Comparison of canonical periods under base change

TL;DR

This paper investigates how the canonical period of a Hilbert modular form changes under base change to a real quadratic extension, showing it differs from the original by a p-adic unit under certain conditions, using Iwasawa theory.

Contribution

It establishes a precise relation between canonical periods under base change for Hilbert modular forms, connecting it to the anticyclotomic Iwasawa main conjecture.

Findings

Canonical period under base change differs by a p-adic unit.

Proves a version of the anticyclotomic Iwasawa main conjecture.

Provides conditions under which the period relation holds.

Abstract

In this paper we prove the canonical period of a Hilbert modular form with respect to the base change of a real quadratic extension differs from the square of its own canonical period only by a $p$-adic unit under some conditions. We prove this by proving a specific version of anticyclotomic Iwasawa main conjecture for Hilbert modular forms.