Integral points on hyperelliptic curves
نویسندگان
چکیده
منابع مشابه
Integral 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...
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We give a formula for the component at p of the p-adic height pairing of a divisor of degree 0 on a hyperelliptic curve. We use this to give a Chabauty-like method for finding p-adic approximations to p-integral points on such curves when the Mordell-Weil rank of the Jacobian equals the genus. In this case we get an explicit bound for the number of such p-integral points, and we are able to use...
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We give an overview over recent results concerning rational points on hyperelliptic curves. One result says that ‘most’ hyperelliptic curves of high genus have very few rational points. Another result gives a bound on the number of rational points in terms of the genus and the Mordell-Weil rank, provided the latter is sufficiently small. The first result relies on work by Bhargava and Gross on ...
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We describe how to prove the Mordell-Weil theorem for Jacobians of hyperelliptic curves over Q and how to compute the rank and generators for the Mordell-Weil group.
متن کاملSieving for rational points on hyperelliptic curves
We give a new and efficient method of sieving for rational points on hyperelliptic curves. This method is often successful in proving that a given hyperelliptic curve, suspected to have no rational points, does in fact have no rational points; we have often found this to be the case even when our curve has points over all localizations Qp. We illustrate the practicality of the method with some ...
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ژورنال
عنوان ژورنال: Algebra & Number Theory
سال: 2008
ISSN: 1937-0652
DOI: 10.2140/ant.2008.2.859