PARALLEL WEIGHT 2 POINTS ON HILBERT MODULAR EIGENVARIETIES AND THE PARITY CONJECTURE

Christian Johansson,James Newton

doi:10.1017/fms.2019.23

Abstract

Let$F$be a totally real field 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 (with a technical assumption if$[F:\mathbb{Q}]$is odd), 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$1+[F:\mathbb{Q}]$) cuspidal Hilbert modular eigenvariety has the property that many (all, if$[F:\mathbb{Q}]$is even) irreducible components contain a classical point with noncritical slopes and parallel weight$2$(with some character at$p$whose conductor can be explicitly bounded), or any other algebraic weight.

Highlights

IntroductionThe idea of the proof is that, using the results of [PX], we can reduce the parity conjecture for a Hilbert modular newform g of general (even) weight to the parity conjecture for parallel weight two Hilbert modular forms, by moving in a p-adic family connecting g to a parallel weight two form
The idea of the proof is that, using the results of [PX], we can reduce the parity conjecture for a Hilbert modular newform g of general weight to the parity conjecture for parallel weight two Hilbert modular forms, by moving in a p-adic family connecting g to a parallel weight two form. This parallel weight two form will have local factors at places dividing p given by ramified principal series representations
The existence of partial eigenvarieties is probably well known to experts, but they have not been utilized so much in the literature. They were discussed and used in works of Chenevier [Che] and Chenevier– Harris [CH13] to remove regularity hypotheses from theorems concerning the existence and local–global compatibility of Galois representations associated to automorphic representations

Summary

Introduction

The idea of the proof is that, using the results of [PX], we can reduce the parity conjecture for a Hilbert modular newform g of general (even) weight to the parity conjecture for parallel weight two Hilbert modular forms, by moving in a p-adic family connecting g to a parallel weight two form This parallel weight two form will have local factors at places dividing p given by ramified principal series representations (moving to the boundary of weight space corresponds to increasing the conductor of the ratio of the characters defining this principal series representation). We use the notation and conventions of [Sai, Section 1] for the weights of Hilbert modular forms, and in particular w = 2 corresponds to the central character of the associated automorphic representation having trivial algebraic part. (3) When [F : Q] is odd we require the existence of the place v0 in the above theorem in order to switch to a totally definite quaternion algebra

A halo for the partial eigenvariety

The parity conjecture

The full eigenvariety