Non annulation des fonctions L des formes modulaires de Hilbert au point central  [ Non-vanishing of L-functions of Hilbert modular forms at the critical point ]
Annales de l'Institut Fourier, Volume 61 (2011) no. 1, p. 187-259
Birch and Swinnerton-Dyer conjecture allows for sharp estimates on the rank of certain abelian varieties defined over Q. in the case of the jacobian of the modular curves, this problem is equivalent to the estimation of the order of vanishing at 1/2 of L-functions of classical modular forms, and was treated, without assuming the Riemann hypothesis, by Kowalski, Michel and VanderKam. The purpose of this paper is to extend this approach in the case of an arbitrary totally real field, which necessitates an appeal of Jacquet-Langlands’ theory and the adelization of the problem. To show that the L-function (resp. its derivative) of a positive density of forms does not vanish at 1/2, we follow Selberg’s method of mollified moments (Iwaniec, Sarnak, Kowalski, Michel and VanderKam among others applied it successfully in the case of classical modular forms). We generalize the Petersson formula, and use it to estimate the first two harmonic moments, this then allows us to match the same unconditional densities as the ones proved over Q by Kowalski, Michel and VanderKam. In this setting, there is an additional term, coming from old forms, to control.
La conjecture de Birch et Swinnerton-Dyer donne des estimations fines sur le rang de certaines variétés abéliennes définies sur Q. Dans le cas des jacobiennes des courbes modulaires, ce problème est équivalent à l’estimation de l’ordre d’annulation en 1/2 des fonctions L des formes modulaires, et a été traité inconditionnellement par Kowalski, Michel et VanderKam. L’objet de ce travail est d’étendre cette approche dans le cas d’un corps totalement réel arbitraire, ce qui nécessite l’utilisation de la théorie adélique. Nous suivons la méthode des moments amolis initiée par Selberg. On généralise la formule de Petersson que l’on utilise pour étudier les deux premiers moments harmoniques, ce qui nous permet d’atteindre inconditionnellement les mêmes proportions de formes dont la fonction L est non nulle en 1/2 que celles établies pour Q. Dans cette situation, il y a un terme additionnel, issu des formes anciennes, à contrôler.
DOI : https://doi.org/10.5802/aif.2601
Classification:  11F41,  11M41,  11F70
Keywords: L-functions, Hilbert Modular Forms, special values, automorphic forms
@article{AIF_2011__61_1_187_0,
     author = {Trotabas, Denis},
     title = {Non annulation des fonctions $L$ des formes modulaires de Hilbert au point central},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {1},
     year = {2011},
     pages = {187-259},
     doi = {10.5802/aif.2601},
     zbl = {pre05899951},
     mrnumber = {2828130},
     language = {fr},
     url = {https://aif.centre-mersenne.org/item/AIF_2011__61_1_187_0}
}
Non annulation des fonctions $L$ des formes modulaires de Hilbert au point central. Annales de l'Institut Fourier, Volume 61 (2011) no. 1, pp. 187-259. doi : 10.5802/aif.2601. https://aif.centre-mersenne.org/item/AIF_2011__61_1_187_0/

[1] Bruggeman, R.; Miatello, R. J. Sum formula for SL (2) over a number field and a Selberg type estimate for exceptional eigenvalues, Geom. Funt. Anal., Tome 8 (1998), pp. 627-655 | Article | MR 1633975 | Zbl 0924.11039

[2] Bruggeman, R.; Miatello, R. J.; Pacharoni, I. Estimates for Kloosterman sums for totally real number fields, J. Reine Angew. Math., Tome 535 (2001), pp. 103-164 | Article | MR 1837097 | Zbl 0970.11032

[3] Bump, D. Automorphic Forms and Representations, Cambridge Studies in Advanced Mathematics, 55 (1997) | MR 1431508 | Zbl 0868.11022

[4] Bushnell, C.; Henniart, G. The local Langlands conjecture for GL ( 2 ) , Grundlehren der Mathematischen Wissenschaften, 335. Springer-Verlag (2006) | MR 2234120 | Zbl 1100.11041

[5] Cassels, J. W. S.; Fröhlich, A. Algebraic Number Theory, Academic Press (1967) | MR 215665 | Zbl 0645.12001

[6] Cogdell, J.; Piatetski-Shapiro, I. The arithmetic and spectral analysis of Poincaré series, Academic Press (1990) | MR 1063847 | Zbl 0714.11032

[7] Gallagher, P. X. The large sieve and probabilistic Galois theory, Proc. Sympos. Pure Math., Vol. XXIV, Amer. Math. Soc. (1973) | MR 332694 | Zbl 0279.10036

[8] Garrett, P. B. Holomorphic Hilbert Modular Forms, Wadsworth Inc. (1990) | MR 1008244 | Zbl 0685.10021

[9] Gelbart, S. Automorphic forms on adele groups, Annals of Math. studies, 83, Princeton University Press (1975) | MR 379375 | Zbl 0329.10018

[10] Godement, R. Notes on Jacquet-Langlands’ Theory, IAS Lecture Notes, Princeton (1970)

[11] Iwaniec, H.; Kowalski, E. Analytic Number Theory, American Mathematical Society Colloquium Publications, 53 (2004) | MR 2061214 | Zbl 1059.11001

[12] Iwaniec, H.; Luo, W.; Sarnak, Ph. Low lying zeros of families of automorphic L-functions, Publ. Math. IHES, Tome 91 (2000), pp. 55-131 | Numdam | Zbl 1012.11041

[13] Iwaniec, H.; Sarnak, Ph. The non-vanishing of central values of automorphic L-functions and Siegel-Landau zeros, Israel J. Math., Tome 120 (2000), pp. 155-177 | MR 1815374 | Zbl 0992.11037

[14] Kim, H.; Shahidi, F. Cuspidality of symmetric powers with applications, Duke Math. J, Tome 112 (2002), pp. 177-197 | Article | MR 1890650 | Zbl 1074.11027

[15] Kowalski, E.; Michel, Ph. The analytic rank of J 0 (q) and zeros of automorphic L-functions, Duke Math. Journal, Tome 100 (1999), pp. 503-542 | Article | MR 1719730 | Zbl 1161.11359

[16] Kowalski, E.; Michel, Ph. A lower bound for the rank of J 0 (q), Acta Arith., Tome 94 (2000), pp. 303-343 | MR 1779946 | Zbl 0973.11065

[17] Kowalski, E.; Michel, Ph.; Vanderkam, J. Mollification of the fourth moment of automorphic L-functions and arithmetic applications, Invent. math., Tome 142 (2000), pp. 95-151 | Article | MR 1784797 | Zbl 1054.11026

[18] Kowalski, E.; Michel, Ph.; Vanderkam, J. Non-vanishing of higher derivatives of automorphic L-functions, J. reine angew. Math., Tome 526 (2000), pp. 1-34 | Article | MR 1778299 | Zbl 1020.11033

[19] Luo, W. Poincaré series and Hilbert modular forms, The Ramanujan Journal, Tome 7 (2003), pp. 129-143 | Article | MR 2035796 | Zbl 1067.11023

[20] Popa, A. Central values of Rankin L-series over real quadratic fields, Compos. Math., Tome 142 (2006), pp. 811-866 | Article | MR 2249532 | Zbl 1144.11041

[21] Rankin, R. Modular forms and functions, Cambridge University Press (1977) | MR 498390 | Zbl 1156.11001

[22] Soudry, D. The L and γ factors for generic representations of GSp (4,k)× GL (2,k) over a local non-Archimedean field k, Duke Math. Journal, Tome 51 (1984), pp. 355-394 | Article | MR 747870 | Zbl 0557.12012

[23] Trotabas, D. Non-annulation des fonctions L des formes modulaires de Hilbert en le point central (preprint), http://arxiv.org/abs/0809.5031

[24] Vanderkam, J. The rank of quotients of J 0 (N), Duke Math. Journal, Tome 97 (1999), pp. 545-577 | Article | MR 1682989 | Zbl 1013.11030

[25] Vanderkam, J. Linear independence in the homology of X 0 (N), Journal London Math. Soc., Tome 61 (2000), pp. 349-358 | Article | MR 1760688 | Zbl 0963.11023

[26] Venkatesh, A. Beyond endoscopy and special forms on GL (2), Journal reine angew. Math., Tome 577 (2004), pp. 23-80 | Article | MR 2108212 | Zbl 1061.22019