Rational and Integral Points on Quadratic Twists of a Given Hyperelliptic Curve
نویسندگان
چکیده
منابع مشابه
Rational and Integral Points on Quadratic Twists of a given Hyperelliptic Curve
We show that the abc-conjecture implies that few quadratic twists of a given hyperelliptic curve have any non-trivial rational or integral points; and indicate how these considerations dovetail with other predictions.
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In this paper, we study bounds for the number of rational points on twists C ′ of a fixed curve C over a number field K, under the condition that the group of K-rational points on the Jacobian J ′ of C ′ has rank smaller than the genus of C ′. The main result is that with some explicitly given finitely many possible exceptions, we have a bound of the form 2r + c, where r is the rank of J ′(K) a...
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THEOREM 1. Let K/Q be a number field, and let C/K be a smooth projective curve of genus at least 2. For each^class ^ e i / 1 (Gal (AT/AT), Aut(C)), let C% be the twist of C by x (cf. [11, X §3]) and le\Jx = J ac (Q be the Jacobian variety of Cx. There is a constant y = y{C/K) such that #C,( /Q^r7 a n k * ( K ) forallxeH{G2\{K/K),A\xi{C)). We illustrate this general theorem by applying it to a p...
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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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ژورنال
عنوان ژورنال: International Mathematics Research Notices
سال: 2010
ISSN: 1073-7928,1687-0247
DOI: 10.1093/imrn/rnm027