Central \(L\)-values of newforms and local polynomials. arXiv:2306.15519
Preprint, arXiv:2306.15519 [math.NT] (2023).
Summary: In this paper, we characterize the vanishing of twisted central \(L\)-values attached to newforms of square-free level in terms of so-called local polynomials and the action of finitely many Hecke operators thereon. Such polynomials are the “local part” of certain locally harmonic Maass forms constructed by Bringmann, Kane and Kohnen in \(2015\). We offer a second perspective on this characterization for weights greater than \(4\) by adapting results of Zagier to higher level. To be more precise, we establish that a twisted central \(L\)-value attached to a newform vanishes if and only if a certain explicitly computable polynomial is constant. We conclude by proving an identity between these constants and generalized Hurwitz class numbers, which were introduced by Pei and Wang in \(2003\). We provide numerical examples in weight \(4\) and levels \(7\), \(15\), \(22\), and offer some questions for future work.
MSC:
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 11F11 | Holomorphic modular forms of integral weight |
| 11F12 | Automorphic forms, one variable |
| 11F25 | Hecke-Petersson operators, differential operators (one variable) |
| 11F37 | Forms of half-integer weight; nonholomorphic modular forms |
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