Weil's converse theorem with poles
type: Item
instance of: scholarly article
zbMATH Open document ID: 1312.11035
publication date: 2014-11-12T00:00:00Z
full work available at URL: weil.pdf
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 11F30
zbMATH DE Number: 6369075
zbMATH Keywords: holomorphic modular forms
zbMATH Keywords: Weil's converse theorem
MaRDI profile type: Publication
OpenAlex ID: W2136333882
DOI: 10.1093/IMRN/RNT127
published in: IMRN. International Mathematics Research Notices
title: Weil's converse theorem with poles
author: Andrew R. Booker
author: M. Krishnamurthy
review text: In the paper under review, improving on their method in [Compos. Math. 147, No. 3, 669--715 (2011; Zbl 1271.11054)], the authors prove a generalization of Weil's converse theorem for classical holomorphic modular forms, allowing the \(L\)-functions twisted by nontrivial Dirichlet characters to have arbitrary poles in \(\mathbb{C}\), as long as they satisfy a suitable functional equation. More precisely, the main result of the paper is the following theorem.NEWLINENEWLINETheorem. Let \(\psi\) be a Dirichlet character modulo \(N\), \(k\) a positive integer satisfying \(\psi(-1)= (-1)^k\) and \(\{f_n\}_{n=1}^{\infty}\), \(\{g_n\}_{n=1}^{\infty}\) be sequences of complex numbers satisfying the bound \(f_n,g_n = O(n^{\sigma})\), for some \(\sigma >0\). Let \(\mathcal{P}\) be the set of primes such that the set \(\{p\in\mathcal{P} : p\equiv u \pmod v\}\) is infinite for all coprime \(u,v \in \mathbb{Z}_{>0}\) and \(p\nmid N\) for any \(p\in \mathcal{P}\). For every primitive Dirichlet character \(\chi\) of modulus \(q \in \mathcal{P} \cup \{1\}\) assume that the twisted \(L\)-functions NEWLINE\[NEWLINE \Lambda_f(s,\chi)= (2\pi)^{-s} \Gamma(s)\sum_{n=1}^{\infty}f_n \chi(n)n^{-s} \text{ and } \Lambda_g(s,\bar{\chi})= (2\pi)^{-s} \Gamma(s)\sum_{n=1}^{\infty}g_n \bar{\chi}(n)n^{-s} NEWLINE\]NEWLINE defined initially for Re\((s)>\sigma +1\), are extended to meromorphic functions on to \(\mathbb{C}\) and satisfy the functional equation NEWLINE\[NEWLINE \Lambda_f(s,\chi)=\varepsilon \psi(q)\chi(N) \frac{\tau(\chi)^2}{q}(q^2N)^{\frac{k}{2} -s} \Lambda_g(k-s,\bar{\chi}), NEWLINE\]NEWLINE where \(\tau(\chi)\) denotes the Gauss sum and complex number \(\varepsilon\) of modulus one is fixed. Let \(\Lambda_f(s)=\Lambda_f(s, \mathbf{1})\), where \(\mathbf{1}\) denotes the character of modulus one and set NEWLINE\[NEWLINE f_0=-\mathrm{Res}_{s=0}\Lambda_f(s), \text{ } f(z)= \sum_{n=0}^{\infty}f_n \exp(2\pi i n z). NEWLINE\]NEWLINE Suppose that there is a nonzero polynomial \(P\in\mathbb{…
reviewed by: Lejla Smajlovic
description: scientific article; zbMATH DE number 6369075