Average analytic ranks of elliptic curves over number fields
Tristan Phillips

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 are modular and have -functions which satisfy the Generalized Riemann Hypothesis, it is shown that the average analytic rank of isomorphism classes of elliptic curves over is bounded above by , 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.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Vietnamese History and Culture Studies · Analytic Number Theory Research
