Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-30T21:19:38.186Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Discontinuous groups and automorphic forms Arithmetic algebraic geometry

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: [email protected])
Håkan Granath
Affiliation:
Department of Mathematics, Karlstad University, 65188 Karlstad, Sweden (email: [email protected])

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1]Abramowitz, M. and Stegun, I., Handbook of mathematical functions with formulas, graphs, and mathematical tables (Dover, New York, 1964).Google Scholar
[2]Alsina, M. and Bayer, P., Quaternion orders, quadratic forms, and Shimura curves, CRM Monograph Series 22 (American Mathematical Society, Providence, RI, 2004).Google Scholar
[3]Baba, S. and Granath, H., ‘Genus 2 curves with quaternionic multiplication’, Canad. J. Math. 60 (2008) no. 4, 734–757.CrossRefGoogle Scholar
[4]Bayer, P. and Travesa, A., ‘Uniformizing functions for certain Shimura curves, in the case D=6’, Acta Arith. 126 (2007) no. 4, 315–339.CrossRefGoogle Scholar
[5]Bayer, P. and Travesa, A., ‘On local constants associated to arithmetical automorphic functions’, Pure Appl. Math. Q. 4 (2008) no. 4, 1107–1132, Special issue: in honor of Jean-Pierre Serre. Part 1.CrossRefGoogle Scholar
[6]Bruinier, J. H., van der Geer, G., Harder, G. and Zagier, D., The 1-2-3 of modular forms, Lectures at a Summer School in Nordfjordeid, Norway (Springer, Berlin, 2008).Google Scholar
[7]Elkies, N. D., ‘Shimura curve computations’, Lecture Notes in Computer Science 1423 (1998) 1–47.CrossRefGoogle Scholar
[8]Maass, H., ‘Die Differentialgleichungen in der Theorie der elliptischen Modulfunktionen’, Math. Ann. 125 (1953) 235–263.CrossRefGoogle Scholar
[9]Nehari, Z., Conformal mapping (McGraw-Hill, New York, Toronto, London, 1952).Google Scholar
[10]Prasanna, K., Periods and special values of L-functions,http://swc.math.arizona.edu/aws/2011/2011PrasannaNotesProject.pdf.Google Scholar
[11]Shimura, G., ‘On canonical models of arithmetic quotients of bounded symmetric domains’, Ann. of Math. (2) 91 (1970) 144–222.CrossRefGoogle Scholar
[12]Shimura, G., ‘Automorphic forms and the periods of abelian varieties’, J. Math. Soc. Japan 31 (1979) no. 3, 561–592.CrossRefGoogle Scholar
[13]Yang, Y., ‘Schwarzian differential equations and Hecke eigenforms on Shimura curves’, Preprint, 2011, arXiv:1110.6284.CrossRefGoogle Scholar
[14]Yoshida, H., Absolute CM-periods, Mathematical Surveys and Monographs 106 (American Mathematical Society, Providence, RI, 2003).CrossRefGoogle Scholar