GLNT: Non-reductive cycles and L-functions in arithmetic geometry

Abstract: There are many fundamental conjectures and programs around L-functions, algebraic cycles and Galois actions on algebraic solutions of polynomial equations, e.g. Langlands and Kudla program. Unlike function field analogs which is more topological (via \ell not = p sheaves), the real story over number fields and their local fields is more analytic involving \ell = p cycles and (g,K)-cohomology, which needs to be further developed.

In this talk, I will firstly give my (naive) understandings of these programs and examples. For central / non-central L-values and p-adic L-functions, in general we must use non-reductive type period integrals and cycles, e.g. L-functions for GLn x GLm. Then I will give some arithmetic analogs, constructions of non-reductive cycles and applications, e.g. a proof of twisted AFL for GL_n. I use two observations: pullback of non-algebraic cycles could be algebraic and useful; raising “the categorical level" by one and applying extra symmetry (e.g. global modularity) is really useful.

Time permitting, I will discuss more aspects of non-reductive cycles (ramifications / archimedes / algebraicity..), based on what we learn from function field analogs (after the work of Ben-Zvi-Sakellaridis-Venkatesh). I will also present a conjecture on Albanese of projective U(n-1,1)-Shimura varieties.