Number Theory Seminar

The relative Langlands duality of Ben-Zvi-Sakellaridis-Venkatesh reinvents L-functions and period integrals via Hamiltonian geometry. In this talk, we will discuss M-cycles (or M-classes) on projective Shimura varieties and provide some conjectural arithmetic frameworks under suitable algebraic assumptions. Here M means a nice G-Hamiltonian space over Q (e.g. from theta liftings). Over C, we use (g,K)-cohomology, revisiting Liu’s formula on H^1 of U(n,1) Shimura varieties and abelianization of arithmetic groups.  Over C_p, we use Fargues-Fontaine curve and introduce local non-reductive M-cycles, e.g. mirabolic cycles and parabolic version local Shimura varieties. We explain arithmetic aspects of M-cycles (e.g. non-reductive AFL of W. Zhang) and proofs of some basic properties.