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The non-vanishing of central values of automorphic $$L$$-functions and Landau-Siegel zeros. (English) Zbl 0992.11037
This paper is a survey and announcement of results, the complete proofs of which are being prepared for publication elsewhere. The authors consider the positivity (even with a given positive lower bound) for the central values of certain families of automorphic $$L$$-functions and the highly interesting relation of this problem with the existence of exceptional zeros for quadratic $$L$$-functions. The families of forms in question are:
(i) the holomorphic cusp forms of even weight $$k\leq K$$ for the full modular group that are Hecke eigenfunctions,
(ii) the holomorphic cusp forms of fixed even weight that are newforms for the congruence group $$\Gamma _0(N)$$ as $$N$$ varies over squarefree positive integers and tends to infinity.
In both cases, for at least one half of the related $$L$$-functions, a certain positive lower bound can be shown, but unfortunately a slightly bigger frequency would be needed for either family in order to eliminate the exceptional zeros. True, in the $$N$$-aspect, the percentage exceeds fifty in mean, but this fact does not entail the same striking consequence as the same hypothetical property for each large $$N$$.

##### MSC:
 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11-02 Research exposition (monographs, survey articles) pertaining to number theory 11M20 Real zeros of $$L(s, \chi)$$; results on $$L(1, \chi)$$
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##### References:
 [1] Bump, D.; Friedberg, S.; Hoffstein, J., Nonvanishing theorems for L-functions of modular forms and their derivatives, Inventiones Mathematicae, 102, 543-618, (1990) · Zbl 0721.11023 [2] H. Bohr and E. Landau, Sur les zéros de la fonction ℓ(s) de Riemann, Comptes Rendus158 (1914), 106-110. · JFM 45.0716.02 [3] Balasubramanian, R.; Murty, V. K., Zeros of Dirichlet $$L$$-functions, Annales Scientifiques de l’École Normale Supérieure, 25, 567-615, (1992) · Zbl 0771.11033 [4] Bombieri, E., On the large sieve, Mathematika, 12, 201-229, (1965) · Zbl 0136.33004 [5] Brumer, A., The rank of $$J$$_{0}($$N$$), Asterisque, 228, 41-68, (1995) · Zbl 0851.11035 [6] A. Brumer and J. Silverman, The number of elliptic curves over Q with conductor N, preprint (1996). · Zbl 0868.11029 [7] Barban, M. B.; Vehov, P. P., On an extremal problem, Transactions of the Moscow Mathematical Society, 18, 91-99, (1968) · Zbl 0195.33101 [8] V. Bykovsky, Trace formula for scalar product of Hecke series and its application (in Russian), preprint (1995) · Zbl 0745.11032 [9] Cohen, H., Sums involving the values at negative integers of $$L$$-functions of quadratic characters, Mathematische Annalen, 217, 171-185, (1975) · Zbl 0311.10030 [10] Coates, J.; Wiles, A., On the conjecture of Birch and Swinnerton-Dyer, Inventiones Mathematicae, 39, 223-251, (1977) · Zbl 0359.14009 [11] Duke, W.; Friedlander, J.; Iwaniec, H., Class group $$L$$-functions, Duke Mathematical Journal, 79, 1-56, (1995) · Zbl 0838.11058 [12] Duke, W.; Friedlander, J.; Iwaniec, H.; Greaves, G. (ed.); Harman, G. (ed.); Huxley, M. (ed.), Representations by the determinant and Mean values of $$L$$-functions, in sievemethods, 109-115, (1997), Cambridge · Zbl 0927.11046 [13] Deshouillers, J.-M.; Iwaniec, H., Kloosterman sums and Fourier coefficients of cusp forms, Inventiones Mathematicae, 70, 219-288, (1982) · Zbl 0502.10021 [14] Duke, W., The critical order of vanishing of automorphic $$L$$-functions with large level, Inventiones Mathematicae, 119, 165-174, (1995) · Zbl 0838.11035 [15] Farmer, D. W., Mean value of Dirichlet series associated with holomorphic cusp forms, Journal of Number Theory, 49, 209-245, (1994) · Zbl 0817.11028 [16] Friedlander, J. B., Bounds for $$L$$-functions, 363-373, (1995), Basel · Zbl 0843.11040 [17] Graham, S., An asymptotic estimate related to selberg’s sieve, Journal of Number Theory, 10, 83-94, (1978) · Zbl 0382.10031 [18] Guo, J., On the positivity of the central critical values of automorphic $$L$$-functions for GL(2), Duke Mathematical Journal, 83, 1-18, (1996) · Zbl 0861.11032 [19] Gross, B.; Zagier, D., Heegner points and derivatives of $$L$$-series, Inventiones Mathematicae, 84, 225-320, (1986) · Zbl 0608.14019 [20] Hafner, J.; Kolesnik, G. (ed.); Vaaler, J. (ed.), On the zeros (à la Selberg) of Dirichlet series attached to certain cusp forms, 125-164, (1985), Austin [21] Hoffstein, J.; Lockhart, P., Coefficients of Maass forms and the Siegel zero, Annals of Mathematics, 140, 177-181, (1994) · Zbl 0814.11032 [22] H. Iwaniec and P. Sarnak, Dirichlet $$L$$-functions at the central point, in Number Theory in Progress (K. Györy, H. Iwaniec and J. Urbanowicz, eds.), Proceedings of the International Conference on Number Theory organized by the Stefan Banach International Mathematical Center in Honor of the 60th Birthday of Andrzej Schinzel, Walter deGruyter Mathematics/Mathematik, 1999, pp. 941-952. · Zbl 0929.11025 [23] Iwaniec, H., Mean values for Fourier coefficients of cusp forms and sums of Kloosterman sums, 306-321, (1982), Cambridge [24] H. Iwaniec, Topics in Classical Automorphic Forms, Graduate Studies in Mathematics 17, American Mathematical Society, Providence, 1997. · Zbl 0905.11023 [25] Katok, S.; Sarnak, P., Heegner points, cycles and Maass forms, Israel Journal of Mathematics, 84, 193-227, (1993) · Zbl 0787.11016 [26] Kolyvagin, V. A.; Lugachev, D. Y., Finiteness of the Shafarevich-Tate group and the group of rational points for some modular abelian varieties, Leningrad Mathematical Journal, 1, 1229-1253, (1990) · Zbl 0728.14026 [27] E. Kowalski and P. Michel, Sur le rang de J_{0}(q), preprint (1997). · Zbl 0819.11038 [28] Kohnen, W., Fourier coefficients of modular forms of half-integral weight, Mathematische Annalen, 271, 237-268, (1985) · Zbl 0542.10018 [29] Katz, N.; Sarnak, P., Zeros of zeta functions and symmetry, Bulletin of the American Mathematical Society, 36, 1-26, (1999) · Zbl 0921.11047 [30] Kohnen, W.; Zagier, D., Values of $$L$$-series of modular forms at the center of the critical strip, Inventiones Mathematicae, 64, 175-198, (1981) · Zbl 0468.10015 [31] E. Landau, Bemerkungen zum Heilbronnschen Satz, Acta Arithmetica1 (1935), 1-18. · JFM 61.0170.01 [32] Luo, W., On the nonvanishing of Rankin-Selberg $$L$$-functions, Duke Mathematical Journal, 69, 411-427, (1993) · Zbl 0789.11032 [33] Luo, W., Zeros of Hecke $$L$$-functions associated with cusp forms, Acta Arithmetica, 71, 139-158, (1995) · Zbl 0818.11033 [34] Mazur, B., Modular curves and the Eisenstein ideal, Publications Mathématiques de l’Institut des Hautes Études Scientifiques, 47, 33-186, (1977) · Zbl 0394.14008 [35] L. Merel, Bornes pour la torsion de courbes elliptiques sur les corps de nombres, Inventiones Mathematicae124 (1996), 437-449. · Zbl 0936.11037 [36] Murty, R.; Murty, V. K., Mean values of derivatives of modular $$L$$-series, Annals of Mathematics, 133, 447-475, (1991) · Zbl 0745.11032 [37] H. Montgomery, Topics in Multiplicative Number Theory, Lecture Notes in Mathematics 227, Springer, New York, 1971. · Zbl 0216.03501 [38] Motohashi, Y., The binary additive divisor problem, Annales Scientifiques de l’École Normale Supérieure, 27, 529-572, (1994) · Zbl 0819.11038 [39] Perelli, A.; Pomykala, J., Averages over twisted elliptic $$L$$-functions, Acta Mathematica, 80, 149-163, (1997) · Zbl 0878.11022 [40] Phillips, R.; Sarnak, P., On cusp forms for cofinite subgroups of PSL(2,ℝ), Inventiones Mathematicae, 80, 339-364, (1985) · Zbl 0558.10017 [41] Rohrlich, D., Nonvanishing of $$L$$-functions for GL(2), Inventiones Mathematicae, 97, 383-401, (1989) · Zbl 0677.10020 [42] A. Selberg, On the zeros of Riemann’s zeta-function, Collected Papers, Vol. 1, Springer-Verlag, Berlin, 1989, pp. 85-141. [43] Shimura, G., On modular forms of half-integral weight, Annals of Mathematics, 97, 440-481, (1973) · Zbl 0266.10022 [44] G. Shimura, in Introduction to the Arithmetic Theory of Automorphic Functions (Iwanomi Shoten, ed.), Princeton University Press, Princeton, NJ, 1971. [45] Shimura, G., On the holomorphy of certain Dirichlet series, Proceedings of the London Mathematical Society, 31, 79-98, (1975) · Zbl 0311.10029 [46] C. L. Siegel, Über die Classenzahl quadratischer Zahlkörper, Acta Arithmetica1 (1935), 83-86. · JFM 61.0170.02 [47] J. M. Vanderkam, The rank of quotients of J_{0}(N), preprint (1997). · Zbl 1013.11030 [48] J. L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, Journal de Mathématiques Pures et Appliquées60 (1981), 375-484. · Zbl 0431.10015
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