Heuristics for the arithmetic of elliptic curves

In this article, we propose a new probabilistic model for the distribution of ranks of elliptic curves in families of fixed Selmer rank, and compare the predictions with previous results, and with the databases of curves over the rationals that we have at our disposal. In addition, we document a phenomenon we refer to as Selmer bias that seems to play an important role in the data and in our models.

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 and Shafarevich-Tate groups of elliptic curves simultaneously, and relies on a theorem counting alternating integer matrices of specified rank. We also discuss analogues for elliptic curves over other global fields.

In this paper, under GRH for elliptic $L$-functions, we give an upper bound for the probability for an elliptic curve with analytic rank $\leq a$ for $a \geq 11$, and also give an upper bound of $n$-th moments of analytic ranks of elliptic curves. These are applications of counting elliptic curves with local conditions, for example, having good reduction at $p$.

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, which in turn follows from an asymptotic formula for the number of binary quartic forms over Z with bounded invariants. We explain their proof, as well as other arithmetic applications.

In this article, we prove that the average rank of elliptic curves over $\mathbb{Q}$, when ordered by height, is less than $1$ (in fact, less than $.885$). As a consequence of our methods, we also prove that at least four fifths of all elliptic curves over $\mathbb{Q}$ have rank either 0 or 1; furthermore, at least one fifth of all elliptic curves in fact have rank 0. The primary ingredient in the proofs of these theorems is a determination of the average size of the $5$-Selm...

The purpose of this paper is to give a "down--to--earth" proof of the well--known fact that a randomly chosen elliptic curve over the rationals is most likely to have trivial torsion.

All the results in this paper are conditional on the Riemann Hypothesis for the L-functions of elliptic curves. Under this assumption, we show that the average analytic rank of all elliptic curves over Q is at most 2, thereby improving a result of Brumer. We also show that the average within any family of quadratic twists is at most 3/2, improving a result of Goldfeld. A third result concerns the density of curves with analytic rank at least R, and shows that the proportion o...

We give some heuristics for counting elliptic curves with certain properties. In particular, we re-derive the Brumer-McGuinness heuristic for the number of curves with positive/negative discriminant up to $X$, which is an application of lattice-point counting. We then introduce heuristics (with refinements from random matrix theory) that allow us to predict how often we expect an elliptic curve $E$ with even parity to have $L(E,1)=0$. We find that we expect there to be about ...

Based on an equation for the rank of an elliptic surface over $\mathbb{Q}$ which appears in the work of Nagao, Rosen, and Silverman, we conjecture that 100% of elliptic surfaces have rank $0$ when ordered by the size of the coefficients of their Weierstrass equations, and present a probabilistic heuristic to justify this conjecture. We then discuss how it would follow from either understanding of certain $L$-functions, or from understanding of the local behaviour of the surfa...

We present experimental evidence to support the widely held belief that one half of all elliptic curves have infinitely many rational points. The method used to gather this evidence is a refinement of an algorithm due to the author which is based upon rigid and crystalline cohomology.