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

Research output: Contribution to journalArticlepeer-review

Christian Johansson, James John Michael Newton

Original languageEnglish
Article numberE27
Pages (from-to)1-36
Number of pages36
Early online date4 Sep 2019
Accepted/In press17 Jun 2019
E-pub ahead of print4 Sep 2019
PublishedSep 2019


King's Authors


Let be a totally real field and let be an odd prime which is totally split in. We define and study one-dimensional 'partial' eigenvarieties interpolating Hilbert modular forms over with weight varying only at a single place above. 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 is odd), by reducing to the case of parallel weight. As another consequence of our results on partial eigenvarieties, we show, still under the assumption that is totally split in, that the 'full' (dimension) cuspidal Hilbert modular eigenvariety has the property that many (all, if is even) irreducible components contain a classical point with noncritical slopes and parallel weight (with some character at whose conductor can be explicitly bounded), or any other algebraic weight.

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