Distribution of toric periods of modular forms on definite quaternion algebras
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 \(\{P_E(\varphi)\}_E\) of toric periods has infinitely many sign changes. Further, let \(W(\varphi)\) be the modular form of weight \(3 / 2\) associated to \(\varphi\), then the authors relates its \(|\operatorname{Disc}(E)|\)-th Fourier coefficient \(a_{\varphi}(|\operatorname{Disc}(E)|)\) to the toric period \(P_E(\varphi)\), and are able to conclude that there are infinitely many sign changes among the Fourier coefficients \(a_{\varphi}(|\operatorname{Disc}(E)|)\) as \(E\) runs through the quadratic fields in \(D\). This result complements the previous work of \textit{T. A. Hulse} et al. [Int. J. Number Theory 8, No. 3, 749--762 (2012; Zbl 1290.11081)] and of \textit{Y. J. Jiang} et al. [Math. Z. 293, No. 1--2, 789--808 (2019; Zbl 1464.11049)] where the levels are confined to \(\Gamma_0(4)\) (but with general half-integral weights), and the work of \textit{S. Lester} and \textit{M. Radziwiłł} [Math. Ann. 379, No. 3--4, 1553--1604 (2021; Zbl 1467.11047)] where the special case of weight \(3 / 2\) seems to have been excluded from their results.
scientific article; zbMATH DE number 7608403
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