zbMATH — the first resource for mathematics

On the values of certain automorphic \(L\)-functions at their center of symmetry. (Sur les valeurs de certaines fonctions \(L\) automorphes en leur centre de symétrie.) (French) Zbl 0567.10021
Let \(f\) be a (holomorphic) cusp form of (even) weight \(k\) which is a newform for the congruence modular group \(\Gamma_0(N)\) and let \(\mathbb{Q}(f)\) be the number field generated by the Hecke eigenvalues \(a_n\) of \(f\). Let \(f'\) be another newform of weight \(kf\) for \(\Gamma_0(N)\) with eigenvalues under Hecke operators \(T_p\) equal to \(\chi(p)a_p\) for almost all primes \(p\), where \(\chi\) is a real Dirichlet character with conductor prime to \(N\) and \(\chi(-1)=1\). For the associated \(L\)-functions \(L(\cdot,s)\), suppose that \(L(f,k/2) L(f',k/2) \ne 0\). A theorem of M.-F. Vigneras [Prog. Math. 12, 331–356 (1981; Zbl 0453.10024)] now asserts that, upto an explicit multiplicative factor, the ratio \(L(f,k/2)/L(f',k/2)\) is a square in \(\mathbb{Q}(f)\). The object of the author is to establish the same result in a more general form and by a different method.
More specifically, the author proves the following, in particular. Let \(F\) be a number field, \(\mathbb{A}\) the ring of \(F\)-adèles and \(\pi\) an automorphic representation of \(\mathrm{GL}_2(\mathbb{A})\) with trivial central character and local components \(\pi_v\) at archimedean primes \(v\) of \(F\) satisfying some special conditions (such as \(\pi_v\) being in the discrete series for real \(v\)). Let \(\chi_1,\chi_2\) be quadratic characters of \(\mathbb{A}^{\times}/F^{\times}\) such that their local components coincide for every archimedean prime \(v\) of \(F\) and every prime \(v\) at which the local component \(\pi_v\) is ramified. Let \(L(\pi \otimes \chi_2, 1/2)\ne 0\) for the \(L\)-function associated to \(\pi \otimes \chi_2\). (There exists a quaternion algebra \(M\) over \(F\), depending on \(\chi\) and an autormorphic representation \(\pi'\) of the adelic group \(G(\mathbb{A})\), corresponding to \(G=M^{\times}\), in an irreducible submodule \(E'\) of the space of automorphic (cusp) forms on \(G(F)\setminus G(\mathbb{A})\) such that \(\pi\) corresponds to \(\pi'\) under the Jacquet-Langlands correspondence.) Then there exist constants \(p(\chi_i)\) depending only on \(\chi_i\) \((i=1,2)\) and on the components \(\pi_v\) of \(\pi\) at archimedean primes \(v\) of \(F\) such that \[ (L(\pi \otimes \chi_1, 1/2)/L(\pi \otimes \chi_2, 1/2))\cdot p(\chi_2)/p(\chi_1) \] is a square in \(\mathbb{Q}(\pi)\), the field of rationality of \(\pi\).
Reviewer: S. Raghavan

11F11 Holomorphic modular forms of integral weight
11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
Full Text: Numdam EuDML
[1] P. Deligne : Formes modulaires et représentations de GL(2) . In: Modular Functions of One Variable II , Springer Lecture Notes 349, Berlin, Heidelberg, New York (1973) pp. 55-106. · Zbl 0271.10032
[2] S. Gelbart et H. Jacquet : A relation between automorphic representations of GL(2) and GL(3) , Ann. Scient. Ec. Norm. Sup. Il (1978) 471-542. · Zbl 0406.10022 · doi:10.24033/asens.1355 · numdam:ASENS_1978_4_11_4_471_0 · eudml:82024
[3] R. Godement : Notes on Jacquet - Langlands’ Theory , preprint IAS, (1970). · Zbl 0298.12004
[4] G. Harder : General aspects in the theory of modular symbols , preprint. · Zbl 0526.10027
[5] H. Jacquet : Automorphic Forms on GL(2), part II , Springer Lecture Notes 278, Berlin, Heidelberg, New York (1972). · Zbl 0243.12005 · doi:10.1007/BFb0058503
[6] H. Jacquet et R.P. Langlands : Automorphic Forms on GL(2) , Springer Lecture Notes 114, Berlin, Heidelberg, New York, (1970). · Zbl 0236.12010 · doi:10.1007/BFb0058988
[7] H. Shimizu : Theta series and automorphic forms on GL(2) , J. Math. Soc. of Japan 24 (1972) 638-683. · Zbl 0241.10016 · doi:10.2969/jmsj/02440638
[8] G. Shimura : On special values of zeta functions associated with cusp forms , Comm. Pure and Applied Math. 29 (1976) 783-804. · Zbl 0348.10015 · doi:10.1002/cpa.3160290618
[9] J. Tate : Fourier analysis in number fields and Hecke’s zeta functions . In: Cassels and Fröhlich (eds.) Algebraic Number Theory , Academic Press (1967) pp. 305-347.
[10] J.B. Tunnell : article à paraître .
[11] M-F. Vigneras : Valeur au centre de symétrie des fonctions L associées aux formes modulaires, Séminaire de Théorie des Nombres, Paris 1979-80 , Progress in Math. 12, Birkhäuser (1981) pp. 331-356. · Zbl 0453.10024
[12] J-L. Waldspurger : Correspondance de Shimura , J. Math. Pures et Appl. 59 (1980) 1-133. · Zbl 0412.10019
[13] J-L. Waldspurger : Correspondances de Shimura et quaternions , preprint. · Zbl 0724.11026 · doi:10.1515/form.1991.3.219 · eudml:186391
[14] J-L. Waldspurger : Quelques propriétés arithmétiques de certaines formes automorphes sur GL(2) , Comp. Math. 54 (1985) 121-171. · Zbl 0567.10022 · numdam:CM_1985__54_2_121_0 · eudml:89701
[15] A. Weil : Sur la formule de Siegel dans la théorie des groupes classiques , Acta Math. 113 (1965) 1-87, ou Oeuvres Sc., vol. III, pp. 71-157. · Zbl 0161.02304 · doi:10.1007/BF02391774
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.