On a problem of Hoffstein and Kontorovich

Abstract

Let $\pi$ be a cuspidal automorphic representation of ${\mathrm{GL}}_2({\mathbb{A}}_{\mathbb{Q}})$ 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 $L(1/2,\pi \otimes {\chi }_d)\ne 0$. 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.

24 pages

References in corpus (1)

• The first non-vanishing quadratic twist of an automorphic L-series