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, pp. 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: 10.5802/aif.2601
Classification: 11F41,  11M41,  11F70
Keywords: L-functions, Hilbert Modular Forms, special values, automorphic forms
Trotabas, Denis 1

1 Stanford University Department of Mathematics Building 380, Stanford, California 94305 (USA)
@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},
     pages = {187--259},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {61},
     number = {1},
     year = {2011},
     doi = {10.5802/aif.2601},
     mrnumber = {2828130},
     language = {fr},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2601/}
}
TY  - JOUR
AU  - Trotabas, Denis
TI  - Non annulation des fonctions $L$ des formes modulaires de Hilbert au point central
JO  - Annales de l'Institut Fourier
PY  - 2011
DA  - 2011///
SP  - 187
EP  - 259
VL  - 61
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2601/
UR  - https://www.ams.org/mathscinet-getitem?mr=2828130
UR  - https://doi.org/10.5802/aif.2601
DO  - 10.5802/aif.2601
LA  - fr
ID  - AIF_2011__61_1_187_0
ER  - 
%0 Journal Article
%A Trotabas, Denis
%T Non annulation des fonctions $L$ des formes modulaires de Hilbert au point central
%J Annales de l'Institut Fourier
%D 2011
%P 187-259
%V 61
%N 1
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2601
%R 10.5802/aif.2601
%G fr
%F AIF_2011__61_1_187_0
Trotabas, Denis. 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/articles/10.5802/aif.2601/

[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., Volume 8 (1998), pp. 627-655 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

[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 | Zbl

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

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

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

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

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

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

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

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

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

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

[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, Volume 51 (1984), pp. 355-394 | DOI | MR | Zbl

[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, Volume 97 (1999), pp. 545-577 | DOI | MR | Zbl

[25] VanderKam, J. Linear independence in the homology of X 0 (N), Journal London Math. Soc., Volume 61 (2000), pp. 349-358 | DOI | MR | Zbl

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

Cited by Sources: