Abstract
We develop an explicit theory of congruence subgroups, their cusps, and Manin symbols for arbitrary number fields. While our motivation is in the application to the theory of modular symbols over imaginary quadratic fields, we give a general treatment which makes no special assumptions on the number field.
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Notes
- 1.
We avoid the common notation (a:b) in order to avoid confusion with M-symbols.
- 2.
Note that the definition marked (*) on [Shi71, p. 25] is not quite correct as stated: in each residue class modulo N/d one must take a representative c which is coprime to d, if one exists, but one cannot restrict to the range 0<c≤N/d.
References
M.T. Aranés, Modular symbols over number fields. PhD thesis, University of Warwick, 2011. See http://wrap.warwick.ac.uk/35128/
J.S. Bygott, Modular forms and elliptic curves over imaginary quadratic number fields. PhD thesis, University of Exeter, 1998. See http://hdl.handle.net/10871/8322
H. Cohen, Advanced Topics in Computational Number Theory. Graduate Texts in Mathematics, vol. 193 (Springer, New York, 2000)
J.E. Cremona, Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields. Compos. Math. 51, 275–323 (1984)
J.E. Cremona, Modular symbols for Γ 1(N) and elliptic curves with everywhere good reduction. Math. Proc. Camb. Philos. Soc. 111(2), 199–218 (1992)
J.E. Cremona, Algorithms for Modular Elliptic Curves, 2nd edn. (Cambridge University Press, Cambridge, 1997). Available from http://www.warwick.ac.uk/staff/J.E.Cremona/book/fulltext/index.html
M. Greenberg, J. Voight, Computing systems of Hecke eigenvalues associated to Hilbert modular forms. Math. Comput. 80(274), 1071–1092 (2011)
M.P. Lingham, Modular forms and elliptic curves over imaginary quadratic fields. PhD thesis, University of Nottingham, 2005. See http://etheses.nottingham.ac.uk/archive/00000138/
A. Mohamed, Universal Hecke L-series associated with cuspidal eigenforms over imaginary quadratic fields (this volume). doi:10.1007/978-3-319-03847-6_9
M.H. Şengün, On the integral cohomology of Bianchi groups. Exp. Math. 20(4), 487–505 (2011)
M.H. Şengün, Arithmetic aspects of Bianchi groups (this volume). doi:10.1007/978-3-319-03847-6_11
G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions. Publications of the Mathematical Society of Japan, vol. 11 (Iwanami Shoten, Publishers, Tokyo, 1971). Kanô Memorial Lectures, No. 1
W. Stein, Modular Forms, a Computational Approach. Graduate Studies in Mathematics, vol. 79 (American Mathematical Society, Providence, 2007)
W. Stein et al., Sage Mathematics Software (Version 5.2). The Sage Development Team, 2012. http://www.sagemath.org
G. van der Geer, Hilbert Modular Surfaces (Springer, Berlin, 1988)
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Cremona, J.E., Aranés, M.T. (2014). Congruence Subgroups, Cusps and Manin Symbols over Number Fields. In: Böckle, G., Wiese, G. (eds) Computations with Modular Forms. Contributions in Mathematical and Computational Sciences, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-03847-6_4
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