Book Review: Rational points on elliptic curves
نویسندگان
چکیده
منابع مشابه
Constructing Rational Points on Elliptic Curves Using Heegner Points
These are the notes I wrote for my candidacy talk. The aims for this talk were to understand Heegner points, examine the different ways they can be characterized, and get an idea of how to construct rational points on an elliptic curve using Heegner points. I cite some good references at the end if you are also trying to begin learning about this beautiful topic. The goal of this talk is to exp...
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The existence of infinitely many elliptic curves with no rational points except the origin oo is proved by refining a theorem of DavenportHeilbronn. The existence of infinitely many quadratic fields with the Iwasawa invariant A3 = 0 is proved at the same time.
متن کاملOn the Denominators of Rational Points on Elliptic Curves
Let x(P ) = AP /B2 P denote the x-coordinate of the rational point P on an elliptic curve in Weierstrass form. We consider when BP can be a perfect power or a prime. Using Faltings’ theorem, we show that for a fixed f > 1, there are only finitely many rational points P with BP equal to an fth power. Where descent via an isogeny is possible, we show that there are only finitely many rational poi...
متن کاملCycles on modular varieties and rational points on elliptic curves
This is a summary of a three-part lecture series given at the meeting on “Explicit methods in number theory” that was held in Oberwolfach from July 12 to 18, 2009. The theme of this lecture series was the explicit construction of algebraic points on elliptic curves from cycles on modular varieties. Given a fixed elliptic curve E over Q, the goal is to better understand the group E(Q̄) of algebra...
متن کاملDescending Rational Points on Elliptic Curves to Smaller Fields
In this paper, we study the Mordell-Weil group of an elliptic curve as a Galois module. We consider an elliptic curve E defined over a number field K whose Mordell-Weil rank over a Galois extension F is 1, 2 or 3. We show that E acquires a point (points) of infinite order over a field whose Galois group is one of Cn×Cm (n = 1, 2, 3, 4, 6, m = 1, 2), Dn×Cm (n = 2, 3, 4, 6, m = 1, 2), A4×Cm (m = ...
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ژورنال
عنوان ژورنال: Bulletin of the American Mathematical Society
سال: 1994
ISSN: 0273-0979
DOI: 10.1090/s0273-0979-1994-00465-3