The main objects of study of the present article are graphs of Hecke operators. These are combinatorial objects, defined in [R. Alvarenga, J. Number Theory 199, 192–228 (2019; Zbl 1454.11093)], which encode the action of Hecke operators on \(GL_n\)-automorphic forms over the function field of a smooth projective geometrically irreducible curve \(X\) over a finite field.
The author reformulates the question of computing graphs of Hecke operators in terms of computing certain products inside the Hall algebra \(\mathbf{H}_X\) of the category of coherent sheaves on \(X\). When \(X\) is an elliptic curve, the Hall algebra \(\mathbf{H}_X\) was extensively studied in [I. Burban and O. Schiffmann, Duke Math. J. 161, No. 7, 1171–1231 (2012; Zbl 1286.16029)], [D. Fratila, Compos. Math. 149, No. 6, 914–958 (2013; Zbl 1355.11061)]. Using rich structural results found in the references above, the author describes an algorithm for computing the graphs of Hecke operators when \(X\) is elliptic. The main idea behind this algorithm is rewriting everything in terms of a nice basis of \(\mathbf{H}_X\) (see Theorem 3.14), doing computations there, and reinterpreting the result in terms of automorphic forms.
The author also illustrates how the algorithm functions by computing the graph of Hecke operators \(\mathcal{G}_{x,1}\) of degree 1 for \(n=2\), and by providing some partial results for \(n>2\).