dc.contributor.author | Elkies, Noam | |
dc.date.accessioned | 2013-09-13T17:08:07Z | |
dc.date.issued | 2006 | |
dc.identifier.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. | en_US |
dc.identifier.isbn | 9783540360759 | en_US |
dc.identifier.isbn | 3540360751 | en_US |
dc.identifier.issn | 1611-3349 | en_US |
dc.identifier.issn | 0302-9743 | en_US |
dc.identifier.uri | http://nrs.harvard.edu/urn-3:HUL.InstRepos:11022283 | |
dc.description.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. | en_US |
dc.description.sponsorship | Mathematics | en_US |
dc.language.iso | en_US | en_US |
dc.publisher | Springer Verlag | en_US |
dc.relation.isversionof | http://dx.doi.org/10.1007/11792086_22 | en_US |
dash.license | LAA | |
dc.subject | algebraic geometry | en_US |
dc.subject | number theory | en_US |
dc.title | Shimura Curves for Level-3 Subgroups of the (2,3,7) Triangle Group and Some Other Examples | en_US |
dc.type | Monograph or Book | en_US |
dc.description.version | Author's Original | en_US |
dc.relation.journal | Lecture Notes in Computer Science | en_US |
dash.depositing.author | Elkies, Noam | |
dc.date.available | 2013-09-13T17:08:07Z | |
dc.identifier.doi | 10.1007/11792086_22 | * |
dash.contributor.affiliated | Elkies, Noam | |