## Abstract

Let X = P_{1} / {0, 1,∞}, and let S denote a finite set of prime numbers. In an article of 2005, Kim gave a new proof of Siegel's theorem for X: the set X(Z[S^{-1}]) of S-integral points of X is finite. The proof relies on a 'nonabelian' version of the classical Chabauty method. At its heart is a modular interpretation of unipotent p-adic Hodge theory, given by a tower of morphisms hn between certain Q_{p}-varieties. We set out to obtain a better understanding of h_{2}. Its mysterious piece is a polynomial in 2|S| variables. Our main theorem states that this polynomial is quadratic, and gives a procedure for writing its coefficients in terms of p-adic logarithms and dilogarithms.

Original language | English |
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Pages (from-to) | 133-171 |

Number of pages | 39 |

Journal | Proceedings of the London Mathematical Society |

Volume | 110 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2015 |

Externally published | Yes |