Cubic relations in Hall algebras and zeroes of zeta functions / Special talk on occasion of Prof. Manin's 75th birthday (special tea at 16h00, tea room MPI)

The Hall algebra is an associative algebra which can be attached to any exact category
A with appropriate finiteness properties. In particular, we can take A to consist of vector bundles on a curve X over a finite field.  In simple instances, the  corresponding Hall algebras
are related to quantum affine algebras (X=P^1) and Cherednik algebras (X elliptic). One can also define the arithmetic analog where X is replaced by spectrum of the ring of integers in a
number field.

In both cases (geometric and arithmetic),  quadratic relations in these Hall algebras are
provided by functional equations of Eisenstein series. Additional relations of higher order
d>2 (if they exist)   are of interest since they correspond to Serre relations in quantum affine algebras. Already the first nontrivial case d=3 is very interesting.

We show that the space of "new", essentially cubic relations (modulo scaling), is identified
with the space spanned by the nontrivial zeroes of the zeta function of X.  Joint work with
O. Schiffmann, E. Vasserot.