Average analytic ranks of elliptic curves over number fields

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.

39 pages. Accuracy and readability greatly improved thanks to suggestions by the referee and others! arXiv admin note: substantial text overlap with arXiv:2201.10624

Cited by in corpus (1)

• Points of bounded height in images of morphisms of weighted projective stacks with applications to counting elliptic curves