Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-13T05:07:14.600Z Has data issue: false hasContentIssue false
  • English
  • Français

On level one cuspidal Bianchi modular forms

Part of: Discontinuous groups and automorphic forms Homology and homotopy of topological groups and related structures

Published online by Cambridge University Press:  01 August 2013

Alexander D. Rahm  and
Mehmet Haluk Şengün
Alexander D. Rahm
Affiliation:
Department of Mathematics,National University of Ireland at Galway, Ireland email Alexander.Rahm@nuigalway.ie
Mehmet Haluk Şengün
Affiliation:
Mathematics Institute, University of Warwick,Coventry CV4 7AL, United Kingdom email M.H.Sengun@warwick.ac.uk

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we present the outcome of vast computer calculations, locating several of the very rare instances of level one cuspidal Bianchi modular forms that are not lifts of elliptic modular forms.

MSC classification

Secondary: 11F55: Other groups and their modular and automorphic forms (several variables) 11F75: Cohomology of arithmetic groups 57T15: Homology and cohomology of homogeneous spaces of Lie groups
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 16 , October 2013 , pp. 187 - 199
Copyright
© The Author(s) 2013 

References

Aranés, M. T., ‘Modular symbols over number fields’, PhD Thesis, University of Warwick, 2010.Google Scholar
Ash, A. and Pollack, D., ‘Everywhere unramified automorphic cohomology for $\mathrm{SL} \_ 3( \mathbb{Z} )$ ’, Int. J. Number Theory 4 (2008) no. 4, 663–675.CrossRefGoogle Scholar
Ash, A. and Stevens, G, ‘Cohomology of arithmetic groups and congruences between systems of Hecke eigenvalues’, J. Reine Angew. Math. 365 (1986) 192–220.Google Scholar
Berger, T., ‘Denominators of Eisenstein cohomology classes for $\mathrm{GL} \_ 2$ over imaginary quadratic fields’, Manuscripta Math. 125 (2008) no. 4, 427–470.CrossRefGoogle Scholar
Bianchi, L., ‘Sui gruppi di sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginarî’, Math. Ann. 40 (1892) no. 3, 332–412.CrossRefGoogle Scholar
Brown, K. S., Cohomology of groups, Graduate Texts in Mathematics 87 (Springer, Berlin, 1982).Google Scholar
Bygott, J., ‘Modular forms and modular symbols over imaginary quadratic fields’, PhD Thesis, University of Exeter, 1998.Google Scholar
Calegari, F. and Mazur, B, ‘Nearly ordinary Galois deformations over arbitrary number fields’, J. Inst. Math. Jussieu 8 (2009) no. 1, 99–177.Google Scholar
Cremona, J. E., ‘Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields’, Compositio Math. 51 (1984) no. 3, 275–324.Google Scholar
Cremona, J. E., ‘Abelian varieties with extra twist, cusp forms, and elliptic curves over imaginary quadratic fields’, J. Lond. Math. Soc. (2) 45 (1992) no. 3, 404–416.Google Scholar
Cremona, J. E. and Lingham, M. P., ‘Finding all elliptic curves with good reduction outside a given set of primes’, Exp. Math. 16 (2007) no. 3, 303–312.CrossRefGoogle Scholar
Doi, K., Hida, H. and Ishii, H., ‘Discriminant of Hecke fields and twisted adjoint $L$ -values for $\mathrm{GL} (2)$ ’, Invent. Math. 134 (1998) no. 3, 547–577.Google Scholar
Elstrodt, J., Grunewald, F. and Mennicke, J., ‘On the group ${\mathrm{PSL} }_{2} ( \mathbb{Z} [i] )$ ’, Number theory days, Exeter, 1980, London Mathematical Society Lecture Note Series 56 (Cambridge University Press, Cambridge, 1982) 255–283.Google Scholar
Finis, T., Grunewald, F. and Tirao, P., ‘The cohomology of lattices in $\mathrm{SL} (2, \mathbb{C} )$ ’, Exp. Math. 19 (2010) no. 1, 29–63.Google Scholar
Flöge, D., ‘Zur Struktur der ${\mathrm{PSL} }_{2} $ über einigen imaginär-quadratischen Zahlringen’, Dissertation, Johann-Wolfgang-Goethe-Universität, Fachbereich Mathematik, 1980.Google Scholar
Flöge, D., ‘Zur Struktur der ${\mathrm{PSL} }_{2} $ über einigen imaginär-quadratischen Zahlringen’, Math. Z. 183 (1983) no. 2, 255–279.Google Scholar
Harder, G., ‘On the cohomology of discrete arithmetically defined groups’, Discrete subgroups of Lie groups and applications to moduli, Internat. Colloq., Bombay, 1973 (Oxford University Press, Oxford, 1975) 129–160.Google Scholar
Humbert, G., ‘Sur la réduction des formes d’Hermite dans un corps quadratique imaginaire’, C. R. Acad. Sci. Paris 16 (1915) 189–196.Google Scholar
Krämer, N., ‘Beiträge zur Arithmetik imaginärquadratischer Zahlkörper’, PhD Thesis, Math.-Naturwiss. Fakultät der Rheinischen Friedrich–Wilhelms-Universität Bonn; Bonn. Math. Schr., 1984.Google Scholar
Lingham, M., ‘Modular forms and elliptic curves over imaginary quadratic fields’, PhD Thesis, University of Nottingham, 2005.Google Scholar
Page, A., ‘Computing arithmetic Kleinian groups’, Preprint, 2012, arXiv:1206.0087 [math.NT].Google Scholar
Poincaré, H., ‘Mémoire sur les groupes Kleinéens’, Acta Math. 3 (1883) no. 1, 49–92.Google Scholar
Rahm, A. D., Bianchi.gp, Open source program (GNU general public license), validated by the CNRS, www.projet-plume.org/fiche/bianchigp part of the GP scripts library of Pari/GP Development Center, 2010.Google Scholar
Rahm, A. D., ‘Homology and $K$ -theory of the Bianchi groups’, C. R. Math. Acad. Sci. Paris 349 (2011) no. 11–12, 615–619.CrossRefGoogle Scholar
Rahm, A. D., ‘Higher torsion in the Abelianization of the full Bianchi groups’, LMS J. Comput. Math., accepted (2013); http://hal.archives-ouvertes.fr/hal-00721690.Google Scholar
Rahm, A. D., ‘On a question of Serre’, C. R. Math. Acad. Sci. Paris 350 (2012) no. 15–16, 741–744.Google Scholar
Rahm, A. D. and Fuchs, M., ‘The integral homology of $\mathrm{PSL} \_ 2$ of imaginary quadratic integers with non-trivial class group’, J. Pure Appl. Algebra 215 (2011) 1443–1472.Google Scholar
Scheutzow, A., ‘Computing rational cohomology and Hecke eigenvalues for Bianchi groups’, J. Number Theory 40 (1992) no. 3, 317–328.Google Scholar
Şengün, M. H., ‘On the integral cohomology of Bianchi groups’, Exp. Math. 20 (2011) no. 4, 487–505.Google Scholar
Şengün, M. H., ‘Arithmetic aspects of Bianchi groups’, Proceedings of Computations with Modular Forms, Heidelberg, 2011, to appear.Google Scholar
Serre, J.-P., ‘Le problème des groupes de congruence pour SL(2)’, Ann. of Math. (2) 92 (1970) 489–527.Google Scholar
Swan, R. G., ‘Generators and relations for certain special linear groups’, Adv. Math. 6 (1971) 1–77.CrossRefGoogle Scholar
Taylor, R., ‘Representations of Galois groups associated to modular forms’, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 Zürich, 1994 (Birkhäuser, Basel, 1995) 435–442.Google Scholar
Whitley, E., ‘Modular symbols and elliptic curves over imaginary quadratic fields’, PhD Thesis, University of Exeter, 1990.Google Scholar
Wiese, G., ‘On the faithfulness of parabolic cohomology as a Hecke module over a finite field’, J. Reine Angew. Math. 606 (2007) 79–103.Google Scholar
Yasaki, D., ‘‘Hyperbolic tessellations associated to Bianchi groups’’, Algorithmic Number Theory, Proceedings of 9th International Symposium ANTS-IX, Nancy, France, July 19–23, 2010 (Springer, Berlin, 2010).Google Scholar