Number Theory Seminar

We construct a "big Hecke action" of the formal multiplicative group on the Katz-Igusa moduli problem over the ordinary locus of the modular curve, and show that the differential operator theta=dq/q on p-adic modular forms is obtained by differentiating this action. The action admits a simple description on q-expansions, but the basic properties of theta (interaction with U_p, relation with the Gauss-Manin connection) can also be deduced directly from the construction without an appeal to q-expansions. We explain two constructions of this action: the first, completely classical, is through Katz's exceptional isomorphisms; the second, better suited for generalization, is via the action of the universal cover on big Igusa varieties of Caraiani-Scholze. This work is based on a suggestion of Peter Scholze.