On Graphs of Hecke operators and Hall algebras

In Bombay 1979, Don Zagier observes that if the kernel of certain
operators on automorphic forms turns out to be an unitarizable
representation, over the field of rational numbers $Q$, a formula of Hecke implies the Riemann hypothesis.

Zagier calls the elements of this kernel toroidal automorphic forms.
Moreover, Zagier asks what happens if $Q$ is replaced by a global function field and remarks that  the space of unramified toroidal automorphic forms can be expected to be finite dimensional.

Motivated by these questions, Oliver Lorscheid  introduces, in 2012,
the  *graphs of Hecke operators* for global function fields.

This theory allowed him to prove, among other things,  that the space
of unramified toroidal automorphic forms for a global function field
is indeed, finite dimensional. The graphs of Hecke operators
introduced by Lorscheid encode the action of Hecke operators on
automorphic forms.

On the other hand, Ringel(1990), Kapranov (1997), Burban and
Schiffmann (2012) et al. have been developing the theory of  *Hall
algebras * of coherent sheaves over a smooth geometric irreducible projective
curve over a finite field (in general for a finitary category).

For this talk we discuss the connection between graphs of Hecke
operators and Hall algebras.

In the elliptic case, Atiyah's work on vector bundles (1957) allow us
to describe (explicitly) these graphs.