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.

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:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1067
Classification: 11F55,  11F75
Keywords: Bianchi modular forms; Bianchi groups; (twists of) base-change forms; CM-forms
Alexander D. Rahm 1; Panagiotis Tsaknias 1

1 Mathematics Research Unit Faculté des Sciences, de la Technologie et de la Communication Université du Luxembourg, Campus Belval, Esch-sur-Alzette, Luxembourg
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Alexander D. Rahm; Panagiotis Tsaknias. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1067/

[1] Tobias Berger; Lassina Dembélé; Ariel Pacetti; Mehmet Haluk Şengün Theta lifts of Bianchi modular forms and applications to paramodularity, J. Lond. Math. Soc., Volume 92 (2015) no. 2, pp. 353-370 | Article | MR: 3404028 | Zbl: 1396.11074

[2] Nicolas Bergeron; Mehmet Haluk Şengün; Akshay Venkatesh Torsion homology growth and cycle complexity of arithmetic manifolds, Duke Math. J., Volume 165 (2016) no. 9, pp. 1629-1693 | MR: 3513571 | Zbl: 1351.11031

[3] Wieb Bosma; John Cannon; Catherine Playoust The Magma algebra system. I. The user language, J. Symb. Comput., Volume 24 (1997) no. 3–4, pp. 235-265 | Article | MR: 1484478 | Zbl: 0898.68039

[4] Wieb Bosma; Michael E. Pohst Computations with finitely generated modules over Dedekind rings, Proceedings of the International Symposium on Symbolic and Algebraic Computation (Bonn, 1991), ACM Press, 1991, pp. 151-156 | Zbl: 0930.11088

[5] Henri Cohen Hermite and Smith normal form algorithms over Dedekind domains, Math. Comput., Volume 65 (1996) no. 216, pp. 1681-1699 | Article | MR: 1361805 | Zbl: 0853.11100

[6] Henri Cohen; Joseph Oesterlé Dimensions des espaces de formes modulaires, Modular functions of one variable VI. Proceedings of an international conference (Bonn, 1976) (Lecture Notes in Mathematics) Volume 627, Springer, 1977, pp. 69-78 | Article | MR: 472703 | Zbl: 0371.10020

[7] Tobias Finis; Fritz Grunewald; Paolo Tirao The cohomology of lattices in SL(2,), Exp. Math., Volume 19 (2010) no. 1, pp. 29-63 | Article | MR: 2649984 | Zbl: 1225.11072

[8] Robert P. Langlands Base Change for GL(2), Annals of Mathematics Studies, Volume 96, Princeton University Press, 1980 | MR: 574808 | Zbl: 0444.22007

[9] Alexander D. Rahm Homology and K-theory of the Bianchi groups, C. R. Math. Acad. Sci. Paris, Volume 349 (2011) no. 11–12, pp. 615-619 | Article | MR: 2817377 | Zbl: 1228.20036

[10] Alexander D. Rahm Higher torsion in the Abelianization of the full Bianchi groups, LMS J. Comput. Math., Volume 16 (2013), pp. 344-365 | Article | MR: 3109616 | Zbl: 1328.11057

[11] Alexander D. Rahm; Mehmet Haluk Şengün On level one cuspidal Bianchi modular forms, LMS J. Comput. Math., Volume 16 (2013), pp. 187-199 | Article | MR: 3091734 | Zbl: 1294.11062

[12] Alexander D. Rahm; Mehmet Haluk Şengün; Panagiotis Tsaknias Dimension tables for spaces of Bianchi modular forms (2016) (homepage with data associated to the present paper, http://math.uni.lu/~rahm/dimensionTables/)

[13] Arthur Ranum The group of classes of congruent quadratic integers with respect to a composite ideal modulus, Trans. Am. Math. Soc., Volume 11 (1910) no. 2, pp. 172-198 | Article | MR: 1500859 | Zbl: 41.0245.01

[14] Panagiotis Tsaknias A possible generalization of Maeda’s conjecture, Computations with modular forms (Wiese Böckle, ed.) (Contributions in Mathematical and Computational Sciences) Volume 6, Springer, 2014, pp. 317-329 | Article | MR: 3381458 | Zbl: 1375.11042

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