Geodesic
Average Analytic Ranks of Elliptic Curves over Number Fields
Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e40

Voir la notice de l'article provenant de la source Cambridge University Press

We give a conditional bound 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, we show that the average analytic rank of isomorphism classes of elliptic curves over K is bounded above by $(9\deg (K)+1)/2$, 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. 
Phillips, Tristan. Average Analytic Ranks of Elliptic Curves over Number Fields. Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e40. doi: 10.1017/fms.2024.127
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