Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-25T04:01:42.579Z Has data issue: false hasContentIssue false
  • English
  • Français

On Elliptic Curves in SL2(ℂ)/Γ.., Schanuel’s Conjecture and Geodesic Lengths

Published online by Cambridge University Press:  22 January 2016

Jörg Winkelmann
Jörg Winkelmann*
Affiliation:
Institut Elie Cartan (Mathématiques), Université Henri PoincaréNancy 1, B.P. 239, F-54506 Vandœuvre-les-Nancy, Cedex, France. jwinkel@member.ams.org
Rights & Permissions [Opens in a new window]

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.

Let Γ be a discrete cocompact subgroup of SL2(ℂ). We conjecture that the quotient manifold X = SL2(ℂ) / Γ contains infinitely many non-isogenous elliptic curves and prove this is indeed the case if Schanuel’s conjecture holds. We also prove it in the special case where Γ ∩ SL2(∝) is cocompact in SL2(ℝ).

Furthermore, we deduce some consequences for the geodesic length spectra of real hyperbolic 2- and 3-folds.

Keywords

22E4032M1032J1753C22
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 176 , 2004 , pp. 159 - 180
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2004

References

[1] Benoist, Y., Propriétés asymptotiques des groupes lineaires (II). http://www.dma.ens.fr/~benoist/prepubli/98asymptII.ps CrossRefGoogle Scholar
[2] Elstrodt, J., Grunewald, F. and Mennicke, J. L., Groups acting on hyperbolic space: Harmonic analysis and number theory, Springer Monographs in Mathematics, Springer, 1998.Google Scholar
[3] Huckleberry, A. T. and Winkelmann, J., Subvarieties of Parallelizable manifolds, Math. Ann., 295 (1993), 469–483.CrossRefGoogle Scholar
[4] Mostow, G. D., Intersections of discrete subgroups with Cartan subgroups, J. Indian Math. Soc., 34 (1970), 203–214.Google Scholar
[5] Raghunathan, M. S., Discrete subgroups of Lie groups, Erg. Math. Grenzgeb. 68, 1972.Google Scholar
[6] Ratcliffe, J., Foundations of Hyperbolic geometry, GTM 149, Springer, New York, 1994.Google Scholar
[7] Reid, A. Isospectrality and commensurability of arithmetic hyperbolic 2- and 3-mani-folds, Duke Math. J., 65 (1992), 215–228.CrossRefGoogle Scholar
[8] Vignéras, M. F., Arithmétique des Algébres de Quaternions, Springer Lecture Notes 800, 1980.Google Scholar
[9] Winkelmann, J., Complex-analytic geometry of complex parallelizable manifolds, Memoirs de la S.M.F. 72/73, 1998.Google Scholar