Weil's converse theorem with poles (Q2929550)

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 \(P\in\mathbb{C}[s]\) such that \(P(s)\Lambda_f(s)\) is extended to an entire function of finite order. ThenNEWLINENEWLINE(i) if \(k\neq 2\) or \( \psi\) is nontrivial, then \(f\in M_k(\Gamma_0(N), \psi)\)NEWLINENEWLINE(ii) if \(k=2\) and \(\psi\) is trivial, then \(f-cE_2 \in M_2(\Gamma_0(N))\), where \(c=\frac{\pi}{6} \mathrm{Res}_{s=1}\Lambda_f(s)\).NEWLINENEWLINEHere, \(M_k(\Gamma_0(N), \psi)\) denotes the set of all weight \(k\) holomorphic modular forms on the congruence group \(\Gamma_0(N)\) twisted by character \(\psi\) and \(E_2(z)= 1- 24 \sum_{n=1}^{\infty} n\exp(2\pi i nz) (1-\exp(2\pi i n z))^{-1}\) is the weight two Eisenstein series on \(\mathrm{PSL}(2, \mathbb{Z})\).