Simultaneous behaviour of the Fourier coefficients of two Hilbert modular cusp forms

TL;DR

This paper investigates the simultaneous sign changes and non-vanishing properties of Fourier coefficients of two Hilbert cusp forms, providing insights into their behavior at prime powers and contributing to the understanding of their arithmetic properties.

Contribution

It introduces new results on the simultaneous sign changes and non-vanishing of Fourier coefficients of Hilbert cusp forms of different weights, extending previous work in the field.

Findings

Proves the existence of simultaneous sign changes of Fourier coefficients.

Establishes conditions for non-vanishing of Fourier coefficients at prime powers.

Analyzes behavior of Fourier coefficients for non-CM eigenforms.

Abstract

In this article, we study the simultaneous sign changes of the Fourier coefficients of two Hilbert cusp forms of different integral weights. We also study the simultaneous non-vanishing of Fourier coefficients, of two distinct non-zero primitive Hilbert cuspidal non-CM eigenforms of integral weights, at powers of a fixed prime ideal.