The distribution of analytic ranks of elliptic curve over prime cyclic number fields

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $C_l$ be the family of prime cyclic extensions of degree $l$ over $\mathbb{Q}$. Under GRH for elliptic L-functions, we give a lower bound for the probability for $K \in C_l$ such that the difference $r_K(E) − r_\mathbb{Q}(E)$ between analytic rank is less than a for $a \asymp l$. This result gives conjectural evidence that the Diophantine Stability problem suggested by Mazur and Rubin holds for most of $K \in C_l$.