Hall algebras and graphs of Hecke operators for elliptic curves
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HALL ALGEBRAS AND GRAPHS OF HECKE OPERATORS FOR ELLIPTIC CURVES
BY
Roberto Alvarenga Instituto de Ciˆencias Matem´ aticas e de Computa¸ca ˜o Universidade de S˜ ao Paulo, S˜ ao Carlos, SP 13566-590, Brazil e-mail: [email protected]
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.
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . 1. Background . . . . . . . . . . . . . . . . . . . 2. Hall algebras and graphs of Hecke operators . 3. Graphs of Hecke operators for elliptic curves 4. The algorithm . . . . . . . . . . . . . . . . . 5. Calculating structure constants . . . . . . . . 6. The case of rank 2 . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .
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Received May 20, 2019 and ion revised form September 2, 2019
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R. ALVARENGA
Isr. J. Math.
Introduction This work is concerned with the graphs of Hecke operators. These graphs are defined from the action of Hecke operators on automorphic forms over a global function field. Motivated by questions of Zagier ([23]) about unramified toroidal automorphic forms (we refer to [14] for full details), Lorscheid develops in [16] a theory of graphs of Hecke operators for PGL2 over a global function field. This theory plays an important role in the proofs of his main theorems. In [15], Lorscheid analyses this theory for elliptic function fields and answers some of Zagier’s questions. In [1], we extend the definition of these graphs from PGL2 to GLn , generalize some of Lorscheid’s results and describe how to obtain these graphs for a rational function field. In this paper, we aim to describe these graphs when the global function field is elliptic. Using the theory of Hall algebras, we exhibit an algorithm to calculate these graphs. Before describing the reformulation in terms of coherent sheaves, let us review the original definition of the graph of a Hecke operator. Graph of a Hecke operator. Let F be the function field of a smooth projective and geometrically irreducible curve X over Fq , A its ring of the adelic integers and K = GLn (OA ), where OA is the set of the adelic integers. For any right K-invariant Hecke operator Φ, there are unique m1 , . . . , mr ∈ C∗ and pairwise distinct [g1 ], . . . , [gr ] ∈ GLn (F ) \ GLn (A)/K such that for all automorphic forms f r Φ(f )(g) = mi f (gi ) i=1
(see [1]
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