Cusp Eigenforms and the Hall Algebra of an Elliptic Curve

TL;DR

This paper constructs cusp eigenforms on elliptic curves over finite fields using Hall algebras and Langlands correspondence, revealing the Hall algebra as an infinite tensor product of universal spherical Hall algebras.

Contribution

It provides an explicit construction of cusp eigenforms and describes the Hall algebra as an infinite tensor product of universal spherical Hall algebras.

Findings

Hall algebra of an elliptic curve is an infinite tensor product of simpler algebras.

All these algebras are specializations of a universal spherical Hall algebra.

Explicit construction of cusp eigenforms using Hall algebra theory.

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 in \cite{BS} and \cite{SV1}).