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Differential equations and expansions for quaternionic modular forms in the discriminant 6 case

Part of: Arithmetic algebraic geometry Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 December 2012

Srinath Baba  and
Håkan Granath
Srinath Baba
Affiliation:
Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montréal, Quebec, H3G 1M8, Canada (email: sbaba1972@gmail.com)
Håkan Granath
Affiliation:
Department of Mathematics, Karlstad University, 65188 Karlstad, Sweden (email: hakan.granath@kau.se)

Abstract

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We study the differential structure of the ring of modular forms for the unit group of the quaternion algebra over ℚ of discriminant 6. Using these results we give an explicit formula for Taylor expansions of the modular forms at the elliptic points. Using appropriate normalizations we show that the Taylor coefficients at the elliptic points of the generators of the ring of modular forms are all rational and 6-integral. This gives a rational structure on the ring of modular forms. We give a recursive formula for computing the Taylor coefficients of modular forms at elliptic points and, as an application, give an algorithm for computing modular polynomials.

MSC classification

Secondary: 11F03: Modular and automorphic functions 11G18: Arithmetic aspects of modular and Shimura varieties
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 15 , May 2012 , pp. 385 - 399
Copyright
© The Author(s) 2012

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