Analytic ranks of elliptic curves over number fields

Abstract

Let $E$ be an elliptic curves over the rational numbers. Let $F$ be a cyclic extension of prime degree $l$. Then, we show that the average of analytic ranks of $E(F)$ over all cyclic extension of prime degree $l$ is at most $2+r_\mathbb{Q}(E)$, where $r_\mathbb{Q}(E)$ is the analytic rank of $E(\mathbb Q)$. This bound is independent of the degree of the cyclic extension. Also, we also obtain some average rank result over $S_d$-fields