The Ubiquity of Elliptic Curves
نویسندگان
چکیده
منابع مشابه
On the ubiquity of trivial torsion on elliptic curves
The purpose of this paper is to give a down-to-earth proof of the well-known fact that a randomly chosen elliptic curve over the rationals is most likely to have trivial torsion. Mathematics Subject Classification (2000). 11G05, 14H52.
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Let $E$ be an elliptic curve over $Bbb{Q}$ with the given Weierstrass equation $ y^2=x^3+ax+b$. If $D$ is a squarefree integer, then let $E^{(D)}$ denote the $D$-quadratic twist of $E$ that is given by $E^{(D)}: y^2=x^3+aD^2x+bD^3$. Let $E^{(D)}(Bbb{Q})$ be the group of $Bbb{Q}$-rational points of $E^{(D)}$. It is conjectured by J. Silverman that there are infinitely many primes $p$ for which $...
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By the Mordell- Weil theorem, the group of rational points on an elliptic curve over a number field is a finitely generated abelian group. This paper studies the rank of the family Epq:y2=x3-pqx of elliptic curves, where p and q are distinct primes. We give infinite families of elliptic curves of the form y2=x3-pqx with rank two, three and four, assuming a conjecture of Schinzel ...
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ژورنال
عنوان ژورنال: Notices of the American Mathematical Society
سال: 2019
ISSN: 0002-9920,1088-9477
DOI: 10.1090/noti1789