CM Values of Green Functions Associated to Special Cycles on Shimura Varieties with Applications to Siegel 3-Fold $X_2(2)$

TL;DR

This paper generalizes CM cycles and provides a uniform formula for Green function values on special cycles, with applications to computing invariants relevant for genus two curve cryptography.

Contribution

It introduces a generalized definition of CM cycles and derives a uniform formula for their Green function values using regularized theta lifts.

Findings

Derived a uniform formula for CM Green function values

Computed special values of theta functions and Rosenhain invariants

Applied results to genus two curve cryptography

Abstract

We generalize the definition of CM cycles beyond the small and big CM ones studied by various authors and give a uniform formula for the CM values of Green functions associated to these special cycles in general using the idea of regularized theta lifts. Finally, as an application to Siegel 3-fold $X_2(2)$, we can compute special values of theta functions and Rosenhain $\lambda$-invariants at a CM cycle, which is useful for genus two curve cryptography.