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Abstract |
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The S-unit equation asks for solutions to a+b=c, where a, b and c are integers, all of whose prime factors lie in a fixed finite set S of primes. A famous theorem attributed to Siegel asserts that the number of solutions to this equation is finite, and work of Evertse shows that moreover the number of solutions is bounded by a constant depending on the size of S only. In 2005, Minhyong Kim re-proved Siegel's theorem by a careful decent argument using the fundamental group of P^1 minus three points. Kim's original proof was ineffective, meaning that it didn't provide upper bound on the number of solutions. In this talk, I will describe how one can refine Kim's method to re-prove Evertse's result bounding the number of solutions to the S-unit equation in terms of #S (our bounds are weaker than Evertse's). The key difference with Kim's original argument is that we replace certain dimension estimates with Hilbert series estimates, and it is this more refined information which allows us to control the number of rational points. Time permitting, I will also sketch how one might generalise this theory to more general curves. |