Comptes Rendus
no. 8
A note on GL2 converse theorems
[Une note sur les théorèmes inverses de GL2]
A. Diaconu1 ; A. Perelli2 ; A. Zaharescu3
1 Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027, USA
2 Dipartimento di Matematica, Via Dodecaneso 35, 16146 Genova, Italy
3 Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801, USA
Comptes Rendus. Mathématique, Volume 334 (2002) no. 8, pp. 621-624.

Le théorème bien connu de Weil montre que les formes modulaires f∈ℳk(Γ0(q)) sont caractérisées par l'équation fonctionnelle des fonctions L tordues attachées à f. Conrey–Farmer ont partiellement réussi à remplacer cette hypothèse par celle où Lf(s) a un produit eulérien. Dans cette Note, on obtient une version hybride des résultats de Weil et de Conrey–Farmer, en prouvant un théorème inverse pour tout q⩾1, sous l'hypothèse d'un produit eulérien et de l'équation fonctionnelle pour les fonctions L tordues par rapport à un seul module.

Weil's well-known converse theorem shows that modular forms f∈ℳk(Γ0(q)) are characterized by the functional equation for twists of Lf(s). Conrey–Farmer had partial success at replacing the assumption on twists by the assumption of Lf(s) having an Euler product of the appropriate form. In this Note we obtain a hybrid version of Weil's and Conrey–Farmer's results, by proving a converse theorem for all q⩾1 under the assumption of the Euler product and, moreover, of the functional equation for the twists to a single modulus.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02277-X

A. Diaconu 1 ; A. Perelli 2 ; A. Zaharescu 3

1 Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027, USA
2 Dipartimento di Matematica, Via Dodecaneso 35, 16146 Genova, Italy
3 Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801, USA 
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A. Diaconu; A. Perelli; A. Zaharescu. A note on GL2 converse theorems. Comptes Rendus. Mathématique, Volume 334 (2002) no. 8, pp. 621-624. doi : 10.1016/S1631-073X(02)02277-X. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02277-X/

[1] J.B. Conrey; D.W. Farmer An extension of Hecke's converse theorem, Internat. Math. Res. Notices (1995), pp. 445-463

[2] H. Halberstam; H.-E. Richert Sieve Methods, Academic Press, 1974

[3] D.R. Heath-Brown Zero-free regions for Dirichlet L-functions, and the least prime in an arithmetic progression, Proc. London Math. Soc. (3), Volume 64 (1992), pp. 265-338

[4] H. Iwaniec Topics in Classical Automorphic Forms, American Mathematical Society, 1997

[5] A. Weil Über die Bestimmung Dirichletscher Reihen durch Funktionengleichungen, Math. Ann., Volume 168 (1967), pp. 149-156

  • Benjamin Bedert; George Cooper; Thomas Oliver; Pengcheng Zhang Twisting moduli for GL(2), Journal of Number Theory, Volume 217 (2020), p. 142 | DOI:10.1016/j.jnt.2020.04.008
  • Andrew R. Booker A CONVERSE THEOREM WITHOUT ROOT NUMBERS, Mathematika, Volume 65 (2019) no. 4, p. 862 | DOI:10.1112/s0025579319000184
  • Sandro Bettin; Jonathan W. Bober; Andrew R. Booker; Brian Conrey; Min Lee; Giuseppe Molteni; Thomas Oliver; David J. Platt; Raphael S. Steiner A conjectural extension of Hecke’s converse theorem, The Ramanujan Journal, Volume 47 (2018) no. 3, p. 659 | DOI:10.1007/s11139-017-9953-y
  • Alberto Perelli Converse theorems: from the Riemann zeta function to the Selberg class, Bollettino dell'Unione Matematica Italiana, Volume 10 (2017) no. 1, p. 29 | DOI:10.1007/s40574-016-0085-x
  • Andrew R. Booker; M. Krishnamurthy Weil's Converse Theorem with Poles, International Mathematics Research Notices, Volume 2014 (2014) no. 19, p. 5328 | DOI:10.1093/imrn/rnt127
  • Władysław Narkiewicz The Last Period, Rational Number Theory in the 20th Century (2012), p. 307 | DOI:10.1007/978-0-85729-532-3_6
  • David W. Farmer; Kevin Wilson Converse theorems assuming a partial euler product, The Ramanujan Journal, Volume 15 (2008) no. 2, p. 205 | DOI:10.1007/s11139-007-9073-1

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