twists should have analytic rank 0 (resp. 1). We show this conjecture holds whenever $E$ has a rational…

We apply one-dimensional convolutional neural networks to the Frobenius traces of elliptic curves over $\mathbb{Q}$ and evaluate and interpret their predictive capacity. In keeping with similar experiments by Kazalicki--Vlah,…

We prove a theorem giving the asymptotic number of binary quartic forms having bounded invariants; this extends, to the quartic case, the classical results of Gauss and Davenport in the quadratic and cubic cases, respectively. Our…

Let C be an algebraic curve in a power of an elliptic curve, both defined over the algebraic numbers. We show that the set of algebraic points of C which satisfy certain conditions is a finite set. This result has implications with the…

The Birch and Swinnerton-Dyer conjecture states that the rank of the Mordell-Weil group of an elliptic curve E equals the order of vanishing at the central point of the associated L-function L(s,E). Previous investigations have focused on…

We show that Lang's conjecture on error terms in Diophantine approximation implies Honda's conjecture on ranks of elliptic curves over number fields. We also show that even a very weak version of Lang's error term conjecture would be enough…

We compute the first murmurations for elliptic curves over function fields F_q(t): oscillatory patterns in average Frobenius traces that separate rank-0 from rank-1 curves, with z-scores up to 256. For the family E_D: y^2 = x^3 + x + D(t)…

We present a heuristic that suggests that ranks of elliptic curves over the rationals are bounded. In fact, it suggests that there are only finitely many elliptic curves of rank greater than 21. Our heuristic is based on modeling the ranks…

Bhargava and Shankar prove that as E varies over all elliptic curves over Q, the average rank of the finitely generated abelian group E(Q) is bounded. This result follows from an exact formula for the average size of the 2-Selmer group,…

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $G=\langle\sigma_1, \dots, \sigma_n\rangle$ be a finitely generated subgroup of $\operatorname{Gal}(\overline{\mathbb{Q}}/ \mathbb{Q})$. Larsen's conjecture claims that the rank of the…

For an elliptic curve $E$ over $K$, the Birch and Swinnerton-Dyer conjecture predicts that the rank of Mordell-Weil group $E(K)$ is equal to the order of the zero of $L(E_{/ K},s)$ at $s=1$. In this paper, we shall give a proof for elliptic…

The solvability in Sobolev spaces is proved for divergence form complex-valued higher order parabolic systems in the whole space, on a half space, and on a Reifenberg flat domain. The leading coefficients are assumed to be merely measurable…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. In 1976, Lang and Trotter conjectured an asymptotic formula for the number $\pi_{E,r}(X)$ of primes $p \leq X$ of good reduction for which the Frobenius trace at $p$ associated to $E$…

We show that under the assumption of Artin's Primitive Root Conjecture, for all primes p there exist ordinary elliptic curves over $\bar F_p(x)$ with arbitrary high rank and constant j-invariant. For odd primes p, this result follows from a…

Let $E_\lambda$ be the Legendre family of elliptic curves. Given $n$ linearly independent points $P_1,\dots , P_n \in E_\lambda\left(\overline{\mathbb{Q}(\lambda)}\right)$ we prove that there are at most finitely many complex numbers…