Roberto Pirisi: An arithmetic theory of characteristic classes for moduli problems
Time: Wed 2019-01-16 13.15 - 15.00
Location: Room 3418, KTH
Participating: Roberto Pirisi (KTH)
In topology, characteristic classes are ways of, given a principal bundle E over a space X, functorially producing an element c(E) in the (singular) cohomology ring of X. Many classical invariants of arithmetic objects (e.g. quadratic forms) do the same, where the space X is replaced by a field F, and singular cohomology is replaced by Galois cohomology.
In the nineties these examples were grouped together under the notion of "cohomological invariants". These are natural transformations between a functor from fields to sets (e.g. isomorphism classes of quadratic forms) and the functor of Galois cohomology with an appropriate choice of coefficients. This created a rich theory which was studied by Serre, Rost, Merkurjev, Garibaldi and many more.
In all the relevant (classical) cases, the functor we try to study can be described as the functor of principal G-bundles for an appropriate algebraic group G (e.g. quadratic forms are in bijection with principal On-bundles). From the point of view of moduli stacks, one can then see cohomological invariants of G as invariants of the classifying stack BG.
Starting from this insight we will construct a theory of cohomological invariants for moduli stacks (e.g. the stack Mg of smooth curves of genus g) extending the classical theory. This theory turns out to be quite rich and connects to the theory of unramified cohomology, which is relevant to rationality problem. We will then explicitly describe the cohomological invariants of the moduli stacks of smooth elliptic and hyperelliptic curves.