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

  • Noam D. Elkies
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4076)


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 \({{\cal X}_0}(3)\), \({{\cal X}_1}(3)\) associated with the congruence subgroups Γ0(3), Γ1(3). Since the rational prime 3 is inert in K, the covering \({{\cal X}_0}(3)/{{\cal X}(1)}\) has degree 28, and its Galois closure \({\cal X}(3)/{{\cal X}(1)}\) has geometric Galois group PSL2(F27). Since \({{\cal X}(1)}\) is rational, the covering \({{\cal X}_0}(3)/{{\cal X}(1)}\) amounts to a rational map of degree 28. We compute this rational map explicitly. We find that \({{\cal X}_0}(3)\) is an elliptic curve of conductor 147=3Open image in new window72 over Q, as is the Jacobian \({{\cal J}_1}(3)\) of \({{\cal X}_1}(3)\); that these curves are related by an isogeny of degree 13; and that the kernel of the 13-isogeny from \({{\cal J}_1}(3)\) to \({{\cal X}_0}(3)\) consists of K-rational points. We also use the map \({{\cal X}_0}(3) \rightarrow {{{\cal X}}(1)}\) to locate some complex multiplication (CM) points on \({{\cal X}(1)}\). We conclude by describing analogous behavior of a few Shimura curves associated with quaternion algebras over other cyclic cubic fields.


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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Noam D. Elkies
    • 1
  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA

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