Applications of G-Zips to the Langlands Program and algebraic geometry

Building on the group-theoretic Hasse invariants constructed with Koskivirta, I´ll develop groundbreaking applications of the up-and-coming theory of G-Zips to the Langlands program and algebraic geometry My project focuses on applications to 3 interrelated frameworks: Moduli spaces, The Calegari-Geraghty Program and Langlands Functoriality.

The Calegari-Geraghty Program is a vast generalization of the Taylor-Wiles modularity method, the crux of Wiles´ proof of Fermat´s Last Theorem.

I´ll begin by proving results in the important test-case of Hodge-type Shimura varieties. lying in their intersection.

G-Zips offer a mod p analogue of Hodge theory and a group-theoretic generalization of the Ekedahl-Oort stratification of the moduli space of abelian varieties mod p.

Using G-Zips and their Hasse invariants, we’ll obtain applications in algebraic geometry to cohomological vanishing, positivity of vector bundles, tautological rings and complete subvarieties of Shimura varieties and other moduli spaces.

The results on Shimura varieties will themselves have important applications to the Calegari-Geraghty Program, generalizing the modularity of elliptic curves over Q to other interesting motives.We will then construct analogues of Hasse invariants for the singular cohomology of locally symmetric spaces.

We’ll use these to establish Langlands functoriality between torsion Hecke classes and associate Galois representations to them beyond Scholze’s method.