Kolyvagin’s Theorem

Please enable javascript in your browser.

Hamidi, Mohammad Masih | 2022

358 Viewed

Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 55377 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Rastegar, Arash
Abstract:
Kolyvagin’s theorem, stated in the late 1980s, is a very important theorem in the theory of elliptic curves, which, together with Gross-Zagier’s work, results in a special case of the Birch and Swinerton-Dyer conjecture, which relates algebraic and analytic properties of the rational elliptic curves to each other. Although the correctness of this conjecture, in general, is an open problem, Kolyvagin’s idea and the concepts he introduced while proving his theorem are very powerful and can be applied to more general situations. The purpose of writing this dissertation is to collect the equivalent forms of Kolyvagin’s theorem with proofs of the main results. We will also give a brief overview of the results obtained after Kolyvagin and to date
Keywords:
Cohomology Group ; Elliptic Curve ; Frobenius Reciprocity ; Eulerian Method ; Birch and Swinerton-Dyer Conjecture ; Gross-Zagier and Kolyvagin Theorem ; Gross-Zagier Formula ; Heegner System