Singular moduli of Shimura curves
by Errthum, Eric Francis, Ph.D., UNIVERSITY OF MARYLAND, COLLEGE PARK, 2007, 140 pages; 3260361

Abstract:

The j-function acts as a parametrization of the classical modular curve. Its values at complex multiplication (CM) points are called singular moduli and are algebraic integers. A Shimura curve is a generalization of the modular curve and, if the Shimura curve has genus 0, a rational parameterizing function evaluated at a CM point is again algebraic over [special characters omitted]. This thesis shows that the coordinate maps given in [5] for the Shimura curves associated to the quaternion algebras with discriminants 6 and 10 are Borcherds lifts of vector-valued modular forms. This property is then used to explicitly compute the rational norms of singular moduli on these curves. This method not only verifies the conjectural values for the rational CM points listed in [5], but also provides a way of algebraically calculating the norms of CM points with arbitrarily large negative discriminant.
 
AdviserStephen S. Kudla
SchoolUNIVERSITY OF MARYLAND, COLLEGE PARK
SourceDAI/B 68-04, p. , Aug 2007
Source TypeDissertation
SubjectsMathematics
Publication Number3260361
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