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 $x2 = -n$. 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.