On automorphic $L$-functions in positive characteristic

Lomelí, Luis Alberto

Annales de l'Institut Fourier, Tome 66 (2016) no. 5, pp. 1733-1771.

Nous donnons une preuve de l’existence des fonctions $L$ locales d’Asai, extérieur et symétrique carré ainsi que des facteurs $\gamma$ et $\epsilon$ correspondants en caractéristique $p$ – le cas $p=2$ étant inclus. Notre étude est possible grâce à la méthode de Langlands-Shahidi sur un corps de fonctions global dans le cas d’un sous-groupe de Siegel Levi d’un groupe classique déployé ou d’un groupe unitaire quasi-déployé. Les fonctions $L$ qui en résultent satisfont une propriété de rationalité et une équation fonctionnelle. Un résultat d’unicité de G. Henniart et de l’auteur permet de montrer que les définitions données dans cet article sont compatibles avec la conjecture de Langlands locale pour ${\mathrm{GL}}_n$. De plus, afin d’être complet, nous décrivons les fonctions $L$ provenant d’un sous-groupe maximal de Levi d’un groupe linéaire général.

We give a proof of the existence of Asai, exterior square, and symmetric square local $L$-functions, $\gamma$-factors and root numbers in characteristic $p$ – the case of $p=2$ included. Our study is made possible by developing the Langlands-Shahidi method over a global function field in the case of a Siegel Levi subgroup of a split classical group or a quasi-split unitary group. The resulting automorphic $L$-functions are shown to satisfy a rationality property and a functional equation. A uniqueness result of G. Henniart and the author allows us to show that the definitions provided in this article are in accordance with the local Langlands conjecture for ${\mathrm{GL}}_n$. Furthermore, in order to be self contained, we include a treatise of $L$-functions arising from maximal Levi subgroups of general linear groups.

Classification : 11F70, 11M38, 22E50, 22E55
Keywords: Automorphic $L$-funcitons, functional equation, Langlands-Shahidi method, local factors
Mot clés : Fonctions $L$-automorphes, équation fonctionnelle, méthode Langlands-Shahidi, facteurs locaux

[1] Borel, A. Automorphic $L$-functions, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2 (Proc. Sympos. Pure Math., XXXIII), Amer. Math. Soc., Providence, R.I., 1979, pp. 27-61

[2] Casselman, W.; Shalika, J. The unramified principal series of $p$-adic groups. II. The Whittaker function, Compositio Math., Volume 41 (1980) no. 2, pp. 207-231

[3] Cogdell, J. W.; Kim, H. H.; Piatetski-Shapiro, I. I.; Shahidi, F. Functoriality for the classical groups, Publ. Math. Inst. Hautes Études Sci. (2004) no. 99, pp. 163-233 | DOI

[4] Conrad, Brian Reductive group schemes, Autour des schémas en groupes. Vol. I (Panor. Synthèses), Volume 42/43, Soc. Math. France, Paris, 2014, pp. 93-444 (SGA3 Summer School, Luminy, August 29–September 9, 2011)

[5] Curtis, Charles W.; Reiner, Irving Methods of representation theory. Vol. I, John Wiley & Sons, Inc., New York, 1981, xxi+819 pages (With applications to finite groups and orders, Pure and Applied Mathematics, A Wiley-Interscience Publication)

[6] Deligne, P. Les constantes des équations fonctionnelles des fonctions $L$, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) (Lecture Notes in Math.), Volume 349, Springer, Berlin, 1973, pp. 501-597

[7] Godement, Roger; Jacquet, Hervé Zeta functions of simple algebras, Lecture Notes in Mathematics, Vol. 260, Springer-Verlag, Berlin-New York, 1972, ix+188 pages

[8] Harder, G. Chevalley groups over function fields and automorphic forms, Ann. of Math. (2), Volume 100 (1974), pp. 249-306 | DOI

[9] Henniart, Guy; Lomelí, Luis Local-to-global extensions for ${\mathrm{GL}}_n$ in non-zero characteristic: a characterization of ${\gamma }_F(s,\pi ,{\mathrm{Sym}}^2,\psi )$ and ${\gamma }_F(s,\pi ,{\wedge }^2,\psi )$, Amer. J. Math., Volume 133 (2011) no. 1, pp. 187-196 | DOI

[10] Henniart, Guy; Lomelí, Luis Characterization of $\gamma$-factors: the Asai case, Int. Math. Res. Not. IMRN (2013) no. 17, pp. 4085-4099

[11] Henniart, Guy; Lomelí, Luis Uniqueness of Rankin-Selberg products, J. Number Theory, Volume 133 (2013) no. 12, pp. 4024-4035 | DOI

[12] Jacquet, Hervé Principal $L$-functions of the linear group, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2 (Proc. Sympos. Pure Math., XXXIII), Amer. Math. Soc., Providence, R.I., 1979, pp. 63-86

[13] Jacquet, Hervé; Piatetskii-Shapiro, I. I.; Shalika, J. A. Rankin-Selberg convolutions, Amer. J. Math., Volume 105 (1983) no. 2, pp. 367-464 | DOI

[14] Jacquet, Hervé; Shalika, Joseph A lemma on highly ramified $ϵ$-factors, Math. Ann., Volume 271 (1985) no. 3, pp. 319-332 | DOI

[15] Keys, C. David; Shahidi, Freydoon Artin $L$-functions and normalization of intertwining operators, Ann. Sci. École Norm. Sup. (4), Volume 21 (1988) no. 1, pp. 67-89

[16] Lafforgue, Laurent Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math., Volume 147 (2002) no. 1, pp. 1-241 | DOI

[17] Langlands, R. P. On the functional equation of Artin’s $L$-functions (mimeographed notes, Yale University, https://publications.ias.edu/sites/default/files/a-ps.pdf)

[18] Laumon, G.; Rapoport, M.; Stuhler, U. $𝒟$-elliptic sheaves and the Langlands correspondence, Invent. Math., Volume 113 (1993) no. 2, pp. 217-338 | DOI

[19] Lomelí, Luis Alberto The Ramanujan Conjecture and the Riemann Hypothesis for the Unitary groups over function fields (preprint)

[20] Lomelí, Luis Alberto Functoriality for the classical groups over function fields, Int. Math. Res. Not. IMRN (2009) no. 22, pp. 4271-4335 | DOI

[21] Lomelí, Luis Alberto The $\mathrm{ℒ𝒮}$ method for the classical groups in positive characteristic and the Riemann hypothesis, Amer. J. Math., Volume 137 (2015) no. 2, pp. 473-496 | DOI

[22] Mœglin, Colette; Waldspurger, Jean-Loup Décomposition spectrale et séries d’Eisenstein, Progress in Mathematics, 113, Birkhäuser Verlag, Basel, 1994, xxx+342 pages (Une paraphrase de l’Écriture. [A paraphrase of Scripture])

[23] Morris, L. E. Eisenstein series for reductive groups over global function fields. I and II, Canad. J. Math., Volume 34 (1982) no. 1 and 5, p. 91-168 and 1112–1182 | DOI

[24] Shahidi, Freydoon On certain $L$-functions, Amer. J. Math., Volume 103 (1981) no. 2, pp. 297-355 | DOI

[25] Shahidi, Freydoon Fourier transforms of intertwining operators and Plancherel measures for $\mathrm{GL}\left(n\right)$, Amer. J. Math., Volume 106 (1984) no. 1, pp. 67-111 | DOI

[26] Shahidi, Freydoon A proof of Langlands’ conjecture on Plancherel measures; complementary series for $p$-adic groups, Ann. of Math. (2), Volume 132 (1990) no. 2, pp. 273-330 | DOI

[27] Shahidi, Freydoon Local coefficients as Mellin transforms of Bessel functions: towards a general stability, Int. Math. Res. Not. (2002) no. 39, pp. 2075-2119 | DOI

[28] Silberger, Allan J. Introduction to harmonic analysis on reductive $p$-adic groups, Mathematical Notes, 23, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1979, iv+371 pages (Based on lectures by Harish-Chandra at the Institute for Advanced Study, 1971–1973)

[29] Waldspurger, J.-L. La formule de Plancherel pour les groupes $p$-adiques (d’après Harish-Chandra), J. Inst. Math. Jussieu, Volume 2 (2003) no. 2, pp. 235-333 | DOI

[30] Zelevinsky, A. V. Induced representations of reductive $𝔭$-adic groups. II. On irreducible representations of $\mathrm{GL}\left(n\right)$, Ann. Sci. École Norm. Sup. (4), Volume 13 (1980) no. 2, pp. 165-210