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Hall algebras and graphs of Hecke operators for elliptic curves

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Abstract

The graph of a Hecke operator encodes all information about the action of this operator on automorphic forms over a global function field. These graphs were introduced by Lorscheid in [16] for PGL2 and generalized to GLn in [1]. After reviewing some general properties, we explain the connection to the Hall algebra of the function field. In the case of an elliptic function field, we can use structure results of Burban-Schiffmann [7] and Fratila [8] to develop an algorithm which explicitly calculates these graphs. We apply this algorithm to determine some structure constants and provide explicitly the rank two case in the last section.

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  1. Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos, SP, 13566-590, Brazil

    Roberto Alvarenga

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  1. Roberto Alvarenga
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Correspondence to Roberto Alvarenga.

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Alvarenga, R. Hall algebras and graphs of Hecke operators for elliptic curves. Isr. J. Math. 239, 215–269 (2020). https://doi.org/10.1007/s11856-020-2056-2

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  • DOI: https://doi.org/10.1007/s11856-020-2056-2

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