Nonvanishing of \(L\)-functions for \(\mathrm{GL}(2)\). (English) Zbl 0677.10020

Let \(F\) be a number field. To each irreducible cuspidal representation \(\pi\) of \(\mathrm{GL}(2)\) over \(F\) one may associate the \(L\)-function \(L(s,\pi)\). It is shown that for any given \(s\in {\mathbb C}\) there are infinitely many primitive ray class characters \(\chi\) of \(F\) such that \(L(s,\pi \otimes \chi)\neq 0\). These characters \(\chi\) may be taken unramified outside a finite set of places of \(F\). The corresponding statement is also proved for ‘automorphic representations of \(\mathrm{GL}(1)'\), i.e. continuous characters of \(F^*_ A/F^*.\)
The proof uses a wide arsenal of number theoretical methods. An important step is a proposition providing for any \(\epsilon >0\) infinitely many integral ideals \({\mathfrak q}\) of \(F\) such that the ray class group of \(F\) modulo \({\mathfrak q}\) has more than \(N({\mathfrak q})^{1-\varepsilon}\) elements, and such that the norm \(N({\mathfrak q})\) over the rationals is the product of different rational primes outside a given finite set of primes.
Consequences of the theorem are discussed in the introduction. They concern the algebraicity of \(L(s,\pi)\) at special points, and even intersection homology of the Baily-Borel compactification of a Hilbert modular variety.
Reviewer: R.W.Bruggeman


11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11R23 Iwasawa theory
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[1] Bump, D., Friedberg, S., Hoffstein, J.: Eisenstein series on the metaplectic group and nonvanishing theorems for automorphicL-functions and their derivatives. Ann. Math. (To appear) · Zbl 0699.10039
[2] Gelbart, S., Jacquet, H.: A relation between automorphic representations of GL (2) and GL (3). Ann. Sci. Ec. Norm. Super., IV. Ser.11, 471-542 (1978) · Zbl 0406.10022
[3] Goldfeld, D., Hoffstein, J., Patterson, S.J.: On automorphic functions of half-integral weight with applications to elliptic curves. In: Number Theory Related to Fermat’s Last Theorem. (Progress in Math., Vol. 26). Boston: Birkhäuser 1982 · Zbl 0499.10029
[4] Gupta, R., Murty, M.R., Murty, V.K.: The Euclidean algorithm forS-integers. In: Number Theory. Proc. 1985 Montreal Conference. (CMS Conf. Proc., Vol. 7). Providence: AMS 1987 · Zbl 0618.12006
[5] Harder, G.: General aspects in the theory of modular symbols. In: Séminaire de Théorie des Nombres, Paris 1981-1982. (Progress in Math., Vol. 38). Boston: Birkhäuser 1983
[6] Harder, G.: Eisenstein cohomology of arithmetic groups. The case GL2. Invent. Math.89, 37-118 (1987) · Zbl 0629.10023
[7] Hooley, C.: An asymptotic formula in the theory of numbers. Proc. London Math. Soc.7, 396-413 (1957) · Zbl 0079.27301
[8] Jacquet, H., Shalika, J.A.: A nonvanishing theorem for zeta functions of GLn. Invent. Math.38, 1-16 (1976) · Zbl 0349.12006
[9] Jacquet, H., Shalika, J.A.: On Euler products and the classification of automorphic representations I. Am. J. Math.103, 499-557 (1981) · Zbl 0473.12008
[10] Murty, M.R., Murty, V.K.: A variant of the Bombieri-Vinogradov theorem. In: Number Theory. Proc. 1985 Montreal Conference. (CMS Conf. Proc., Vol. 7). Providence: AMS 1987 · Zbl 0619.10039
[11] Oda, T.: Periods of Hilbert Modular Surfaces. (Progress in Math., Vol. 19). Boston: Birkhäuser 1982 · Zbl 0489.14014
[12] Piatetski-Shapiro, I.: Work of Waldspurger. In: Lie Group Representations II. (Lect. Notes Math., Vol. 1041). Berlin-Heidelberg-New York: Springer 1983 · Zbl 0539.22017
[13] Rohrlich, D.E.: OnL-functions of elliptic curves and cyclotomic towers. Invent. Math.75, 409-423 (1984) · Zbl 0565.14006
[14] Rohrlich, D.E.:L-functions and division towers. Math. Ann.281, 611-632 (1988) · Zbl 0656.14013
[15] Shahidi, F.: On the Ramanujan conjecture and finiteness of poles for certainL-functions. Ann. Math.127, 547-584 (1988) · Zbl 0654.10029
[16] Shimura, G.: The special values of the zeta functions associated with Hilbert modular forms. Duke Math. J.45, 637-679 (1978) · Zbl 0394.10015
[17] Shimura, G.: On the periods of modular forms. Math. Ann.229, 211-221 (1977) · Zbl 0363.10019
[18] Shintani, T.: A remark on zeta functions of algebraic number fields. In: Automorphic Forms, Representation Theory and Arithmetic. (Proc. Bombay Colloquium 1979). Berlin-Heidelberg-New York: Springer 1981
[19] Tate, J.T.: Local constants. In: Algebraic Number Fields, Fröhlich, A. (ed.). New York: Academic Press 1977 · Zbl 0425.12019
[20] Waldspurger, J.-L.: Sur les valeurs de certaines fonctionsL automorphes en leur centre de symétrie. Compos. Math.54, 173-242 (1985) · Zbl 0567.10021
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