Analytic rank growth of elliptic curves over cyclic extensions

Abstract

Let E be an elliptic curve defined over ℚ. For an odd prime l, we consider the family of degree l cyclic extensions K over ℚ. When we view the elliptic curve E as a curve over K, the analytic rank of the L-function LK(s, E) of E over K may increase compared to that of the L-function Lℚ(s, E) of E over ℚ. Under the generalized Riemann hypothesis, we demonstrate the rarity of significant increases in analytic ranks.