An Euler system for characters over an imaginary biquadratic field

TL;DR

This paper constructs an Euler system for characters over an imaginary biquadratic field and uses it to bound a Selmer group related to Rankin--Selberg convolutions of modular forms with complex multiplication.

Contribution

It introduces an Euler system approach for characters over an imaginary biquadratic field, extending previous methods to new Galois representations.

Findings

Bounded the Selmer group over the $Z_p^3$-extension of the biquadratic field

Connected modular forms with complex multiplication to Galois representations over biquadratic fields

Applied Euler system techniques to control Selmer groups in this setting

Abstract

Given a pair of modular forms with complex multiplication by distinct imaginary quadratic fields, the four dimensional Galois representation associated to their Rankin--Selberg convolution is induced from a character over an imaginary biquadratic field $F$. Using the Euler system of Lei, Loeffler and Zerbes we bound a Selmer group associated to this character, over the unique $\mathbb{Z}_{p}^{3}$-extension of $F$.