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On Serre's uniformity conjecture for semistable elliptic curves over totally real fields
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Anni, Samuele and Siksek, Samir (2015) On Serre's uniformity conjecture for semistable elliptic curves over totally real fields. Mathematische Zeitschrift, 281 (1-2). pp. 193-199. doi:10.1007/s00209-015-1478-8 ISSN 0025-5874.
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Official URL: https://doi.org/10.1007/s00209-015-1478-8
Abstract
Let K be a totally real field, and let S be a finite set of non-archimedean places of K . It follows from the work of Merel, Momose and David that there is a constant BK,S so that if E is an elliptic curve defined over K , semistable outside S , then for all p>BK,S , the representation ρ¯¯¯E,p is irreducible. We combine this with modularity and level lowering to show the existence of an effectively computable constant CK,S , and an effectively computable set of elliptic curves over K with CM E1,⋯,En such that the following holds. If E is an elliptic curve over K semistable outside S , and p>CK,S is prime, then either ρ¯¯¯E,p is surjective, or ρ¯¯¯E,p∼ρ¯¯¯Ei,p for some i=1,…,n .
| Item Type: | Journal Article | |||||||||
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| Subjects: | Q Science > QA Mathematics | |||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | |||||||||
| Library of Congress Subject Headings (LCSH): | Curves, Elliptic , Galois theory, Hilbert modules | |||||||||
| Journal or Publication Title: | Mathematische Zeitschrift | |||||||||
| Publisher: | Springer | |||||||||
| ISSN: | 0025-5874 | |||||||||
| Official Date: | October 2015 | |||||||||
| Dates: |
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| Volume: | 281 | |||||||||
| Number: | 1-2 | |||||||||
| Page Range: | pp. 193-199 | |||||||||
| DOI: | 10.1007/s00209-015-1478-8 | |||||||||
| Status: | Peer Reviewed | |||||||||
| Publication Status: | Published | |||||||||
| Access rights to Published version: | Restricted or Subscription Access | |||||||||
| Date of first compliant deposit: | 22 August 2019 | |||||||||
| Date of first compliant Open Access: | 30 August 2019 | |||||||||
| RIOXX Funder/Project Grant: |
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