Determination of modular forms by twists of critical \(L\)-values
Let \(f\) be a normalized holomorphic newform of weight \(2k\), level \(N\), and trivial character. Let \(\chi\) be a primitive Dirichlet character of conductor \(M_\chi\), and let \(L(f,\chi,s)\) denoted the twisted \(L\)-function for \(f\), normalized to have functional equation under \(s\mapsto 1-s\). Then the authors show that for any \(s_0\in{\mathbb C}\) the \(L\)-values \(L(f,\chi,s_0)\) as \(\chi\) ranges over the characters whose conductor and order are powers of a fixed odd prime \(p\), \((p,N)=1\), determine \(f\). They also show that the central \(L\)-values \(L(f,\chi,1/2)\) as \(\chi\) ranges over the quadratic primitive Dirichlet characters with \((M_\chi,N)=1\) determine \(f\). These results have many implications. The first result implies that the fundamental periods of \(f\) may be recovered by taking a limit of averages of the algebraic parts of the central critical values for twists of \(f\), the twists being of order a power of the fixed prime \(p\), \((p,N)=1\). Next, the authors show that the field \(K_f\) generated over \({\mathbb Q}\) by the Fourier coefficients of \(f\) is also generated over \({\mathbb Q}\) by the set of ratios \(g(\chi_0)g(\chi)^{-1}L(f,\chi,1/2) /L(f,\chi_0,1/2)\) where \(\chi_0\) is a fixed quadratic character such that \(L(f,\chi_0,1/2)\neq 0\), \(g(\nu)\) denotes the Gauss sum associated to the character \(\nu\), and \(\chi\) ranges over the quadratic characters of the same parity as \(\chi_0\). A result of \textit{G. Shimura} [Commun. Pure Appl. Math. 29, 783-804 (1976; Zbl 0348.10015)] implies that \(K_f\) contains this field; the authors apply Shimura's reciprocity law to obtain equality. The authors also give an application to \(p\)-adic \(L\)-functions. One says that a finite order \(p\)-adic character \(\eta\) is of tame type \(i\pmod {p-1}\) if \(\eta=\omega^i\nu\) where \(\omega\) denotes the Teichmüller character and \(\nu\) is a wild character. Then the authors show that for any such \(\eta\), \(0\leq i\leq 2k-2\), \(f\) is determined by the associated one-variable \(p\)-adic \(L\)-function \(L_p(f,\eta,s)\). (If \(p\) is supersingular for \(f\), then there are two such \(L\)-functions, depending on the choice of a root of the \(p\)-th Hecke polynomial, and the result holds for either.) The weaker statement that the collection of such \(L\)-functions for varying \(\eta\) determines \(f\) was previously known, by work of Rohrlich. The authors' result implies that \(L_p(f,\eta,n)\neq 0\) for all but a finite number of positive integers \(n\), \((n,p)=1\). Also, specializing to the weight 2 case, the authors deduce that a modular elliptic curve \(E\) over \({\mathbb Q}\) is determined up to \({\mathbb Q}\)-isogeny by \(L_p(E,s)\) for any prime \(p\), \((p,2N)=1\). The application is obtained by combining their work with some results on modular symbols of Mazur, Tate and Teitelbaum and on \(p\)-adic analytic functions with logarithmic growth of Vishik. The final application presented concerns the Shimura correspondence; it answers a question posed by Kohnen. Let \(N\geq 1\) now be an odd square-free integer, let \(k\geq 1\), and let \(S_{k+1/2}^+(4N)\) be the Kohnen subspace of the space of holomorphic cusp forms of weight \(k+1/2\), level \(4N\), and trivial character. This space is isomorphic as a Hecke module to \(S_{2k}(N)^{\text{ new}}\), by the Shimura correspondence. The authors show that if \(g_1,g_2\) are two newforms in \(S_{k+1/2}^+(4N)\), with Fourier coefficients \(b_1(n)\), \(b_2(n)\), resp., such that \(b_1^2(| D|)=b_2^2(| D|)\) for almost all fundamental discriminants \(D\) with \((-1)^kD>0\), then \(g_1=\pm g_2\). The authors' main idea is to analyze variants of the twisted average of the \(L\)-values \[ T^{-1}\sum_{M_\chi<T}L(f,\chi,s_0) \chi(m) \] where the sum is restricted to those characters \(\chi\) of the appropriate type, \(s_0=1/2\) in the quadratic twist case. The authors show that a suitable limit of this average may be expressed in terms of the Fourier coefficients of \(f\); by the strong multiplicity one theorem their results follow. For the quadratic twist case, the authors also use the Ramanujan bound, proved by Deligne. The analysis of the twisted average relies on the method of \textit{H. Iwaniec} [Sémin. Théor. Nombres Bordx., Sér. II 2, No. 2, 365-376 (1990; Zbl 0719.11029)] who studied the sum (over quadratic characters) in the untwisted case (and studied derivatives as well). After these results were obtained, \textit{H. Stark} [in: Analytic number theory, Vol. 2, Proceedings of a conference in honor of Heini Halberstam, May 16-20, 1995, Urbana, Prog. Math. 139, 737--743 (1996; Zbl 0870.11036)] showed by using transcendence arguments that a modular elliptic curve \(E\) over \({\mathbb Q}\) is determined up to \({\mathbb Q}\)-isogeny by its central value \(L(E,1)\), provided this value is non-zero. It is not apparent how to generalize this approach to higher weight. Also, the authors have obtained a result for derivatives \(L'(f,\chi,1/2)\) in the case that the sign in the functional equations is \(-1\) [Determination of modular elliptic curves by Heegner points, Olga Taussky-Todd: In memoriam. Cambridge, MA: International Press. Pac. J. Math., Spec. Issue, 251--258 (1998; Zbl 1012.11052)].
scientific article; zbMATH DE number 1096056