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Moscow Mathematical Journal, 2005, Volume 5, Number 4, Pages 829–856
DOI: https://doi.org/10.17323/1609-4514-2005-5-4-829-856
(Mi mmj224)
 

This article is cited in 6 scientific papers (total in 6 papers)

Ramanujan modular forms and the Klein quartic

G. Lachaud

Institut de Mathématiques de Luminy
Full-text PDF Citations (6)
References:
Abstract: In one of his notebooks, Ramanujan gave some algebraic relations between three theta functions of order 7. We describe the automorphic character of a vector-valued mapping constructed from these theta series. This provides a systematic way to establish old and new identities on modular forms for the congruence subgroup of level 7, above all, a parametrization of the Klein quartic. From a historical point of view, this shows that Ramanujan discovered the main properties of this curve with his own means. As an application, we introduce an $L$-series in four different ways, generating the number of points of the Klein quartic over finite fields. From this, we derive the structure of the Jacobian of a suitable form of the Klein quartic over finite fields and some congruence properties on the number of its points.
Key words and phrases: Ramanujan, Klein quartic, modular form, theta series, curve over a finite field, $L$-series, Jacobian, zeta function.
Received: December 16, 2005
Bibliographic databases:
MSC: 11G25, 11M38
Language: English
Citation: G. Lachaud, “Ramanujan modular forms and the Klein quartic”, Mosc. Math. J., 5:4 (2005), 829–856
Citation in format AMSBIB
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\by G.~Lachaud
\paper Ramanujan modular forms and the Klein quartic
\jour Mosc. Math.~J.
\yr 2005
\vol 5
\issue 4
\pages 829--856
\mathnet{http://mi.mathnet.ru/mmj224}
\crossref{https://doi.org/10.17323/1609-4514-2005-5-4-829-856}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2266461}
\zmath{https://zbmath.org/?q=an:1131.11040}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000208595600007}
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  • This publication is cited in the following 6 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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