Parallel weight 2 points on Hilbert modular eigenvarieties and the parity conjecture

TL;DR

This paper constructs and analyzes partial Hilbert modular eigenvarieties over totally real fields, proving their structure near boundary points and applying these results to verify the parity conjecture for certain Hilbert modular forms.

Contribution

It introduces new partial eigenvarieties interpolating Hilbert modular forms with varying weights at a single place and applies these to prove the parity conjecture in specific cases.

Findings

Partial eigenvarieties decompose into finite components over boundary annuli.

Many irreducible components contain classical points with non-critical slopes and algebraic weights.

The parity conjecture is proved for finite slope Hilbert modular forms with trivial central character.

Abstract

Let F be a totally real field of degree d and let p be an odd prime which is totally split in F. We define and study one-dimensional partial eigenvarieties interpolating Hilbert modular forms over F with weight varying only at a single place v above p. For these eigenvarieties, we show that methods developed by Liu, Wan and Xiao apply and deduce that, over a boundary annulus in weight space of sufficiently small radius, the partial eigenvarieties decompose as a disjoint union of components which are finite over weight space. We apply this result to prove the parity version of the Bloch--Kato conjecture for finite slope Hilbert modular forms with trivial central character (under some assumptions), by reducing to the case of parallel weight 2. As another consequence of our results on partial eigenvarieties, we show, still under the assumption that p is totally split in F, that the full (dimension 1 + d) cuspidal Hilbert modular eigenvariety has the property that many (all, if d is even) irreducible components contain a classical point with non-critical slopes and parallel weight 2 (with some character at p whose conductor can be explicitly bounded), or any other algebraic weight.