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Selmer Group and Proof of Catalan’s Conjecture

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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.

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Notes

  1. 1.

    We do not assume from our reader any knowledge of the theory of elliptic curves.

  2. 2.

    Recall that the q-torsion subgroup of a multiplicatively written abelian group A is \(A[q] =\{ x \in A: x^{q} = 1\}\).

  3. 3.

    Alternatively, one could determine both dimensions using Proposition D.13.

References

  1. Thaine, F.: On the ideal class groups of real abelian number fields. Ann. Math. 128, 1–18 (1988)

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© 2014 Springer International Publishing Switzerland

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Bilu, Y.F., Bugeaud, Y., Mignotte, M. (2014). Selmer Group and Proof of Catalan’s Conjecture. In: The Problem of Catalan. Springer, Cham. https://doi.org/10.1007/978-3-319-10094-4_11

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