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

We discuss the-adic case of Mazur's 'Program B'over: the problem of classifying the possible
images of-adic Galois representations attached to elliptic curves E over, equivalently …

This paper investigates the detection of the rank of elliptic curves with ranks 0 and 1,
employing a heuristic known as the Mestre–Nagao sum S (B)= 1 log B∑ p< B good …

This is an introduction to a probabilistic model for the arithmetic of elliptic curves, a model
developed in a series of articles of the author with Bhargava, Kane, Lenstra, Park, Rains …

We present a conjecture on the average number of Galois orbits of newforms when fixing the
weight and varying the level. This conjecture implies, for instance, that the central $ L …

In this paper, we explicitly classify the minimal discriminants of all elliptic curves E/QE/Q with
a non-trivial torsion subgroup. This is done by considering various parameterized families of …

In this note we obtain effective lower bounds for the canonical heights of non-torsion points
on $ E (\mathbb {Q}) $ by making use of suitable elliptic curve ideal class pairings\begin …

We explicitly construct the Dirichlet series where is the proportion of elliptic curves in short
Weierstrass form with Tamagawa product m. Although there are no with everywhere good …

We prove the existence of secondary terms of order $ X^{3/4} $, with power saving error
terms, in the counting functions of $|{\rm Sel} _2 (E)| $, the 2-Selmer group of E, for elliptic …

Let E be an elliptic curve over Q, p an odd prime number and na positive integer. In this
article, we investigate the ideal class group Cl (Q (E [pn])) of the p n-division field Q (E [pn]) …