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 twistminimal 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 subWeyl 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 MordellWeil rank equal to zero.
 Publication:

arXiv eprints
 Pub Date:
 July 2020
 DOI:
 10.48550/arXiv.2007.07765
 arXiv:
 arXiv:2007.07765
 Bibcode:
 2020arXiv200707765D
 Keywords:

 Mathematics  Number Theory;
 Primary 11F03;
 11F66;
 11F68
 EPrint:
 24 pages