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Uniformization of triangle modular curves.

Pilar Bayer, Artur Travesa (2007)

Publicacions Matemàtiques

In the present article, we determine explicit uniformizations of modular curves attached to triangle Fuchsian groups with cusps. Their Hauptmoduln are obtained by integration of non-linear differential equations of the third order. Series expansions involving integral coefficients are calculated around the cusps as well as around the elliptic points. The method is an updated form of a differential construction of the elliptic modular function j, first performed by Dedekind in 1877. Subtle differences...

Unipotent vector bundles and higher-order non-holomorphic Eisenstein series

Jay Jorgenson, Cormac O’Sullivan (2008)

Journal de Théorie des Nombres de Bordeaux

Higher-order non-holomorphic Eisenstein series associated to a Fuchsian group Γ are defined by twisting the series expansion for classical non-holomorphic Eisenstein series by powers of modular symbols. Their functional identities include multiplicative and additive factors, making them distinct from classical Eisenstein series. In this article we prove the meromorphic continuation of these series and establish their functional equations which relate values at s and 1 - s . In addition, we construct...

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