Parallel weight 2 points on Hilbert modular eigenvarieties and the parity conjecture
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...
| Main Authors: | , |
|---|---|
| Format: | Journal article |
| Language: | English |
| Published: |
Cambridge University Press
2019
|
| _version_ | 1797093848849055744 |
|---|---|
| author | Newton, J Johansson, C |
| author_facet | Newton, J Johansson, C |
| author_sort | Newton, J |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-07T04:06:06Z |
| format | Journal article |
| id | oxford-uuid:c6350b31-03ae-4673-974c-d78316b9fe2a |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T04:06:06Z |
| publishDate | 2019 |
| publisher | Cambridge University Press |
| record_format | dspace |
| spelling | oxford-uuid:c6350b31-03ae-4673-974c-d78316b9fe2a2022-03-27T06:36:30ZParallel weight 2 points on Hilbert modular eigenvarieties and the parity conjectureJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:c6350b31-03ae-4673-974c-d78316b9fe2aEnglishSymplectic ElementsCambridge University Press2019Newton, JJohansson, CLet 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. |
| spellingShingle | Newton, J Johansson, C Parallel weight 2 points on Hilbert modular eigenvarieties and the parity conjecture |
| title | Parallel weight 2 points on Hilbert modular eigenvarieties and the parity conjecture |
| title_full | Parallel weight 2 points on Hilbert modular eigenvarieties and the parity conjecture |
| title_fullStr | Parallel weight 2 points on Hilbert modular eigenvarieties and the parity conjecture |
| title_full_unstemmed | Parallel weight 2 points on Hilbert modular eigenvarieties and the parity conjecture |
| title_short | Parallel weight 2 points on Hilbert modular eigenvarieties and the parity conjecture |
| title_sort | parallel weight 2 points on hilbert modular eigenvarieties and the parity conjecture |
| work_keys_str_mv | AT newtonj parallelweight2pointsonhilbertmodulareigenvarietiesandtheparityconjecture AT johanssonc parallelweight2pointsonhilbertmodulareigenvarietiesandtheparityconjecture |