Average Rank of Elliptic Curves Over Q

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Amiri, Farahnaz | 2016

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 48705 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Rastegar, Arash; Rajaei, Ali
Abstract:
In elliptic curves equation, if the number of solution is infinite, complication of calculation increase quickly. Even the simplest solutioncan be large. y2 + y = x35115523309x − 140826120488927 The x-coordinate number of smallest solution has 5454 digits. The theorem of Mordell and Weil states that “The set E(Q) of rational solutions has the structure of an infinitely generated abelian group.” E(Q) = (Z)rankE(Q) ⊕T Can the rank be arbitrary large? The current record is rank(E)=28.Manjul Behargava has recently made progress on the study of the average rank for all elliptic curves with rational coefficient. Every such curve has a unique equation of the form y2 = x3 +Ax+B, where A and B are integers and △ = 4A3 − 27B2 ̸= 0.We define the height of H(E) as the maximum of the positive integers jAj3 and jBj2. For any positive real number X. there are only finitely many curves with H(E) ≤ X call this number N(X).Define the average rank by limit as X −! 1 of limX−!1 1 X ∑H(E)≤X rank(E) Recently proved that this limit exists, and is equal to ½ . In fact, on average half the elliptic curves have rank zero, and half the elliptic curves have rank one
Keywords:
Elliptic Curve ; Rank ; Selmer Group ; Algebraic Rank ; Analytic Rank