Slopes of modular forms and geometry of eigencurves
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866908847672655872 |
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| author | Liu, Ruochuan Truong, Nha Xuan Xiao, Liang Zhao, Bin |
| author_facet | Liu, Ruochuan Truong, Nha Xuan Xiao, Liang Zhao, Bin |
| contents | Under a stronger genericity condition, we prove the local analogue of ghost conjecture of Bergdall and Pollack. As applications, we deduce in this case (a) a folklore conjecture of Breuil--Buzzard--Emerton on the crystalline slopes of Kisin's crystabelian deformation spaces, (b) Gouvea's $\lfloor\frac{k-1}{p+1}\rfloor$-conjecture on slopes of modular forms, and (c) the finiteness of irreducible components of the eigencurve. In addition, applying combinatorial arguments by Bergdall and Pollack, and by Ren, we deduce as corollaries in the reducible and strongly generic case, (d) Gouvea--Mazur conjecture, (e) a variant of Gouvea's conjecture on slope distributions, and (f) a refined version of Coleman's spectral halo conjecture. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2302_07697 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Slopes of modular forms and geometry of eigencurves Liu, Ruochuan Truong, Nha Xuan Xiao, Liang Zhao, Bin Number Theory 11F33, 11F85 Under a stronger genericity condition, we prove the local analogue of ghost conjecture of Bergdall and Pollack. As applications, we deduce in this case (a) a folklore conjecture of Breuil--Buzzard--Emerton on the crystalline slopes of Kisin's crystabelian deformation spaces, (b) Gouvea's $\lfloor\frac{k-1}{p+1}\rfloor$-conjecture on slopes of modular forms, and (c) the finiteness of irreducible components of the eigencurve. In addition, applying combinatorial arguments by Bergdall and Pollack, and by Ren, we deduce as corollaries in the reducible and strongly generic case, (d) Gouvea--Mazur conjecture, (e) a variant of Gouvea's conjecture on slope distributions, and (f) a refined version of Coleman's spectral halo conjecture. |
| title | Slopes of modular forms and geometry of eigencurves |
| topic | Number Theory 11F33, 11F85 |
| url | https://arxiv.org/abs/2302.07697 |