Arithmetic and geometry on triangular Shimura curves

Citation

Ji, Shujuan (1995) Arithmetic and geometry on triangular Shimura curves. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/xyjv-sv57. https://resolver.caltech.edu/CaltechETD:etd-10052007-134336

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. By a triangular Shimura curve, we mean the canonical model [...] of [...], the quotient of the upper half plane [...] by a cocompact arithmetic subgroup [...] of [...] with a triangular fundamental domain. To be concise, let F be a totally real algebraic number field of degree d, and B a quaternion algebra over F, with [...], where H is the Hamilton quaternion algebra. Let O be an order of B, and [...] = {[...] is totally positive}. A Fuchsian group [...] of the first kind is called arithmetic if it is commensurable with [...] for some B and O. Here we are only interested in the arithmetic triangular groups, i.e., those generated by three elliptic elements. If the three generators [...], [...], [...] are of order [...], [...], [...], then we call ([...], [...], [...]) its signature. Our main results are the follows: We first exhibit, for each arithmetic triangle group [...], positive integers k such that the space [...] of modular forms for [...] of weight k is 1-dimensional (cf. Theorem A, Chapter 2). Then we establish a class of modular functions on a family of coverings of triangular Shimura curve [...], satisfying some arithmetic properties analogous to those of the classical functions [...] (cf. Theorem B, Chapter 4). Finally, we provide two explicit examples and illustrate the properties proved.

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