Corralesrodriganez 1997 support problem elliptic analogue

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theorem_1: Corrales-Rodrigáñez and Schoof's theorem that if, for almost all prime ideals of a number field and all positive n, x to the n congruent to one forces y to the n congruent to one, then y is a power of x.

theorem_2: Corrales-Rodrigáñez and Schoof's elliptic analogue of their Theorem 1: if nP vanishing modulo almost every good prime forces nQ to vanish, then Q is an F-rational endomorphism image of P or both points are torsion.

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Corrales-Rodrigáñez, Capi and Schoof, René, The support problem and its elliptic analogue. J. Number Theory 64 (1997), 276--290. DOI: 10.1006/jnth.1997.2114.

The paper answers a question Erdos asked at the 1988 Banff number theory conference: if positive integers x, y satisfy Supp(x^n - 1) = Supp(y^n - 1) for all n > 0, must x = y? Theorem 1 proves the general multiplicative statement: for a number field F and x, y in F*, if y^n = 1 (mod p) whenever x^n = 1 (mod p), for all n and almost all prime ideals p of the ring of integers of F, then y is a power of x; specializing to F = Q gives Erdos's answer, since the two-sided hypothesis forces x = y^{±1} or both to be roots of unity. Theorem 2 is the elliptic analog: for an elliptic curve E over F and F-rational points P, Q, if nQ = 0 in E(F_p) whenever nP = 0 in E(F_p) for every n and almost every prime of good reduction, then either Q = fP for some F-rational endomorphism f of E or both P and Q are torsion. The proof of Theorem 1 combines the Frobenius density theorem, Kummer theory and Dirichlet's unit theorem; the proof of Theorem 2 follows the same three steps with division points of E in place of roots of unity and closes with Siegel's theorem on integral points. The authors note that the straightforward generalization fails for the additive group and GL_n with n > 1, and remark that an analogue of Theorem 2 for abelian varieties would be interesting. The paper is the source cited for Problem 1214, Erdos's support problem.

Source: https://reneschoof.github.io/papers.html. The copy read for this card, from the author's page named here, is the publisher's PDF and prints "Copyright © 1997 by Academic Press All rights of reproduction in any form reserved." on printed p. 276, every other right reserved.

Bears on. #1214: the paper says it follows easily from Theorem 1 that the two-sided hypothesis forces x = y^{±1} or both x and y roots of unity, and that applying this with F = Q to positive integers x, y answers the problem's question (p. 277); it calls the answer affirmative (p. 276).

Results.

• Theorem 1 (p. 277): for a number field F and x, y in F*, if for almost all prime ideals p and all positive integers n, x^n = 1 (mod p) implies y^n = 1 (mod p), then y is a power of x. The paper deduces that Supp(x^n-1) = Supp(y^n-1) for all n >= 1 forces x = y for positive integers x, y.
• Theorem 2 (p. 277): elliptic analogue: for F-rational points P, Q on an elliptic curve E over F, if for every integer n and almost every prime of good reduction nP = 0 in E(F_p) implies nQ = 0 in E(F_p), then Q = fP for an F-rational endomorphism f of E, or P and Q are both torsion.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.