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Applications of the Chebotarev density theorem to elliptic curves

Resource type
Thesis type
(Thesis) M.Sc.
Date created
2024-03-12
Authors/Contributors
Abstract
A natural question to ask in the study of ℓ-adic and mod-ℓ representations attached to elliptic curves over Q is what conditions guarantee two such representations will be isomorphic. Related to this question is if we suppose two such representations are not isomorphic, does there exist a 'certificate' which proves they are not isomorphic? One way to show that two representations are not isomorphic is to show that they have different trace values on a group element. For an elliptic curve E over Q and a prime p of good reduction, one defines the trace of Frobenius ap(E), which is an integer independent of ℓ and arises as a trace value. In this thesis, we are interested in giving upper bounds for the smallest prime p such that ap(E) ̸= ap(E′). Serre [30] gives a classical asymptotic bound, although the constants are quite large. Recent work of Mayle-Wang [18] provides an explicit bound with smaller constants. We expand on Mayle-Wang's work and further reduce the constants appearing in their result. Both methods use explicit forms of the Chebotarev density theorem which assume the generalized Riemann hypothesis.
Document
Extent
70 pages.
Identifier
etd22956
Copyright statement
Copyright is held by the author(s).
Permissions
This thesis may be printed or downloaded for non-commercial research and scholarly purposes.
Supervisor or Senior Supervisor
Thesis advisor: Chen, Imin
Language
English
Member of collection
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