On Elliptic Curves in SL_2(C)/Γ, Schanuel's conjecture and geodesic lengths

Details On Elliptic Curves in SL_2(C)/Γ, Schanuel's conjecture and geodesic lengths On Elliptic Curves in SL_2(C)/Γ, Schanuel's conjecture and geodesic lengths

2002-04-15

arxiv.org/abs/math/0204195v3

Mathematics

Algebraic Geometry

20 pages; LaTeX; lemma 2 corrected, some minor improvements in presentation

Scientific paper

Let H be a discrete cocompact subgroup of SL_2(C). We conjecture that the quotient manifold X=SL_2(C)/H contains infinitely many non-isogeneous elliptic curves and prove that this is indeed the case if Schanuel's conjecture holds. We also prove it in the special case where the intersection of H and SL_2(R) is cocompact in SL_2(R). Furthermore, we deduce some consequences for the geodesic length spectra of real hyperbolic 2- and 3-folds.

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