# Interpolation of Generalized Heegner Cycles in Coleman Families

## TL;DR

This paper extends the interpolation of generalized Heegner cycles to Coleman families, facilitating proofs of $p$-adic Gross--Zagier formulas and a conjecture of Perrin-Riou, advancing the understanding of $p$-adic L-functions and automorphic forms.

## Contribution

It introduces a new interpolation of Heegner cycles along Coleman families, connecting geometric cycles with $p$-adic families of automorphic forms.

## Key findings

- Constructed distribution-valued cohomology classes in Coleman families.
- Applied to prove $p$-adic Gross--Zagier formulas for non-ordinary eigenforms.
- Supported a conjecture of Perrin-Riou through this interpolation.

## Abstract

Kobayashi recently proved that the generalized Heegner cycles of Bertolini--Darmon--Prasanna can be interpolated along the anticyclotomic tower, giving rise to distribution valued cohomology classes with expected growth rate. We interpolate these classes along Coleman families. This construction plays a role in the proofs of $p$-adic Gross--Zagier formulae at for non-ordinary eigenforms and consequentially, also in the proof of a conjecture of Perrin-Riou.