PARALLEL WEIGHT 2 POINTS on HILBERT MODULAR EIGENVARIETIES and the PARITY CONJECTURE
Journal article, 2019

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 : 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 : Q]) cuspidal Hilbert modular eigenvariety has the property that many (all, if [F : 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.

11F33 (primary)

2010 Mathematics Subject Classification:

11G40 (secondary)

Author

Christian Johansson

Chalmers, Mathematical Sciences, Algebra and geometry

James Newton

King's College London

Forum of Mathematics, Sigma

20505094 (eISSN)

Vol. 7

Subject Categories

Algebra and Logic

Geometry

Mathematical Analysis

DOI

10.1017/fms.2019.23

More information

Latest update

11/10/2019