Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks

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.

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...

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 study the family of elliptic curves $E/\mathbb{Q}$, having good reduction at $2$ and $3$, and whose $j$-invariants are small. Within this set of elliptic curves, we consider the following two subfamilies: first, the set of elliptic curves $E$ such that the ratio $\Delta(E)/C(E)$ is squarefree; and second, the set of elliptic curves $E$ such that $\Delta(E)/C(E)$ is bounded by a small power $(<3/4)$ of $C(E)$. Both these families are conjectured to contain ...

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, Voight, and Wood. We discuss the theoretical evidence for the model, and we make predictions about elliptic curves based on corresponding theorems proved about the model. In particular, the model suggests that all but finitely many elliptic curves over $\mathbb{Q}$ have rank $\le 21$, which w...

The aim of this article is to give some numerical data related to the order of the Selmer groups in twist families of elliptic curves. To do this we assume the Birch and Swinnerton-Dyer conjecture is true and we use a celebrated theorem of Waldspurger to get a fast algorithm to compute $% L_{E}(1)$. Having an extensive amount of data we compare the distribution of the order of the Selmer groups by functions of type $\alpha \frac{(\log \log (X))^{1+\varepsilon}}{\log (X)}$ wit...

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

We determine average sizes/bounds for the $2$- and $3$-Selmer groups in various families of elliptic curves with marked points, thus confirming several cases of the Poonen--Rains heuristics. As a consequence, we deduce that the average ranks of the elliptic curves in all of these families are bounded. Our proofs are uniform and make use of parametrizations involving various forms of $2 \times 2 \times 2 \times 2$ and $3 \times 3 \times 3$ matrices that we studied in a previou...

We investigate the average rank in the family of quadratic twists of a given elliptic curve defined over $\mathbb{Q}$, when the curves are ordered using the canonical height of their lowest non-torsion rational point.

In 2016, Balakrishnan-Ho-Kaplan-Spicer-Stein-Weigandt produced a database of elliptic curves over $\mathbb{Q}$ ordered by height in which they computed the rank, the size of the $2$-Selmer group, and other arithmetic invariants. They observed that after a certain point, the average rank seemed to decrease as the height increased. Here we consider the family of elliptic curves over $\mathbb{Q}$ whose rational torsion subgroup is isomorphic to $\mathbb{Z}/2\mathbb{Z} \times \ma...