Average rank of elliptic curves

In this paper we refine recent work due to A. Shankar, A. N. Shankar, and X. Wang on counting elliptic curves by conductor to the case of elliptic curves with a rational 2-torsion point. This family is a small family, as opposed to the large families considered by the aforementioned authors. We prove the analogous counting theorem for elliptic curves with so-called square-free index as well as for curves with suitably bounded Szpiro ratios. We note that our assumptions on the...

In this paper, we prove a function field-analogue of Poonen-Rains heuristics on the average size of $p$-Selmer group. Let $E$ be an elliptic curve defined over $\mathbb{Z}[t]$. Then $E$ is also defined over $\mathbb{F}_q$ for any $q$ of prime power. We show that for large enough $q$, the average size of the $p$-Selmer groups over the family of quadratic twists of $E$ over $\mathbb{F}_q[t]$ is equal to $p+1$ for all but finitely many primes $p$. Namely, if we twist the curve i...

Let $E$ be an elliptic curve over $\mathbb{Q}$. Then, we show that the average analytic rank of $E$ over cyclic extensions of degree $l$ over $\mathbb{Q}$ with $l$ a prime not equal to $2$, is at most $2+r_{\mathbb{Q}}(E)$, where $r_{\mathbb{Q}}(E)$ is the analytic rank of the elliptic curve $E$ over $\mathbb{Q}$. This bound is independent of the degree $l$ Also, we also obtain some average analytic rank results over $S_d$-fields.

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.

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 prove that, when elliptic curves $E/\mathbb{Q}$ are ordered by height, the average number of integral points $\#|E(\mathbb{Z})|$ is bounded, and in fact is less than $66$ (and at most $\frac{8}{9}$ on the minimalist conjecture). By "$E(\mathbb{Z})$" we mean the integral points on the corresponding quasiminimal Weierstrass model $E_{A,B}: y^2 = x^3 + Ax + B$ with which one computes the na\"{\i}ve height. The methods combine ideas from work of Silverman, Helfgott, and Helfgo...

We study the average behaviour of the Iwasawa invariants for the Selmer groups of elliptic curves, setting out new directions in arithmetic statistics and Iwasawa theory.

Let $K$ be the function field of a smooth curve $B$ over a finite field $k$ of arbitrary characteristic. We prove that the average size of the $2$-Selmer groups of elliptic curves $E/K$ is at most $1+2\zeta_B(2)\zeta_B(10)$, where $\zeta_B$ is the zeta function of the curve $B/k$. In particular, in the limit as $q=\#k\to\infty$ (with the genus $g(B)$ fixed), we see that the average size of 2-Selmer is bounded above by $3$, even in "bad" characteristics.

The elliptic curve $E_k \colon y^2 = x^3 + k$ admits a natural 3-isogeny $\phi_k \colon E_k \to E_{-27k}$. We compute the average size of the $\phi_k$-Selmer group as $k$ varies over the integers. Unlike previous results of Bhargava and Shankar on $n$-Selmer groups of elliptic curves, we show that this average can be very sensitive to congruence conditions on $k$; this sensitivity can be precisely controlled by the Tamagawa numbers of $E_k$ and $E_{-27k}$. As consequences, we...

We prove several results regarding some invariants of elliptic curves on average over the family of all elliptic curves inside a box of sides $A$ and $B$. As an example, let $E$ be an elliptic curve defined over $\mathbb{Q}$ and $p$ be a prime of good reduction for $E$. Let $e_{E}(p)$ be the exponent of the group of rational points of the reduction modulo $p$ of $E$ over the finite field $\mathbb{F}_p$. Let $\mathcal{C}$ be the family of elliptic curves $$E_{a,b}:~y^2=x^3+ax+...