Analytic Ranks of Elliptic Curves over Cyclotomic Fields

For an elliptic curve $E$ defined over $\mathbb{Q}$, the Birch and Swinnerton Dyer conjecture relates its analytic rank - the order of vanishing of the associated $L$-function (over $\mathbb{Q}$) at the central point, to its algebraic rank - the rank of the group of $\mathbb{Q}$-rational points of $E$.  An analogous conjecture holds for cyclotomic extensions $\mathbb{Q}(e^{2 \pi i/ q})$. In this talk, I will discuss a recent result where we obtain an upper bound for the analytic rank of $E$ over $\mathbb{Q}(e^{2 \pi i/ q})$, as $q$ varies over primes. This bound follows from a more general nonvanishing result of $L$-functions of newforms twisted by Dirichlet characters in sufficiently large Galois orbits. This is joint work with Rizwanur Khan.