Distribution of toric periods of modular forms on definite quaternion algebras
type: Item
instance of: scholarly article
title: Distribution of toric periods of modular forms on definite quaternion algebras
author: Miyu Suzuki
author: Satoshi Wakatsuki
author: Shun'ichi Yokoyama
published in: Research in Number Theory
publication date: 2022-10-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/2203.07606
review text: Let \(D\) be a quaternion algebra over \(\mathbb{Q}\), let \(G = \operatorname{PGL}_1(D)\), and let \(\varphi\) be a newform on \(G_{\mathbb{A}}\). In this paper, the authors study the non-vanishing and sign changes of the toric integrals of \(\varphi\) given by \[ P_E(\varphi) = \int_{\mathbb{A}^{\times} E^{\times} \setminus (E \otimes_{\mathbb{Q}} \mathbb{A})^{\times}} \varphi(h) \ \mathrm{d}h \] as \(E\) runs over suitable set of quadratic fields in \(D\). Further, through the work of \textit{J. L. Waldspurger} [Compos. Math. 54, 121--171 (1985; Zbl 0567.10022)], these findings are also applied to derive the corresponding results for the associated automorphic forms over \(\operatorname{PGL}_2(\mathbb{A})\). For the non-vanishing problem, under certain technical conditions the authors show in Theorem 4.6 that the toric periods \(P_E(\varphi)\) are non-vanishing for a positive proportion of imaginary quadratic fields \(E \subseteq D\). With the help of Waldspurger's formula, they also obtain in Theorem 4.7 an explicit lower bound for the proportion of the non-vanishing quadratic twists \(L(1 / 2, \pi' \otimes \eta_E)\), where \(\pi'\) is the Jacquet-Langlands transfer to \(\operatorname{PGL}_2(\mathbb{A})\) of the irreducible cuspidal automorphic representation of \(G_{\mathbb{A}}\) generated by \(\varphi\), and \(\eta_E\) is the quadratic character on \(\mathbb{A}^{\times} / \mathbb{Q}^{\times}\) attached to \(E\). In particular, the authors apply their results to the study of Goldfeld's conjecture on average ranks of quadratic twists of elliptic curves, and in some cases (such as the Cremona elliptic curve 19A1) improve upon the existing records. It should be noted, as remarked in the paper, that we still do not know whether the above mentioned technical conditions (namely Condition 4.4 in the paper) would allow Theorems 4.6 and 4.7 to cover infinitely many cases. For the sign change problem, the authors show in Theorem 5.1 that the sequence \(\{P_E(\varphi)\}_E…
reviewed by: Qiao Zhang
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 11F37
Mathematics Subject Classification ID: 11F30
zbMATH DE Number: 7608403
zbMATH Keywords: toric integrals
zbMATH Keywords: non-vanishing
zbMATH Keywords: sign changes
describes a project that uses: Magma
MaRDI profile type: Publication
OpenAlex ID: W4304989353
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DOI: 10.1007/S40993-022-00389-8
description: scientific article; zbMATH DE number 7608403