Cusp eigenforms and the hall algebra of an elliptic curve

2011/09/20 by Dragos Fratila
Mathematics · #Advanced Algebra and Geometry #Algebra over a field #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Cusp (singularity) #Elliptic curve #Hecke algebra #Modular elliptic curve #Preprint #Product (mathematics) #Tensor product #math.AG #math.NT #math.RT

paper · pdf · doi:10.1112/s0010437x12000784

published as Compositio Math. 149 (2013) 914-958

arxiv created 2011/09/20 · openalex publication_date 2013/03/04 · openalex created_date 2016/06/24 · arxiv updated 2019/02/20 · openalex updated_date 2026/08/05

Abstract

Abstract 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 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]).

Citations

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