Algebra and Topology Seminar

4pm Tuesday 9 December 2003

Victor Scharaschkin
University of Queensland

Elliptic Curves and the ABC Conjecture

An abc triple is a triple a, b, c of relatively prime integers satisfying a + b = c. Let rad(n) denote the product of distinct primes dividing n, without multiplicity. The quality q of the triple is defined to be log max (|a|, |b|, |c|) / log rad abc, and the abc conjecture is that the lim sup of q values over all triples is 1. If true, this would imply many famous results such as Fermat's Last Theorem (up to a finite number of exceptions), and Faltings' Theorem.

Following an idea of Elkies, we show how maps to the projective line can be used to generate interesting abc triples, and discuss what further ingredients would be needed to prove that their quality approaches 1.




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