Near Counterexamples to Weil’s Converse Theorem
Raphael S. Steiner1
1 FH317 Institute for Advanced Study 1 Einstein Drive Princeton, New Jersey 08540, USA
Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 2, pp. 313-321.

We show that in Weil’s converse theorem the functional equations of multiplicative twists of the first p-24 3 moduli are not sufficient to conclude modularity for the group Γ 0 (p), where p is a prime number.

Nous démontrons que dans le théorème réciproque de Weil les équations fonctionnelles pour les caractères de Dirichlet modulo ≤p-24 3 ne suffisent pas à assurer la modularité relativement au groupe Γ 0 (p), où p est un nombre premier.

Received:
Accepted:
Published online:
DOI: 10.5802/jtnb.1082
Classification: 11F66, 11F11
Keywords: Converse Theorem, Modular Forms, Multiplier System

Raphael S. Steiner 1

1 FH317 Institute for Advanced Study 1 Einstein Drive Princeton, New Jersey 08540, USA
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Raphael S. Steiner. Near Counterexamples to Weil’s Converse Theorem. Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 2, pp. 313-321. doi : 10.5802/jtnb.1082. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1082/

[1] Salomon Bochner Some properties of modular relations, Ann. Math., Volume 53 (1951), pp. 332-363 | DOI | MR | Zbl

[2] Erich Hecke Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung, Math. Ann., Volume 112 (1936) no. 1, pp. 664-699 | DOI | MR | Zbl

[3] Ha Huy Khoai Sur les séries L associées aux formes modularies, Bull. Soc. Math. Fr., Volume 120 (1992) no. 1, pp. 1-13 | DOI | MR | Zbl

[4] Jacobus Hendricus van Lint On the multiplier system of the Riemann–Dedekind function η, Nederl. Akad. Wet., Proc., Ser. A, Volume 61 (1958), pp. 522-527 | Zbl

[5] Hans Rademacher Über die Erzeugenden von Kongruenzuntergruppen der Modulgruppe, Abh. Math. Semin. Univ. Hamb., Volume 7 (1929) no. 1, pp. 134-148 | DOI | Zbl

[6] Robert Alexander Rankin Modular forms and functions, Cambridge University Press, 1977 | Zbl

[7] André Weil Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann., Volume 168 (1967), pp. 149-156 | DOI | Zbl

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