Lifting of Modular Forms

Jitendra Bajpai

doi:10.5802/pmb.27

Lifting of Modular Forms
[Relévement de Formes Modulaires]

Jitendra Bajpai ¹

¹ Mathematisches Institut, Georg-August Universität Göttingen, D-37073 Germany.

Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 1 (2019), pp. 5-20.

Résumé
Abstract

L’existence et construction de formes modulaires vectorielles (vvmf) pour un groupe Fuchsien arbitraire $G$ et pour une représentation $ρ : G ⟶ {GL}_{d} (ℂ)$ d’image finie peut être établie en relevant des formes modulaires scalaires pour le sous-groupe d’indice fini $ker (ρ)$ de $G$ . Dans cet article, des vvmf sont explicitement construites pour tout multiplicateur admissible (représentation) $ρ$ (voir paragraphe 3 pour la définition du multiplicateur admissible). En d’autres termes, on a partiellement répondu à la question suivante : Pour quelles représentations $ρ$ d’un groupe $G$ donné, existe-t-il une vvmf avec au moins une composante non nulle ?

The existence and construction of vector-valued modular forms (vvmf) for any arbitrary Fuchsian group $G$ , for any representation $ρ : G ⟶ {GL}_{d} (ℂ)$ of finite image can be established by lifting scalar-valued modular forms of the finite index subgroup $ker (ρ)$ of $G$ . In this article vvmf are explicitly constructed for any admissible multiplier (representation) $ρ$ , see Section 3 for the definition of admissible multiplier. In other words, the following question has been partially answered: For which representations $ρ$ of a given $G$ , is there a vvmf with at least one nonzero component?

Reçu le : 2017-06-02
Publié le : 2019-10-15

DOI : 10.5802/pmb.27

Classification : 11F03, 11F55, 30F35
Mots clés : Fuchsian group, Vector-valued modular form, Induced representation

Affiliations des auteurs :

Jitendra Bajpai ¹

¹ Mathematisches Institut, Georg-August Universität Göttingen, D-37073 Germany.

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Jitendra Bajpai. Lifting of Modular Forms. Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 1 (2019), pp. 5-20. doi : 10.5802/pmb.27. https://pmb.centre-mersenne.org/articles/10.5802/pmb.27/

Bibliographie
Cité par

[1] Tom M. Apostol Modular functions and Dirichlet series in number theory, Graduate Texts in Mathematics, 41, Springer, 1976, x+198 pages | MR | Zbl

[2] A. Oliver L. Atkin; Henry P. F. Swinnerton-Dyer Modular forms on noncongruence subgroups, Combinatorics (Univ. California, 1968) (Proceedings of Symposia in Pure Mathematics), Volume XIX, American Mathematical Society, 1971, pp. 1-25 | Zbl

[3] Jitendra Bajpai On Vector Valued Automorphic Forms (2015) (Ph. D. Thesis)

[4] Peter Bantay; Terry Gannon Vector-valued modular functions for the modular group and the hypergeometric equation, Commun. Number Theory Phys., Volume 1 (2007) no. 4, pp. 651-680 | MR | Zbl

[5] Alan F. Beardon The geometry of discrete groups, Graduate Texts in Mathematics, 91, Springer, 1995, xii+337 pages (Corrected reprint of the 1983 original) | MR

[6] Jan H. Bruinier Borcherds products on O(2, $l$ ) and Chern classes of Heegner divisors, Lecture Notes in Mathematics, 1780, Springer, 2002, viii+152 pages | DOI | MR

[7] Jan H. Bruinier On the converse theorem for Borcherds products, J. Algebra, Volume 397 (2014), pp. 315-342 | DOI | MR

[8] Luca Candelori; Cameron Franc Vector-valued modular forms and the modular orbifold of elliptic curves, Int. J. Number Theory, Volume 13 (2017) no. 1, pp. 39-63 | DOI | MR

[9] A. Cappelli; Claude Itzykson; Jean-Bernard Zuber The A-D-E classification of minimal and $A_{1}^{(1)}$ conformal invariant theories, Commun. Math. Phys., Volume 113 (1987) no. 1, pp. 1-26 | MR | Zbl

[10] Chris J. Cummins Cusp forms like $Δ$ , Can. Math. Bull., Volume 52 (2009) no. 1, pp. 53-62 | DOI | Zbl

[11] Martin Eichler; Don Zagier The theory of Jacobi forms, Progress in Mathematics, 55, Birkhäuser, 1985, v+148 pages | MR

[12] Cameron Franc; Geoffrey Mason Fourier coefficients of vector-valued modular forms of dimension 2, Can. Math. Bull., Volume 57 (2014) no. 3, pp. 485-494 | DOI | MR

[13] Cameron Franc; Geoffrey Mason Three-dimensional imprimitive representations of the modular group and their associated modular forms, J. Number Theory, Volume 160 (2016), pp. 186-214 | DOI | MR

[14] Terry Gannon The theory of vector-valued modular forms for the modular group, Conformal field theory, automorphic forms and related topics (Contributions in Mathematical and Computational Sciences), Volume 8, Springer, 2014, pp. 247-286 | MR

[15] Svetlana Katok Fuchsian groups, Chicago Lectures in Mathematics, University of Chicago Press, 1992, x+175 pages | MR

[16] Marvin I. Knopp Modular functions in analytic number theory, Markham Publishing Co., 1970, x+150 pages | MR | Zbl

[17] Marvin I. Knopp; Geoffrey Mason On vector-valued modular forms and their Fourier coefficients, Acta Arith., Volume 110 (2003) no. 2, pp. 117-124

[18] Marvin I. Knopp; Geoffrey Mason Vector-valued modular forms and Poincaré series, Ill. J. Math., Volume 48 (2004) no. 4, pp. 1345-1366 | MR

[19] Chris A. Kurth; Ling Long On modular forms for some noncongruence subgroups of ${SL}_{2} (ℤ)$ , J. Number Theory, Volume 128 (2008) no. 7, pp. 1989-2009 | DOI

[20] Chris A. Kurth; Ling Long On modular forms for some noncongruence subgroups of ${SL}_{2} (ℤ)$ . II, Bull. Lond. Math. Soc., Volume 41 (2009) no. 4, pp. 589-598 | DOI | MR

[21] Wen-Ching Winnie Li; Ling Long Fourier coefficients of noncongruence cuspforms, Bull. Lond. Math. Soc., Volume 44 (2012) no. 3, pp. 591-598 | DOI | MR

[22] Christopher Marks Irreducible vector-valued modular forms of dimension less than six, Ill. J. Math., Volume 55 (2011) no. 4, pp. 1267-1297 | MR | Zbl

[23] Christopher Marks; Geoffrey Mason Structure of the module of vector-valued modular forms, J. Lond. Math. Soc., Volume 82 (2010) no. 1, pp. 32-48 | DOI | MR

[24] Geoffrey Mason On the Fourier coefficients of 2-dimensional vector-valued modular forms, Proc. Am. Math. Soc., Volume 140 (2012) no. 6, pp. 1921-1930 | DOI

[25] Samir D. Mathur; Sunil Mukhi; Ashoke Sen On the classification of rational conformal field theories, Phys. Lett. B, Volume 213 (1988) no. 3, pp. 303-308 | DOI | MR

[26] Robert A. Rankin Modular forms and functions, Cambridge University Press, 1977, xiii+384 pages | MR

[27] Hicham Saber; Abdellah Sebbar Vector-valued automorphic forms and vector bundles (2014) (https://arxiv.org/abs/1312.2992v3)

[28] A. J. Scholl Modular forms on noncongruence subgroups, Séminaire de Théorie des Nombres, Paris 1985–86 (Progress in Mathematics), Volume 71, Birkhäuser, 1987, pp. 199-206

[29] Atle Selberg On the estimation of Fourier coefficients of modular forms, Proc. Sympos. Pure Math., Vol. VIII, American Mathematical Society, 1965, pp. 1-15 | MR

[30] Goro Shimura Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, 11, Iwanami Shoten, 1971, xiv+267 pages | Zbl

[31] Alexei B. Venkov Spectral theory of automorphic functions and its applications, Mathematics and its Applications (Soviet Series), 51, Kluwer Academic Publishers, 1990, xiv+176 pages (Translated from the Russian by N. B. Lebedinskaya) | MR | Zbl

[32] Shaul Zemel On quasi-modular forms, almost holomorphic modular forms, and the vector-valued modular forms of Shimura, Ramanujan J., Volume 37 (2015) no. 1, pp. 165-180 | DOI | MR

[33] Sander P. Zwegers Mock theta functions (2002) (Ph. D. Thesis) | Zbl

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Algèbre et Théorie des Nombres