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On the equation a p + 2 α b p + c p = 0

Kenneth A. Ribet — 1997

Acta Arithmetica

We discuss the equation a p + 2 α b p + c p = 0 in which a, b, and c are non-zero relatively prime integers, p is an odd prime number, and α is a positive integer. The technique used to prove Fermat’s Last Theorem shows that the equation has no solutions with α < 1 or b even. When α=1 and b is odd, there are the two trivial solutions (±1, ∓ 1, ±1). In 1952, Dénes conjectured that these are the only ones. Using methods of Darmon, we prove this conjecture for p≡ 1 mod 4.

Galois theory and torsion points on curves

Matthew H. BakerKenneth A. Ribet — 2003

Journal de théorie des nombres de Bordeaux

In this paper, we survey some Galois-theoretic techniques for studying torsion points on curves. In particular, we give new proofs of some results of A. Tamagawa and the present authors for studying torsion points on curves with “ordinary good” or “ordinary semistable” reduction at a given prime. We also give new proofs of : (1) the Manin-Mumford conjecture : there are only finitely many torsion points lying on a curve of genus at least 2 embedded in its jacobian by an Albanese map; and (2) the...

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