Faltings heights of CM cycles and derivatives of L-functions

TL;DR

This paper investigates the relationship between Faltings heights of CM cycles on Shimura varieties and derivatives of L-functions, providing new proofs and conjectures in the context of arithmetic geometry.

Contribution

It introduces a conjecture linking height pairings to L-function derivatives and proves it in low-dimensional cases, offering new insights into the Gross-Zagier formula.

Findings

Computed the Archimedian contribution to height pairings.

Established a conjecture relating height pairings to L-function derivatives.

Provided new proofs of the Gross-Zagier formula in specific cases.

Abstract

We study the Faltings height pairing of arithmetic Heegner divisors and CM cycles on Shimura varieties associated to orthogonal groups. We compute the Archimedian contribution to the height pairing and derive a conjecture relating the total pairing to the central derivative of a Rankin L-function. We prove the conjecture in certain cases where the Shimura variety has dimension 0, 1, or 2. In particular, we obtain a new proof of the Gross-Zagier formula.