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

Artikel i vetenskaplig tidskrift
Författare Christian Johansson
J. Newton
Publicerad i Forum of Mathematics Sigma
Volym 7
ISSN 2050-5094
Publiceringsår 2019
Publicerad vid Institutionen för matematiska vetenskaper
Språk en
Länkar dx.doi.org/10.1017/fms.2019.23
Ämnesord galois representations, extended eigenvarieties, families, forms
Ämneskategorier Matematik

Sammanfattning

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.

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