On the conductor of cohomological transforms

Fouvry, Étienne; Kowalski, Emmanuel; Michel, Philippe

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 30 (2021) no. 1, pp. 203-254.

Dans l’étude analytique des fonctions traces de faisceaux $\ell$–adiques sur les corps finis, un problème crucial est de contrôler la taille du conducteur de faisceaux construits de façons variées. Nous considérons les transformées cohomologiques sur la droite affine sur un corps fini dont les fonctions traces sont données par des opérateurs linéaires dont la matrice est un caractère additif évalué sur une fonction rationnelle en deux variables. Nous prouvons que le conducteur de telles transformées est borné en fonction du conducteur du faisceau de départ et de la fraction rationnelle définissant le noyau. Enfin nous présentons des applications et des exemples, en particulier des exemples provenant du projet Polymath8.

In the analytic study of trace functions of $\ell$-adic sheaves over finite fields, a crucial issue is to control the conductor of sheaves constructed in various ways. We consider cohomological transforms on the affine line over a finite field which have trace functions given by linear operators with an additive character of a rational function in two variables as a kernel. We prove that the conductor of such transforms is bounded in terms of the complexity of the input sheaf and of the rational function defining the kernel, and discuss applications of this result, including motivating examples arising from the Polymath8 project.

Classification : 11T23, 14F20, 11G20
Mots clés : Étale cohomology, conductor, $\ell $-adic sheaves, Riemann Hypothesis over finite fields, exponential sums

[1] Conrey, John B.; Iwaniec, Henryk The cubic moment of central values of automorphic $L$-functions, Ann. Math., Volume 151 (2000) no. 3, pp. 1175-1216 | DOI | MR | Zbl

[2] Deligne, Pierre Cohomologie étale, Lecture Notes in Mathematics, 569, Springer, 1977 (S.G.A, $4\frac{1}{2}$)

[3] Deligne, Pierre La conjecture de Weil. II, Publ. Math., Inst. Hautes Étud. Sci., Volume 52 (1980), pp. 137-252 | DOI | Numdam

[4] Fouvry, Étienne; Kowalski, Emmanuel; Michel, Philippe Algebraic trace weights over the primes, Duke Math. J., Volume 163 (2014) no. 9, pp. 1683-1736

[5] Fouvry, Étienne; Kowalski, Emmanuel; Michel, Philippe Algebraic twists of modular forms and Hecke orbits, Geom. Funct. Anal., Volume 25 (2015) no. 2, pp. 580-657 | DOI | MR | Zbl

[6] Fouvry, Étienne; Kowalski, Emmanuel; Michel, Philippe On the exponent of distribution of the ternary divisor function, Mathematika, Volume 61 (2015) no. 1, pp. 121-144 | DOI | MR | Zbl

[7] Fu, Lei Etale cohomology theory, Nankai Tracts in Mathematics, 13, World Scientific, 2011 | MR | Zbl

[8] Katz, Nicholas M. Sommes exponentielles, Astérisque, 79, Société Mathématique de France, 1980 | Numdam | MR | Zbl

[9] Katz, Nicholas M. Gauss sums, Kloosterman sums, and monodromy groups, Annals of Mathematics Studies, Princeton University Press, 1988 no. 116 | Zbl

[10] Katz, Nicholas M. Exponential Sums and Differential Equations, Annals of Mathematics Studies, 124, Princeton University Press, 1990 | MR | Zbl

[11] Katz, Nicholas M. Affine cohomological transforms, perversity, and monodromy, J. Am. Math. Soc., Volume 6 (1993) no. 1, pp. 149-222 | DOI | MR | Zbl

[12] Katz, Nicholas M. Rigid local systems, Annals of Mathematics Studies, 139, Princeton University Press, 1996 | MR | Zbl

[13] Katz, Nicholas M. L-functions and monodromy: four lectures on Weil II, Adv. Math., Volume 160 (2001) no. 1, pp. 81-132 | DOI | MR | Zbl

[14] Katz, Nicholas M. Sums of Betti numbers in arbitrary characteristic, Finite Fields Appl., Volume 7 (2001) no. 1, pp. 29-44 | DOI | MR | Zbl

[15] Kowalski, Emmanuel; Michel, Philippe; Sawin, Will Bilinear forms with Kloosterman sums and applications, Ann. Math.(2), Volume 186 (2017) no. 2, pp. 413-500 | DOI | MR | Zbl

[16] Laumon, Gérard Semi-continuité du conducteur de Swan (d’après P. Deligne), Caractéristique d’Euler-Poincaré (Astérisque), Volume 83, Société Mathématique de France, 1981, pp. 173-219 | Numdam | Zbl

[17] Laumon, Gérard Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil, Publ. Math., Inst. Hautes Étud. Sci., Volume 65 (1987), pp. 131-210 | DOI | Zbl

[18] Milne, James S. Étale cohomology, Princeton Mathematical Series, 33, Princeton University Press, 1980 | MR | Zbl

[19] Petrow, Ian; Young, Matthew P. The Weyl bound for Dirichlet $L$-functions of cube-free conductor, Ann. Math., Volume 192 (2020) no. 2, pp. 437-486 | DOI | MR | Zbl

[20] Polymath, G. H. J. New equidistribution estimates of Zhang type, Algebra Number Theory, Volume 8 (2014) no. 9, pp. 2067-2199 | MR

[21] Rotman, Joseph J. An introduction to homological algebra, Springer Universitext, 2009 | Zbl

[22] Sawin, Will Quantitative sheaf theory (drafted by A. Forey, J. Fresán and E. Kowalski, https://arxiv.org/abs/2101.00635)