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On the Curves Associated to Certain Rings of Automorphic Forms
- Source :
- Canadian Journal of Mathematics. 53:98-121
- Publication Year :
- 2001
- Publisher :
- Canadian Mathematical Society, 2001.
-
Abstract
- In a 1987 paper, Gross introduced certain curves associated to a definite quaternion algebra B over Q; he then proved an analog of his result with Zagier for these curves. In Gross’ paper, the curves were defined in a somewhat ad hoc manner. In this article, we present an interpretation of these curves as projective varieties arising from graded rings of automorphic forms on B×, analogously to the construction in the Satake compactification. To define such graded rings, one needs to introduce a “multiplication” of automorphic forms that arises from the representation ring of B×. The resulting curves are unions of projective lines equipped with a collection of Hecke correspondences. They parametrize two-dimensional complex tori with quaternionic multiplication. In general, these complex tori are not abelian varieties; they are algebraic precisely when they correspond to CM points on these curves, and are thus isogenous to a product E × E, where E is an elliptic curve with complex multiplication. For these CM points one can make a relation between the action of the p-th Hecke operator and Frobenius at p, similar to the well-known congruence relation of Eichler and Shimura.
- Subjects :
- Pure mathematics
Quaternion algebra
General Mathematics
010102 general mathematics
Automorphic form
Complex multiplication
Congruence relation
01 natural sciences
Algebra
Elliptic curve
0103 physical sciences
Representation ring
010307 mathematical physics
Compactification (mathematics)
0101 mathematics
Hecke operator
Mathematics
Subjects
Details
- ISSN :
- 14964279 and 0008414X
- Volume :
- 53
- Database :
- OpenAIRE
- Journal :
- Canadian Journal of Mathematics
- Accession number :
- edsair.doi...........8993b4a55009c2d6c3f3bbd6be5e4b2e