Surface bundles and the section conjecture

TL;DR

This paper formulates a tropical analogue of Grothendieck's section conjecture for stable graphs and proves many cases, producing examples of curves satisfying the conjecture over various fields and introducing new Galois cohomology obstructions.

Contribution

It introduces a tropical version of the section conjecture, constructs new obstructions to rational points, and proves the conjecture for generic curves of certain genera.

Findings

Proved the section conjecture for generic curves of genus g>2.

Constructed new Galois cohomology obstructions to rational points.

Produced examples of curves satisfying the section conjecture over p-adic and number fields.

Abstract

We formulate a tropical analogue of Grothendieck's section conjecture: that for every stable graph G of genus g>2, and every field k, the generic curve with reduction type G over k satisfies the section conjecture. We prove many cases of this conjecture. In so doing we produce many examples of curves satisfying the section conjecture over fields of geometric interest, and then over p-adic fields and number fields via a Chebotarev argument. We construct two Galois cohomology classes o_1 and o_2, which obstruct the existence of pi_1-sections and hence of rational points. The first is an abelian obstruction, closely related to the period of a curve and to a cohomology class on the moduli space of curves M_g studied by Morita. The second is a 2-nilpotent obstruction and appears to be new. We study the degeneration of these classes via topological techniques, and we produce examples of surface bundles over surfaces where these classes obstruct sections. We then use these constructions to produce curves over p-adic fields and number fields where each class obstructs pi_1-sections and hence rational points. Among our geometric results are a new proof of the section conjecture for the generic curve of genus g>2, and a proof of the section conjecture for the generic curve of even genus with a rational divisor class of degree one (where the obstruction to the existence of a section is genuinely non-abelian).