The Mordell-Weil Sieve
نویسنده
چکیده
We discuss the Mordell-Weil sieve as a general technique for proving results concerning rational points on a given curve. In the special case of curves of genus 2, we describe quite explicitly how the relevant local information can be obtained if one does not want to restrict to mod p information at primes of good reduction. We describe our implementation of the Mordell-Weil sieve algorithm and discuss its efficiency.
منابع مشابه
Chabauty and the Mordell-Weil Sieve
These notes are based on lectures given at the “Arithmetic of Hyperelliptic Curves” workshop, Ohrid, Macedonia, 28 August–5 September 2014. They offer a brief (if somewhat imprecise) sketch of various methods for computing the set of rational points on a curve, focusing on Chabauty and the Mordell–Weil sieve.
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 کاملIntegral Points on Hyperelliptic Curves
Let C : Y 2 = anX + · · · + a0 be a hyperelliptic curve with the ai rational integers, n ≥ 5, and the polynomial on the right irreducible. Let J be its Jacobian. We give a completely explicit upper bound for the integral points on the model C, provided we know at least one rational point on C and a Mordell–Weil basis for J(Q). We also explain a powerful refinement of the Mordell–Weil sieve whic...
متن کاملHow to Obtain Global Information From Computations Over Finite Fields
This is an extended version of the talk I gave at the summer school in Göttingen in July 2007. We discuss the Mordell-Weil Sieve and some applications.
متن کاملOn a Problem of Hajdu and Tengely
We answer a question asked by Hajdu and Tengely: The only arithmetic progression in coprime integers of the form (a, b, c, d) is (1, 1, 1, 1). For the proof, we first reduce the problem to that of determining the sets of rational points on three specific hyperelliptic curves of genus 4. A 2-cover descent computation shows that there are no rational points on two of these curves. We find generat...
متن کامل