Publications
Computing integral points on genus 2 hyperelliptic curves estimating hyperelliptic logarithms
Abstract
Let C:y2 = f(x) be a hyperelliptic curve with f(x) in QQ[x] monic and irreducible over QQ of degree n = 5. Let J be its Jacobian. We give an algorithm to explicitly determine the set of integral points on C provided we know at least one rational point on C, a Mordell--Weil basis for J(QQ) and an upper bound for the height of the integral points. We estimate the hyperelliptic logarithms of the elements on the Mordell--Weil basis to reduce the bound for the height of the integral points to manageable proportions using linear forms. We then find all the integral points on C by a direct search. We illustrate the practicality of the method by finding all integral points on the rank 5, genus 2 curve y2 = x5 + 105x4 + 4405x3 + 92295x2 + 965794x + 4038280.
S-integral points on hyperelliptic curves
Abstract
Let C : Y^2 = a_n X^n +...+ a_0 be a hyperelliptic curve with the a_i rational integers, n \geq 5, and the polynomial on the right irreducible. Let J be its Jacobian. Let S be a finite set of rational primes. We give a completely explicit upper bound for the size of the S-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 use a refinement of the Mordell-Weil sieve which, combined with the upper bound, is capable of determining all the S-integral points. The method is illustrated by determining the S-integral points on the genus 2 hyperelliptic model Y^2 - Y = X^5 - X for the set S of the first 22 primes.
International Journal of Number Theory, Vol.07 (No.03) 2011. p. 803. ISSN 1793-0421. 22 pages.
PhD Thesis. S-integral points on hyperelliptic curves
Abstract
Let C : Y^2 = a_n X^n + ... + a_0 be a hyperelliptic curve with the a_i rational integers, n \geq 5, and the polynomial on the right irreducible. Let J be its Jacobian. Let S be a finite set of rational primes. In this thesis we give explicit methods for finding all of the integral and S-integral points on C. The work consists of the following parts.
- We give a completely explicit upper bound for the size of the S-integral points on the model C, provided we know at least one rational point on C and a Mordell-Weil basis for J(Q). There is a refinement of the Mordell-Weil sieve that can then be used to determine all the S-integral points on the curve.
- In the case the curve has genus 2 and the polynomial defining the curve has real roots only, we reduce the upper bound for the size of the integral points to manageable proportions using linear forms in hyperelliptic logarithms. We then find all of the integral points on C by a direct search.
- We give an algorithm for the computation of hyperelliptic logarithms of real points on genus 2 curves defined by a polynomial having real roots only. This is needed for 2.