2025 Theses Doctoral

The modularity of arithmetic generating series of special cycles on 𝑋₀(𝑁)

Zhu, Baiqing

In the late 1990s, Stephen S. Kudla initiated an influential program to study the special cycles on orthogonal and unitary Shimura varieties. A major conjecture is the modularity of generating series of special cycles. In this thesis we focus on the case of modular curve 𝑋₀(𝑁) where 𝑁 is an arbitrary positive integer. We prove the modularity of the generating series of special cycles values in the arithmetic Chow groups.

For the generating series valued in the codimension 2 arithmetic Chow group, the modularity is proved by establishing the arithmetic Siegel–Weil formula on the modular curve 𝑋₀(𝑁), i.e., we relate the arithmetic degrees of special cycles on 𝑋₀(𝑁) to the derivatives of Fourier coefficients of a genus 2 Eisenstein series. The identity is proved by combining “difference formulae” on both the geometric and analytic sides. When 𝑁 is odd and square-free, our work gives a different proof of the main results of Sankaran, Shi and Yang. For the generating series valued in the codimension 1 arithmetic Chow group, the modularity is proved by combining the known results in codimension 2 and analyzing the reduction to primes 𝑝|𝑁 of cusps of 𝑋₀(𝑁). This generalizes the previous work of Du and Yang.