Twisting eigensystems of Drinfeld Hecke eigenforms by characters
Journal de Théorie des Nombres de Bordeaux, Volume 29 (2017) no. 3, pp. 903-929.

We address some questions posed by Goss related to the modularity of Drinfeld modules of rank 1 defined over the field of rational functions in one variable with coefficients in a finite field.

For each positive characteristic valued Dirichlet character, we introduce certain projection operators on spaces of Drinfeld modular forms with character of a given weight and type such that when applied to a Hecke eigenform return a Hecke eigenform whose eigensystem has been twisted by the given Dirichlet character. Unlike the classical case, however, the effect on Goss’ u-expansions for these eigenforms (and even on Petrov’s A-expansions) is more complicated than a simple twisting of the u- (or A-) expansion coefficients by the given character.

We also introduce Eisenstein series with character for irreducible levels 𝔭 and show that they and their Fricke transforms are Hecke eigenforms with a new type of A-expansion and A-expansion in the sense of Petrov, respectively. We prove congruences between certain cuspforms in Petrov’s special family and the Eisenstein series and their Fricke transforms introduced here, and we show that in each weight there are as many linearly independent Eisenstein series with character as there are cusps for Γ 1 (𝔭).

Nous considérons des questions posées par Goss concernant la modularité des modules de Drinfeld de rang un définis sur le corps des fonctions rationnelles en une variable, avec coefficients dans un corps fini.

Pour chaque caractère de Dirichlet à valeurs dans un corps fini, nous introduisons des opérateurs de projection sur des espaces de formes modulaires de Drinfeld avec caractère, de poids et type donnés. Ces opérateurs envoient des formes propres pour les opérateurs de Hecke sur des formes propres de Hecke, dont le système de valeurs propres est tordu par le caractère de Dirichlet. À la différence du cas classique, l’effet de ces opérateurs sur les u-expansions à la Goss pour ces formes propres, et même sur les A-expansions au sens de Petrov, est plus compliqué que le simple fait de tordre les coefficients des u- (ou A-) expansions par le caractère donné.

Nous introduisons aussi des séries d’Eisenstein avec caractère, avec niveau irréductible 𝔭, et nous démontrons qu’avec leurs transformées de Fricke, elles sont des formes propres possédant un nouveau type d’A-expansion. Nous démontrons des congruences entre certaines formes paraboliques dans la famille spéciale de Petrov, et les séries d’Eisenstein et leurs transformées de Fricke introduites ici, et nous démontrons que pour tout poids, il y a autant de séries d’Eisenstein avec caractère, linéairement indépendantes, que de formes paraboliques pour Γ 1 (𝔭).

Published online:
DOI: 10.5802/jtnb.1006
Classification: 11F52
Keywords: A-expansions, twisting, congruences, Eisenstein series, Drinfeld modular forms, modularity
     author = {Rudolph Perkins},
     title = {Twisting eigensystems of {Drinfeld} {Hecke} eigenforms by characters},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {903--929},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {29},
     number = {3},
     year = {2017},
     doi = {10.5802/jtnb.1006},
     language = {en},
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Rudolph Perkins. Twisting eigensystems of Drinfeld Hecke eigenforms by characters. Journal de Théorie des Nombres de Bordeaux, Volume 29 (2017) no. 3, pp. 903-929. doi : 10.5802/jtnb.1006.

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