Special cohomology classes for modular Galois representations

TL;DR

This paper extends Cornut's nonvanishing results from Heegner points on elliptic curves to special cohomology classes associated with modular forms, broadening the understanding of Galois representations in number theory.

Contribution

It generalizes Cornut's theorem by replacing elliptic curves with modular form Galois cohomology and introduces analogous special cohomology classes.

Findings

Proves nonvanishing of special cohomology classes in anticyclotomic extensions.

Extends nonvanishing results from elliptic curves to modular forms.

Provides new tools for studying Galois representations and automorphic forms.

Abstract

Building on ideas of Vatsal, Cornut proved a conjecture of Mazur asserting the generic nonvanishing of Heegner points on an elliptic curve E as one ascends the anticyclotomic Z_p-extension of a quadratic imaginary extension K/Q. In the present article Cornut's result is extended by replacing the elliptic curve E with the Galois cohomology of Deligne's 2-dimensional l-adic representation attached to a modular form of weight 2k>2, and replacing the family of Heegner points with an analogous family of special cohomology classes.