Bounding the number of rational points on certain curves of high rank
نویسنده
چکیده
Let K be a number field and let C be a curve of genus g > 1 defined over K. In this dissertation we describe techniques for bounding the number of K-rational points on C. In Chapter I we discuss Chabauty techniques. This is a review and synthesis of previously known material, both published and unpublished. We have tried to eliminate unnecessary restrictions, such as assumptions of good reduction or the existence of a known rational point on the curve. We have also attempted to clearly state the circumstances under which Chabauty techniques can be applied. Our primary goal is to provide a flexible and powerful tool for computing on specific curves. In Chapter II we develop a technique which, given a K-rational isogeny to the Jacobian of C, produces a positive integer n and a collection of covers of C with the property that the set of K-rational points in the collection is in n-to-1 correspondence with the set of K-rational points on C. If Chabauty is applicable to every curve in the collection, then we can use the covers to bound the number of K-rational points on C. The examples in Chapters I and II are taken from problem VI.17 in the Arabic text of the Arithmetica. Chapter III is devoted to the background calculations for this problem. When we assemble the pieces, we discover that the solution given by Diophantus is the only positive rational solution to this problem.
منابع مشابه
On the elliptic curves of the form $ y^2=x^3-3px $
By the Mordell-Weil theorem, the group of rational points on an elliptic curve over a number field is a finitely generated abelian group. There is no known algorithm for finding the rank of this group. This paper computes the rank of the family $ E_p:y^2=x^3-3px $ of elliptic curves, where p is a prime.
متن کاملOn the rank of certain parametrized elliptic curves
In this paper the family of elliptic curves over Q given by the equation Ep :Y2 = (X - p)3 + X3 + (X + p)3 where p is a prime number, is studied. Itis shown that the maximal rank of the elliptic curves is at most 3 and someconditions under which we have rank(Ep(Q)) = 0 or rank(Ep(Q)) = 1 orrank(Ep(Q))≥2 are given.
متن کاملOn the Elliptic Curves of the Form $y^2 = x^3 − pqx$
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 ...
متن کاملComplete characterization of the Mordell-Weil group of some families of elliptic curves
The Mordell-Weil theorem states that the group of rational points on an elliptic curve over the rational numbers is a finitely generated abelian group. In our previous paper, H. Daghigh, and S. Didari, On the elliptic curves of the form $ y^2=x^3-3px$, Bull. Iranian Math. Soc. 40 (2014), no. 5, 1119--1133., using Selmer groups, we have shown that for a prime $p...
متن کاملOn the Number of Rational Iterated Pre-images of the Origin under Quadratic Dynamical Systems
For a quadratic endomorphism of the affine line defined over the rationals, we consider the problem of bounding the number of rational points that eventually land at the origin after iteration. In the article “Uniform Bounds on Pre-Images Under Quadratic Dynamical Systems,” by the present authors and five others, it was shown that the number of rational pre-images of the origin is bounded as on...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2001