On a problem of Hoffstein and Kontorovich

Abstract

Let $\pi$ be a cuspidal automorphic representation of $\operatorname{GL}_2(\mathbb{A}_{\mathbb{Q}})$ and $d$ be a fundamental discriminant. Hoffstein and Kontorovich ask for a bound on the least $|d|$ (if it exists) such that the central value $L(1/2, \pi \otimes \chi_d) \neq 0$. The bound should be given in terms of the weight, Laplace eigenvalue and/or level of $\pi$. Let $f$ be a holomorphic twist-minimal newform of even weight $\ell$, odd cubefree level $N$, and trivial nebentypus. When $\pi \cong \pi_f$ and the squarefree part of $N$ is of appropriate size, we conditionally improve upon level aspect results of Hoffstein and Kontorovich under subconvexity (with a sub-Weyl exponent) for automorphic $L$-functions. As a consequence we conditionally prove that given an elliptic curve $E/\mathbb{Q}$ of conductor $N$, there exists a small twist that has Mordell--Weil rank equal to zero.