Integral Points on Certain Elliptic Curves
نویسندگان
چکیده
منابع مشابه
Computing Integral Points on Elliptic Curves
By a famous theorem of Siegel [S], the number of integral points on an elliptic curve E over an algebraic number field K is finite. A conjecture of Lang and Demjanenko [L3] states that, for a quasiminimal model of E over K, this number is bounded by a constant depending only on the rank of E over K and on K (see also [HSi], [Zi4]). This conjecture was proved by Silverman [Si1] for elliptic curv...
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1. Introduction. By a famous theorem of Siegel [S], the number of integral points on an elliptic curve E over an algebraic number field K is finite. A conjecture of Lang and Demyanenko (see [L3], p. 140) states that, for a quasiminimal model of E over K, this number is bounded by a constant depending only on the rank of E over K and on K (see also [HSi], [Zi4]). This conjecture was proved by Si...
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If E is a minimal elliptic curve defined over Z, we obtain a bound C, depending only on the global Tamagawa number of E, such that for any point P ∈ E(Q), nP is integral for at most one value of n > C. As a corollary, we show that if E/Q is a fixed elliptic curve, then for all twists E′ of E of sufficient height, and all torsion-free, rank-one subgroups Γ ⊆ E′(Q), Γ contains at most 6 integral ...
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We investigate a problem considered by Zagier and Elkies, of finding large integral points on elliptic curves. By writing down a generic polynomial solution and equating coefficients, we are led to suspect four extremal cases that still might have nondegenerate solutions. Each of these cases gives rise to a polynomial system of equations, the first being solved by Elkies in 1988 using the resul...
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0. In the course of another investigation we came across a sequence of polynomials Pd ∈ Z[x, y], such that Pd is absolutely irreducible, of degree d, has low height and at least d + 2d + 3 integral solutions to Pd(x, y) = 0. We know of no other family of polynomials of increasing degree with as many integral (or even rational) solutions in terms of their degree, except of course those with infi...
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ژورنال
عنوان ژورنال: Rendiconti del Seminario Matematico della Università di Padova
سال: 2008
ISSN: 0041-8994
DOI: 10.4171/rsmup/119-1