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Nonvanishing theorems for L-functions of modular forms and their derivatives. (English) Zbl 0721.11023

Let f be a cuspidal normalized newform of even weight k on \(\Gamma_ 0(M)\), let D be a fundamental discriminant with associated quadratic character \(\chi_ D\) and suppose that \((D,M)=1\). Let \(L(s,f,\chi_ D)\) be the L-series of f twisted with \(\chi_ D\) and put (INVALID INPUT)\(\Lambda\) (s,f,\(\chi\) \({}_ D):=(D^ 2M)^{s/2}(2\pi)^{- s}\Gamma (s)L(s,f,\chi_ D).\)
One knows that \(\Lambda (s,f,\chi_ D)\) has a holomorphic continuation to \({\mathbb{C}}\) and is \(\epsilon \chi_ D(-M)\)-invariant under \(s\mapsto k- s\) where \(\epsilon =\pm 1\) is the sign in the functional equation of \(L(s,f)=L(s,f,\chi_ 1)\). Suppose that every prime dividing M splits in \({\mathbb{Q}}(\sqrt{D})\). Then the sign in the functional equation of \(L(s,f,\chi_ D)\) is equal to \(\epsilon \chi_ D(-1).\)
The main result of the paper states that if S is a finite set of primes including those dividing M, then there exists a quadratic field \({\mathbb{Q}}(\sqrt{D})\) of discriminant D such that \(\epsilon \chi_ D(- 1)<0\), every prime in S splits in \({\mathbb{Q}}(\sqrt{D})\) and \(L(s,f,\chi_ D)\) has a simple zero at \(s=k/2\). A similar result was proved by M. R. Murty and V. K. Murty [Mean values of derivatives of modular L- series Ann. Math., II. Ser. 133, No.3, 447-475 (1991)] by a different method. Combined with the work of V. A. Kolyvagin [Math. USSR, Izv. 33, 473-499 (1989); translation from Izv. Akad. Nauk. SSSR, Ser. Mat. 52, 1154-1180 (1988; Zbl 0681.14016)] and B. Gross and D. Zagier [Invent. Math. 84, 225-320 (1986; Zbl 0608.14019)] the above result implies that if E/\({\mathbb{Q}}\) is a modular elliptic curve with L(E,1)\(\neq 0\), then the Mordell-Weil group and the Tate-Shafarevich group of E/\({\mathbb{Q}}\) are finite.
To prove their result the authors investigate the Fourier-Jacobi expansions of certain Jacobi-Eisenstein series on the Jacobi group \(GSp_ 4({\mathbb{R}})\ltimes {\mathbb{H}}({\mathbb{R}})\) (where \({\mathbb{H}}({\mathbb{R}})\) is a Heisenberg group) attached to f and show that they involve certain Eisenstein series of half-integral weight attached to f. Then they apply a certain integral transform which was first introduced by Novodvorsky in a different context, to these Eisenstein series, obtain a Dirichlet series in a new variable u whose D-th coefficient is \(L(s,f,\chi_ D)\) and study its poles in u.

MSC:

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F11 Holomorphic modular forms of integral weight
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11G05 Elliptic curves over global fields

References:

[1] Atkin, O., Li, W.: Twists of newforms and pseudoeigenvalues ofW-operators. Invent. Math.48, 221-243 (1978) · doi:10.1007/BF01390245
[2] Bump, D.: The Rankin-Selberg method: a survey. To appear in the proceedings of the Selberg Symposium, Oslo (1987) · Zbl 0668.10034
[3] Bump, D., Friedberg, S., Hoffstein, J.: Eisenstein series on the metaplectic group and nonvanishing theorems for automorphicL-functions and their derivatives. Ann. Math.131, 53-127 (1990) · Zbl 0699.10039 · doi:10.2307/1971508
[4] Bump, D., Friedberg, S., Hoffstein, J.: A nonvanishing theorem for derivatives of automorphicL-functions with applications to elliptic curves. Bull. Am. Math. Soc. (to appear) (1989) · Zbl 0699.10038
[5] Eichler, M., Zagier, D.: The theory of Jacobi forms. Boston: Birkh?user 1985 · Zbl 0554.10018
[6] Goldfeld, D., Hoffstein, J.: Eisenstein series of 1/2-integral weight and the mean value of real DirichletL-series. Invent. Math.80, 185-208 (1985) · Zbl 0564.10043 · doi:10.1007/BF01388603
[7] Gross, D., Zagier, D.: Heegner points and derivatives ofL-series. Invent. Math.84, 225-320 (1986) · Zbl 0608.14019 · doi:10.1007/BF01388809
[8] Gradshteyn, I., Ryzhik, I.: Table of integrals, series and products. New York: Academic Press 1980 · Zbl 0521.33001
[9] Jacquet, H.: Fonctions de Whittaker associc?s aux groupes de Chevalley. Bull. Soc. Math. France95, 243-309 (1967) · Zbl 0155.05901
[10] Jacquet, H.: On the nonvanishing of someL-functions. Proc. Ind. Acad. Sci.97, 117-155 (1987) · Zbl 0659.10031 · doi:10.1007/BF02837819
[11] Jacquet, H., Piatetski-Shapiro, I., Shalika, J.: Automorphic forms onGL(3), I, II. Ann. Math.109, 169-258 (1979) · Zbl 0401.10037 · doi:10.2307/1971270
[12] Kohnen, W.: Newforms of half-integral weight. J. Reine Angew. Math.333, 32-72 (1982) · Zbl 0475.10025 · doi:10.1515/crll.1982.333.32
[13] Kolyvagin, V.: On groups of Mordell-Weil and Shafarevich-Tate and Weil elliptic curves. Preprint, in Russian (1988) · Zbl 0681.14016
[14] Maass, H.: Siegel’s Modular Forms and Dirichlet Series. Berlin-Heidelberg. New York: (Lecture Notes in Mathematics, (216). Springer 1971 · Zbl 0224.10028
[15] Murty, M.R., Murty, V.K.: Mean values of derivatives of modularL-series. Preprint 1989 · Zbl 0745.11032
[16] Novodvorsky, M.: AutomorphicL-functions for the symplectic groupGSp 4. In: Automorphic Forms, Representations andL-functions. AMS Proc. Symp. Pure Math.,33, 2, 87-95 (1979)
[17] Ogg, A.: On a convolution ofL-series. Invent. Math.7, 297-312 (1969) · Zbl 0205.50902 · doi:10.1007/BF01425537
[18] Shimura, G.: Introduction to the arithmetic theory of automorphic forms. Iwanami Shoten and Princeton University Press, 1971
[19] Waldspurger, J.-L.: Correspondences de Shimura. In: Proceedings of the International Congress of Mathematicians, Warzawa (1983) · Zbl 0567.10020
[20] Waldspurger, J.-L.: Correspondences de Shimura et Quaternions. Preprint 1984 · Zbl 0567.10020
[21] Whittaker, E., Watson, G.: A course of Modern Analysis, fourth edition. Cambridge, 1927 · JFM 53.0180.04
[22] Ziegler, C.: Jacobi forms of higher degree. Abh. Math. Sem. Univ. Hamburg59, 191-224 (1989) · Zbl 0707.11035 · doi:10.1007/BF02942329
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