Weighted completion of Galois groups and rational points

Abstract: The goal of this talk is to explain how one can use weighted and relative completion to generalize some recent results (arXiv:2010.07331) of Li, Litt, Salter and Srinivasan. In particular, we will explain how algebraic completions of mapping class groups and their arithmetic analogues can be used to give examples, for each $g > 3$, $n \ge 0$, $r > 0$, of a smooth projective curve of genus $g > 3$ over a finitely generated field $K$ of char 0 where $\#C(K) = n$, $\mathrm{Pic}^1(C)(K)$ is non-empty and $\mathrm{Pic}^0(C)(K)$ contains a free abelian subgroup of rank $n + r - 1$.

The talk will begin with a review of how one uses weighted (unipotent) completion of the Galois group of the function field of a moduli spaces of curves (plus other structure) to study rational points of the the universal curve over its generic point. If there is sufficient time, I will explain how this leads to a theory of characteristic classes of rational points.

number theory

Audience: researchers in the topic

( paper | slides | video )

Comments: REFERENCES:

[HM2003] R. Hain, M. Matsumoto: Weighted completion of Galois groups and Galois actions on the fundamental group of P^1-{0,1,infty}. Compositio Math. 139 (2003), 119--167. arXiv:math/0006158

[H2011] R. Hain: Rational points of universal curves, J. Amer. Math. Soc. 24 (2011), 709--769. www.ams.org/journals/jams/2011-24-03/S0894-0347-2011-00693-0/home.html

[LLSP] W. Li, D. Litt, N. Salter, P. Srinivasan: Surface bundles and the section conjecture, arXiv:2010.07331.

[W2019] T. Watanabe: Rational points of universal curves in positive characteristics, Trans. Amer. Math. Soc. 372 (2019), 7639--7676. www.ams.org/journals/tran/2019-372-11/S0002-9947-2019-07842-X/

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Rational Points and Galois Representations

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