منابع مشابه
Ranks of Elliptic Curves
This paper gives a general survey of ranks of elliptic curves over the field of rational numbers. The rank is a measure of the size of the set of rational points. The paper includes discussions of the Birch and SwinnertonDyer Conjecture, the Parity Conjecture, ranks in families of quadratic twists, and ways to search for elliptic curves of large rank.
متن کاملRanks of elliptic curves over function fields
We present experimental evidence to support the widely held belief that one half of all elliptic curves have infinitely many rational points. The method used to gather this evidence is a refinement of an algorithm due to the author which is based upon rigid and crystalline cohomology.
متن کاملRanks of Elliptic Curves in Cubic Extensions
For an elliptic curve over the rationals, Goldfeld’s conjecture [4] asserts that the analytic rank ords=1 L(Ed/Q, s) of quadratic twists Ed of E is positive for squarefree d’s with density 1/2. In other words, the analytic rank of E goes up in quadratic extensions Q( √ d)/Q half of the time. In particular, for every E/Q there are (a) infinitely many quadratic extensions where the rank goes up, ...
متن کاملRanks of Elliptic Curves in Families of Quadratic Twists
In this paper we reformulate the question of whether the ranks of the quadratic twists of an elliptic curve over Q are bounded, into the question of the whether certain infinite series converge. Our results were inspired by ideas in a paper of Gouvêa and Mazur [2]. Fix a, b, c ∈ Z such that f(x) = x + ax + bx+ c has 3 distinct complex roots, and let E be the elliptic curve y = f(x). For D ∈ Z−{...
متن کاملRanks of Quadratic Twists of Elliptic Curves over Fq(t)
Some notes on the analogy between number theory over Z and Fq[t] and an attempt to translate a paper of Gouvêa and Mazur on ranks of quadratic twists of elliptic curves over Q to elliptic curves over Fq(t).
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ژورنال
عنوان ژورنال: Bulletin of the American Mathematical Society
سال: 2002
ISSN: 0273-0979
DOI: 10.1090/s0273-0979-02-00952-7