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Arithmetic of p-irregular modular forms : families and p-adic L-functions
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Betina, Adel and Williams, Christopher David (2021) Arithmetic of p-irregular modular forms : families and p-adic L-functions. Mathematika, 67 (4). pp. 917-948. doi:10.1112/mtk.12107
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Official URL: https://doi.org/10.1112/mtk.12107
Abstract
Let fnew be a classical newform of weight ≥2 and prime to p level. We study the arithmetic of fnew and its unique p-stabilisation f when fnew is p-irregular, that is, when its Hecke polynomial at p admits a single repeated root. In particular, we study p-adic weight families through f and its base-change to an imaginary quadratic field F where p splits, and prove that the respective eigencurves are both Gorenstein at f. We use this to construct a two-variable p-adic L-function over a Coleman family through f, and a three-variable p-adic L-function over the base-change of this family to F. We relate the two- and three-variable p-adic L-functions via p-adic Artin formalism. These results are used in work of Xin Wan to prove the Iwasawa Main Conjecture in this case. In an appendix, we prove results towards Hida duality for modular symbols, constructing a pairing between Hecke algebras and families of overconvergent modular symbols and proving that it is non-degenerate locally around any cusp form. This allows us to control the sizes of (classical and Bianchi) Hecke algebras in families.
| Item Type: | Journal Article | ||||||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||||||
| Divisions: | Faculty of Science > Mathematics | ||||||||||||
| Library of Congress Subject Headings (LCSH): | Forms, Modular, Hecke operators, p-adic numbers, L-functions | ||||||||||||
| Journal or Publication Title: | Mathematika | ||||||||||||
| Publisher: | London Mathematical Society | ||||||||||||
| ISSN: | 0025-5793 | ||||||||||||
| Official Date: | October 2021 | ||||||||||||
| Dates: |
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| Volume: | 67 | ||||||||||||
| Number: | 4 | ||||||||||||
| Page Range: | pp. 917-948 | ||||||||||||
| DOI: | 10.1112/mtk.12107 | ||||||||||||
| Status: | Peer Reviewed | ||||||||||||
| Publication Status: | Published | ||||||||||||
| Publisher Statement: | This is the peer reviewed version of the following article: Betina, A. and Williams, C. (2021), ARITHMETIC OF p-IRREGULAR MODULAR FORMS: FAMILIES AND p-ADIC L-FUNCTIONS. Mathematika, 67: 917-948., which has been published in final form at https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.12107 . This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions. | ||||||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||||||
| RIOXX Funder/Project Grant: |
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