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Cusp eigenforms and the hall algebra of an elliptic curve

Part of:Discontinuous groups and automorphic formsAlgebraic number theory: global fieldsLie algebras and Lie superalgebrasLie groups

Published online by Cambridge University Press:  04 March 2013

Dragos Fratila
Dragos Fratila*
Affiliation:
Université Paris Denis-Diderot - Paris 7, Institut de Mathématiques de Jussieu, UMR 7586 du CNRS, Batiment Chevaleret, 75205 Paris Cedex 13, France email fratila@math.jussieu.fr
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Abstract

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We give an explicit construction of the cusp eigenforms on an elliptic curve defined over a finite field, using the theory of Hall algebras and the Langlands correspondence for function fields and ${\mathrm{GL} }_{n} $. As a consequence we obtain a description of the Hall algebra of an elliptic curve as an infinite tensor product of simpler algebras. We prove that all these algebras are specializations of a universal spherical Hall algebra (as defined and studied by Burban and Schiffmann  [On the Hall algebra of an elliptic curve I, Preprint (2005), arXiv:math/0505148 [math.AG]] and Schiffmann and Vasserot [The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials, Compositio Math. 147 (2011), 188–234]).

Keywords

Hall algebraselliptic curvesautomorphic forms

MSC classification

Secondary:11F70: Representation-theoretic methods; automorphic representations over local and global fields11R39: Langlands-Weil conjectures, nonabelian class field theory22E55: Representations of Lie and linear algebraic groups over global fields and adèle rings17B99: None of the above, but in this section

Information

Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 6 , June 2013, pp. 914 - 958
Copyright
© The Author(s) 2013 

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