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

Part of: Discontinuous groups and automorphic forms Arithmetic algebraic geometry

Published online by Cambridge University Press:  04 September 2019

CHRISTIAN JOHANSSON Open the ORCID record for CHRISTIAN JOHANSSON [Opens in a new window]  and
JAMES NEWTON Open the ORCID record for JAMES NEWTON [Opens in a new window]
CHRISTIAN JOHANSSON
Affiliation:
Department of Mathematical Sciences, Chalmers University of Technology and The University of Gothenburg, SE-412 96 Gothenburg, Sweden; [email protected]
JAMES NEWTON
Affiliation:
Department of Mathematics, King’s College London, Strand, London WC2R 2LS, UK; [email protected]

Abstract

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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.

MSC classification

Primary: 11F33: Congruences for modular and $p$-adic modular forms
Secondary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e27
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

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