1995
DOI: 10.2140/pjm.1995.167.263
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S-integer points on elliptic curves

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Cited by 34 publications
(25 citation statements)
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“…Silverman [16,Section 4,Theorem] showed that Lang's conjecture holds for any elliptic curve with j-invariant non-integral for at most R places of K (note that this includes our curves, E b , since their j-invariant is 0), but with C 1 dependent on K and R. Gross and Silverman [9,Proposition 3(3)] proved an explicit version of this result from which it follows that for non-torsion points, P , on E b , we have h(P ) > 3 · 10 −14 log |∆ (E b )| .…”
Section: Lower Boundsmentioning
confidence: 99%
“…Silverman [16,Section 4,Theorem] showed that Lang's conjecture holds for any elliptic curve with j-invariant non-integral for at most R places of K (note that this includes our curves, E b , since their j-invariant is 0), but with C 1 dependent on K and R. Gross and Silverman [9,Proposition 3(3)] proved an explicit version of this result from which it follows that for non-torsion points, P , on E b , we have h(P ) > 3 · 10 −14 log |∆ (E b )| .…”
Section: Lower Boundsmentioning
confidence: 99%
“…However he did not compute the constants involved. Gross and Silverman [1995] used Roth's theorem to obtain an explicit bound. To state their theorem, let us write the Weierstrass equation of the elliptic curve E as…”
Section: Introductionmentioning
confidence: 99%
“…The best result is due to Silverman [27], who shows the conjecture is true when the j-invariant j(E) is integral, and in fact proves that for every number field K there is a constant C K such that the number of S-integral points over K is bounded by C (1+r)(1+δ)+|S| K , where δ is the number of primes of K at which j(E) is nonintegral. Explicit constants appear in [14]. Szpiro's conjecture [31], which is equivalent to the Masser-Oesterlé ABC conjecture [24], states that ∆ ≪ N 6+o(1) ; Hindry and Silverman [15] show this implies that the number of S-integral points on a quasi-minimal model of E/K is bounded by C (1+r)σ E/K +|S| K where σ E/K is the Szpiro ratio, which is the ratio of the logarithms of the norms of the discriminant and the conductor of E/K.…”
Section: Introductionmentioning
confidence: 99%