Ranks of Elliptic Curves in Cubic Extensions

For an elliptic curve over the rationals, Goldfeld’s conjecture [4] asserts that the analytic rank ords=1 L(Ed/Q, s) of quadratic twists Ed of E is positive for squarefree d’s with density 1/2. In other words, the analytic rank of E goes up in quadratic extensions Q( √ d)/Q half of the time. In particular, for every E/Q there are (a) infinitely many quadratic extensions where the rank goes up, and (b) infinitely many ones where it does not. In fact, both (a) and (b) are known for the analytic rank and also for the algebraic rank rk(E/K) = dimQ E(K)⊗Z Q. On the other hand, root number formulas in [2, 7] show that the situation is somewhat different for extensions Q( r √ m)/Q with r > 2 and varying m > 1. We will be concerned with the case r = 3, and there are examples of curves (such as E =19A3, see [2] Cor. 7) for which the analytic rank goes up in every non-trivial extension Q( 3 √ m)/Q; so (b) fails for cubic extensions. As for (a), the formulas do imply that the analytic rank goes up in infinitely many cubic extensions if E/Q is semistable. It turns out that the same is true of the algebraic rank for any E over a number field K. Thus we have