Average analytic ranks of elliptic curves over number fields

TL;DR

This paper establishes a conditional upper bound on the average analytic rank of elliptic curves over any number field, assuming modularity and the Generalized Riemann Hypothesis, using novel point-counting techniques on weighted projective stacks.

Contribution

It provides the first conditional bound on the average analytic rank over arbitrary number fields, based on new asymptotic results for counting elliptic curves with local conditions.

Findings

Average analytic rank bounded by (9*deg(K)+1)/2 under assumptions

Asymptotics for counting elliptic curves with local conditions

Development of point-counting methods on weighted projective stacks

Abstract

A conditional bound is given for the average analytic rank of elliptic curves over an arbitrary number field. In particular, under the assumptions that all elliptic curves over a number field $K$ are modular and have $L$-functions which satisfy the Generalized Riemann Hypothesis, it is shown that the average analytic rank of isomorphism classes of elliptic curves over $K$ is bounded above by $(9\ \text{deg}(K)+1)/2$, when ordered by naive height. A key ingredient in the proof is giving asymptotics for the number of elliptic curves over an arbitrary number field with a prescribed local condition; these results are obtained by proving general results for counting points of bounded height on weighted projective stacks with a prescribed local condition, which may be of independent interest.