Rational points of universal curves

TL;DR

This paper proves a version of Grothendieck's section conjecture for universal curves over moduli spaces, focusing on the fundamental group's unipotent completion over function fields, advancing understanding in algebraic geometry.

Contribution

It introduces a new version of the section conjecture for universal curves, replacing the fundamental group with its ell-adic unipotent completion for n > 1.

Findings

Proves a version of Grothendieck's section conjecture for universal curves

Replaces fundamental group with ell-adic unipotent completion

Applicable over function fields of moduli spaces

Abstract

In this paper we prove a version of Grothendieck's section conjecture for the restriction of the universal complete curve over M_{g,n}, g > 4, to the function field k(M_{g,n}) where k is, for example, a number field. In this version, the fundamental group of the closed fiber is replaced by its ell-adic unipotent completion when n > 1.