Shimura Curves for Level-3 Subgroups of the (2,3,7) Triangle Group and Some Other Examples

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Shimura Curves for Level-3 Subgroups of the (2,3,7) Triangle Group and Some Other Examples

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Title: Shimura Curves for Level-3 Subgroups of the (2,3,7) Triangle Group and Some Other Examples
Author: Elkies, Noam
Citation: Elkies, Noam D. 2006. Shimura curves for level-3 subgroups of the (2,3,7) triangle group and some other examples. In Algorithmic number theory: 7th International Symposium, ANTS-VII, Berlin, Germany, July 23-28, 2006: Proceedings, ed. Florian Hess, Sebastian Pauli, Michael Pohst, 302-316. Lecture Notes in Computer Science 4076. Berlin: Springer Verlag.
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Abstract: The (2,3,7) triangle group is known to be associated with a quaternion algebra A/K ramified at two of the three real places of K=Q(cos2π/7) and unramified at all other places of K. This triangle group and its congruence subgroups thus give rise to various Shimura curves and maps between them. We study the genus-1 curves X_0(3), X_1(3) associated with the congruence subgroups Γ_0(3), Γ_1(3). Since the rational prime 3 is inert in K, the covering X_0(3)/X(1) has degree 28, and its Galois closure X(3)/X(1) has geometric Galois group PSL2(F27). Since X(1) is rational, the covering X_0(3)/X(1) amounts to a rational map of degree 28. We compute this rational map explicitly. We find that X_0(3) is an elliptic curve of conductor 147=3·72 over Q, as is the Jacobian J_1(3) of X_1(3); that these curves are related by an isogeny of degree 13; and that the kernel of the 13-isogeny from J_1(3) to X_0(3) consists of K-rational points. We also use the map X_0(3) --> X(1) to locate some complex multiplication (CM) points on X(1). We conclude by describing analogous behavior of a few Shimura curves associated with quaternion algebras over other cyclic cubic fields.
Published Version: http://dx.doi.org/10.1007/11792086_22
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Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:11022283
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