Stark-Heegner cycles attached to Bianchi modular forms

TL;DR

This paper constructs Stark--Heegner cycles for Bianchi modular forms over imaginary quadratic fields, linking special values of L-functions to Selmer groups via p-adic Abel--Jacobi maps, extending classical modular form techniques.

Contribution

It introduces a new construction of Stark--Heegner cycles for Bianchi modular forms using p-adic Abel--Jacobi maps, generalizing previous approaches from classical modular forms.

Findings

Construction of Stark--Heegner cycles for Bianchi forms

Connection between L-values and Selmer groups

Extension of classical modular form methods

Abstract

Let f be a Bianchi modular form, that is, an automorphic form for GL(2) over an imaginary quadratic field F, and let P be a prime of F at which f is new. Let K be a quadratic extension of F, and L(f/K,s) the L-function of the base-change of f to K. Under certain hypotheses on f and K, the functional equation of L(f/K,s) ensures that it vanishes at the central point. The Bloch--Kato conjecture predicts that this should force the existence of non-trivial classes in an appropriate global Selmer group attached to f and K. In this paper, we use the theory of double integrals developed by Barrera Salazar and the second author to construct certain P-adic Abel--Jacobi maps, which we use to propose a construction of such classes via "Stark--Heegner cycles". This builds on ideas of Darmon and in particular generalises an approach of Rotger and Seveso in the setting of classical modular forms.