no. 4
Non-vanishing of class group L-functions at the central point
[Non-annulation des fonctions L de groupes de classes au point central]
Blomer, Valentin1
1 University of Toronto, Department of Mathematics, 100 St. George Street, Toronto M5S 3G3, Ontario, (Canada)
Annales de l'Institut Fourier, Tome 54 (2004) no. 4, pp. 831-847.

Étant donné un corps quadratique imaginaire K=ℚ(-D), notons h son nombre de classes. Nous montrons qu’il existe une constante c telle que pour D assez grand, au moins c·h∏ p∣D (1-p -1 ) des h fonctions L distinctes L K (s,χ) ne s’annulent pas au point central s=1/2.

Let K=ℚ(-D) be an imaginary quadratic field, and denote by h its class number. It is shown that there is an absolute constant c>0 such that for sufficiently large D at least c·h∏ p∣D (1-p -1 ) of the h distinct L-functions L K (s,χ) do not vanish at the central point s=1/2.

DOI : 10.5802/aif.2035
Classification : 11R42, 11M41, 11F67
Keywords: non-vanishing results, $L$-functions, imaginary quadratic fields, mollifier
Mot clés : théorèmes de non-annulation, fonctions $L$, corps quadratique imaginaire, fonction de mollification

Blomer, Valentin 1

1 University of Toronto, Department of Mathematics, 100 St. George Street, Toronto M5S 3G3, Ontario, (Canada) 
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Blomer, Valentin. Non-vanishing of class group $L$-functions at the central point. Annales de l'Institut Fourier, Tome 54 (2004) no. 4, pp. 831-847. doi : 10.5802/aif.2035. https://aif.centre-mersenne.org/articles/10.5802/aif.2035/

[1] D. Bump; S. Friedberg; J. Hoffstein Eisenstein series on the metaplectic group and nonvanishing theorems for automorphic L-functions and their derivatives, Ann. of Math (2), Volume 131 (1990), pp. 53-127 | MR | Zbl

[2] D.A. Burgess On character sums and L-series, Proc. London Math. Soc (2), Volume 12 (1962), pp. 193-206 | MR | Zbl

[3] J.-M. Deshouillers; H. Iwaniec The nonvanishing of Rankin-Selberg zeta-functions at special points, Contemp. Math, Volume 53 (1986), pp. 51-95 | MR | Zbl

[4] W. Duke Hyperbolic distribution problems and half-integral weight Maass forms, Invent. Math, Volume 112 (1988), pp. 73-90 | MR | Zbl

[5] W. Duke; J. Friedlander; H. Iwaniec Class group L-functions, Duke Math. J, Volume 79 (1995), pp. 1-56 | MR | Zbl

[6] E. Fouvry; H. Iwaniec Low-lying zeros of dihedral L-functions, Duke Math. J, Volume 116 (2003), pp. 189-217 | MR | Zbl

[7] H. Iwaniec; P. Sarnak The non-vanishing of central values of automorphic L-functions and the Landau-Siegel zero, Isr. J. Math, Volume 120 (2000), pp. 155-177 | MR | Zbl

[8] N. Katz; P. Sarnak Zeros of zeta-functions and symmetry, Bull. AMS, Volume 36 (1999), pp. 1-26 | MR | Zbl

[9] E. Kowalski; P. Michel; J. VanderKam Mollification of the fourth moment of automorphic L-functions and arithmetic applications, Invent. Math, Volume 142 (2000), pp. 95-151 | MR | Zbl

[10] E. Kowalski; P. Michel; J. VanderKam Non-vanishing of high derivatives of automorphic L-functions at the center of the critical strip, J. Reine Angew. Math, Volume 526 (2000), pp. 1-34 | MR | Zbl

[11] M.R. Murty; V.K. Murty Mean values of derivatives of modular L-series, Ann. of Math (2), Volume 133 (1991), pp. 447-475 | MR | Zbl

[12] M.R. Murty; V.K. Murty Non-vanishing of L-functions and applications, Progress in Mathematics, 157, Birkhäuser, Basel, 1997 | MR | Zbl

[13] A. Perelli; J. Pomykala Averages over twisted elliptic L-functions, Acta Arith, Volume 80 (1997), pp. 149-163 | EuDML | MR | Zbl

[14] J. Pintz Elementary methods in the theory of L-functions II, Acta Arith, Volume 31 (1976), pp. 273-306 | MR | Zbl

[15] D. Rohrlich Nonvanishing of L-functions for GL(2), Invent. Math, Volume 97 (1989), pp. 383-401 | MR | Zbl

[16] G. Shimura On modular forms of half-integral weight, Ann of Math (2), Volume 97 (1973), pp. 440-481 | MR | Zbl

[17] K. Soundararajan Non-vanishing of quadratic Dirichlet L-functions at s=1 2, Ann. of Math (2), Volume 152 (2000), pp. 447-488 | EuDML | Zbl

[18] J.-L. Waldspurger Sur le coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl., Volume 60 (1981), pp. 375-484 | MR | Zbl

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