The Ramanujan Journal

, Volume 17, Issue 1, pp 107–121 | Cite as

Further properties of a function of Ogg and Ligozat

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Abstract

Certain identities of Ramanujan may be succinctly expressed in terms of the rational function \(\breve{g}_{\chi}=\breve{f}_{\chi}-\frac{1}{\breve {f}_{\chi}}\) on the modular curve X0(N), where \(\breve{f}_{\chi}=w_{N}f_{\chi}\) and fχ is a certain modular unit on the Nebentypus cover Xχ(N) introduced by Ogg and Ligozat for prime \(N\equiv1\allowbreak\mkern5mu(\mathrm{mod}~4)\) and wN is the Fricke involution. These correspond to levels N=5,13, where the genus gN of X0(N) is zero. In this paper we study slightly more general kind of relations for each \(\breve{g}_{\chi}\) such that X0(N) has genus gN=1,2, and also for each \(h_{\chi}=g_{\chi}+\breve{g}_{\chi}\) such that the Atkin–Lehner quotient X0+(N) has genus gN+=1,2. Finally we study the normal closure of the field of definition of the zeros of the latter.

Keywords

Modular units Nebentypus cover 

Mathematics Subject Classification (2000)

11E16 20H05 

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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeCambridgeUK

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