Selmer Group and Proof of Catalan’s Conjecture

Yuri F. Bilu, Yann Bugeaud and Maurice Mignotte
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Yuri F. Bilu: University of Bordeaux and CNRS, Institute of Mathematics of Bordeaux
Yann Bugeaud: University of Strasbourg and CNRS, IRMA, Mathematical Institute
Maurice Mignotte: University of Strasbourg and CNRS, IRMA, Mathematical Institute

Chapter Chapter 11 in The Problem of Catalan, 2014, pp 135-144 from Springer

Abstract: Abstract As we have seen before, Catalan’s problem would be solved if we show that a solution of Catalan’s equation gives rise to a nontrivial element in the real part of Mihăilescu’s ideal. It is natural to look for such elements in the annihilator of the class group. (More precisely, we want to annihilate a related group, called here the qth Selmer group.) Unfortunately, Stickelberger’s theorem is not suitable for this purpose, because the real part of Stickelberger’s ideal is uninteresting. In 1988 Thaine discovered a partial “real” analogue of Stickelberger’s theorem, and Mihăilescu showed that Thaine’s theorem is sufficient for solving Catalan’s problem. In this chapter we reproduce Mihăilescu’s argument.

Keywords: Unit Group; Elliptic Curf; Galois Group; Number Field; Galois Extension (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/978-3-319-10094-4_11

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