Genuine Bianchi modular forms of higher level at varying weight and discriminant
Alexander D. Rahm; Panagiotis Tsaknias
Journal de Théorie des Nombres de Bordeaux, Volume 31 (2019) no. 1, p. 27-48

Bianchi modular forms are automorphic forms over an imaginary quadratic field, associated to a Bianchi group. Those of the cuspidal Bianchi modular forms which are relatively well understood, namely (twists of) base-change forms and CM-forms, are what we call non-genuine forms; the remaining forms are what we call genuine. In a preceding paper by Rahm and Şengün, an extreme paucity of genuine forms has been reported, but those and other computations were restricted to level One. In this paper, we are extending the formulas for the non-genuine Bianchi modular forms to higher levels, and we are able to spot the first, rare instances of genuine forms at higher level and higher weight.

Les formes modulaires de Bianchi sont des formes automorphes sur un corps quadratique imaginaire associées à un groupe de Bianchi. Nous appelons formes non génuines les formes de Bianchi qu’on connait relativement bien, c’est-à-dire les (twists des) formes obtenues par changement de base et les formes CM. Les autres formes sont appelées génuines. Dans un précédent article de Rahm et Şengün, il a été constaté que les formes génuines sont extrêmement rares, mais ces calculs ont été restreints au cas des groupes de Bianchi entiers. Dans ce travail, nous généralisons les formules pour les formes de Bianchi non génuines au cas de niveau supérieur, et nous sommes capables d’observer les premiers, rares exemples de formes génuines de niveau et poids supérieurs.

Received : 2017-12-15
Revised : 2018-08-28
Accepted : 2018-10-23
Published online : 2019-07-29
DOI : https://doi.org/10.5802/jtnb.1067
Classification:  11F55,  11F75
Keywords: Bianchi modular forms; Bianchi groups; (twists of) base-change forms; CM-forms
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     author = {Alexander D. Rahm and Panagiotis Tsaknias},
     title = {Genuine Bianchi modular forms of higher level  at varying weight and discriminant},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {1},
     year = {2019},
     pages = {27-48},
     doi = {10.5802/jtnb.1067},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_2019__31_1_27_0}
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Rahm, Alexander D.; Tsaknias, Panagiotis. Genuine Bianchi modular forms of higher level  at varying weight and discriminant. Journal de Théorie des Nombres de Bordeaux, Volume 31 (2019) no. 1, pp. 27-48. doi : 10.5802/jtnb.1067. jtnb.centre-mersenne.org/item/JTNB_2019__31_1_27_0/

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