Indecomposable vector-valued modular forms and periods of modular curves
We classify the three-dimensional representations of the modular group that are reducible but indecomposable, and their associated spaces of holomorphic vector-valued modular forms. We then demonstrate how such representations may be employed to compute periods of modular curves. This technique obviates the use of Hecke operators, and therefore provides a method for studying noncongruence modular curves as well as congruence.
KeywordsIndecomposable representations Modular forms Periods
Mathematics Subject Classification11F12 11F23
We would like to acknowledge Bill Hoffman, Ling Long and Geoff Mason for helpful discussions. The first author would like to thank the Mathematics Department at Chico State for the hospitality enjoyed during the brief visit in which this article was initiated. We would also like to thank the referees for numerous comments and a correction to an earlier version of the manuscript.
- 2.Candelori, L., Franc, C.: Vector bundles and modular forms for fuchsian groups of genus zero (2017). arXiv:1707.01693 [math.AG.NT]
- 6.Farkas, H.M., Kra, I.: Riemann Surfaces, Volume 71 of Graduate Texts in Mathematics. Springer, New York (1980)Google Scholar
- 8.Franc, C., Mason, G.: Hypergeometric series, modular linear differential equations and vector-valued modular forms. Ramanujan J., 1–35 (2014)Google Scholar
- 14.Mertens, M.H., Raum, M.: Modular forms of virtually real-arithmetic type (2017). arXiv:1712.03004 [math.NT]
- 16.Selberg, A: On the estimation of Fourier coefficients of modular forms. In: Proc. Sympos. Pure Math., Vol. VIII, pp. 1–15. American Mathematical Society, Providence (1965)Google Scholar
- 17.Thompson, J.G.: Hecke operators and noncongruence subgroups. In: Group theory (Singapore, 1987), pp. 215–224. de Gruyter, Berlin (1989). Including a letter from J.-P. SerreGoogle Scholar