Venue： Lecture Hall

Speaker： Andreas Johannes Mihatsch (University of Bonn)

Abstract：Let $c_n$ be the number of isomorphism classes of pairs $(E, x)$  consisting of an elliptic curve $E$ over $\mathbb{C}$ and an  endomorphism $x$ that satisfies $x^2 = -n$. A classical theorem of  Zagier states that the series $\sum_{n = 1}^\infty c_n q^n$ is the  positive part of the $q$-expansion of a non-holomorphic modular form.  Its arithmetic version, due to Kudla--Rapoport--Yang, states that the  generating series of complex multiplication (CM) divisors on the  integral modular curve has a similar modularity property.  

In my talk, I will define CM cycle generating series for symplectic and  unitary Shimura varieties, and present first results on their  modularity. This adds a new facet to the Kudla program, which aims to  systematically relate special cycles on Shimura varieties with Fourier  expansions of automorphic forms. My talk is based on joint work with  Lucas Gerth, Siddarth Sankaran, and Tonghai Yang.