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Showing 1 to 5 of 5 for “"Birch and Swinnerton-Dyer conjecture"”.
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The Birch and Swinnerton-Dyer Conjecture for elliptic curves
… dissertation is to provide an exposition of the Birch and Swinnerton-Dyer Conjecture, considered by many to be one of the most important unsolved problems in modern Mathematics. A review of topics in Algebraic Number Theory and Algebraic Geometry is provided in order to provide a characterisation …
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Solutions of the Cubic Fermat Equation in Quadratic Fields
… $d$, but not for all. Our argument assumes the Birch and Swinnerton-Dyer conjecture and follows a similar argument as Tunnell's solution to the congruent number problem.
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Coleman integration for hyperelliptic curves : algorithms and applications
… Coleman integrals on hyperelliptic curves and discuss some immediate applications. I give algorithms to compute single and iterated integrals on odd models of hyperelliptic curves, as well as the necessary modifications to iplemieit these algorithms for even models. Furthermore, I show how …
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Computing canonical heights on Jacobians
… Building on an existing algorithm due to Flynn and Smart with modifications by Stoll we generalize efficient methods for the computation of canonical heights on elliptic curves to the case of Jacobian surfaces. The main tools are the explicit theory of the Kummer surface associated to a Jacobian …
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On the main conjectures of Iwasawa theory for certain elliptic curves with complex multiplication
The conjecture of Birch and Swinnerton-Dyer is unquestionably one of the most important open problems in number theory today. Let $E$ be an elliptic curve defined over an imaginary quadratic field $K$ contained in $\mathbb{C}$, and suppose that $E$ has complex multiplication by the ring of integers …