Various Versions of the Sato-Tate Conjecture

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Shavali , Alireza | 2021

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 53964 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Rastegar, Arash; Gholamzadeh Mahmoudi, Mohammad
Abstract:
The Sato-Tate conjecture is an important conjecture regarding the distribution of the Frobenius traces of a family of elliptic curves over finite fields obtained from the reductions of an elliptic curve without CM over a number field modulo the prime ideals of its ring of integers. The statement is that the sequence of normalized Frobenius traces should follow a semicircle distribution. It was discovered by Mikio Sato and reformulated by John Tate in terms of L-functions around 1960. A complete proof of the conjecture for elliptic curves over totally real fields was published in 2008 by R. Taylor et al. under some mild technical assumptions. In addition to the original Sato-Tate conjecture, there are various versions and generalizations of the conjecture in different contexts. The goal of this thesis is to collect these different versions with the proofs or the basic ideas of the proofs if there is any proof and also to formulate some new versions of the conjecture
Keywords:
Elliptic Curve ; Galois Representations ; Frobenius Algebra ; Equidistribution ; Sato-Tate Conjecture