On the central derivatives of l-functions and modularity of heenger cycles

TL;DR

This paper links the derivatives of L-functions at the center to heights of higher Heegner cycles, extending the Gross-Zagier-Zhang formula and exploring the modularity of Heegner cycle series.

Contribution

It establishes an arithmetic intersection formula for L-derivatives in higher weights and investigates the modularity of Heegner cycle generating series.

Findings

Proved the representation of L-derivative by height pairing for general cusp forms.

Provided a framework extending the Gross-Zagier-Zhang formula.

Reduced the modularity conjecture to a vanishing conjecture with supporting evidence.

Abstract

This paper establishes an arithmetic intersection formula for central L-derivatives in higher weights.We prove that for a general cusp form (extending the previous result for newforms), the derivative is represented by the global height pairing between higher Heegner cycles. This result provides a framework for the Gross-Zagier-Zhang formula and its generalizations.Furthermore, we investigate the modularity of the generating series of Heegner cycles,proving a weak version of the conjecture and reducing the full modularity to a vanishing conjecture,for which we provide supporting evidence.