Galois theory, functional Lindemann–Weierstrass, and Manin maps

Daniel Bertrand and Anand Pillay

Vol. 281 (2016), No. 1, 51–82

DOI: 10.2140/pjm.2016.281.51

Abstract

We prove several new results of Ax–Lindemann type for semiabelian varieties over the algebraic closure $K$ of $ℂ\left(t\right)$, making heavy use of the Galois theory of logarithmic differential equations. Using related techniques, we also give a generalization of the theorem of the kernel for abelian varieties over $K$. This paper is a continuation of earlier work by Bertrand and Pillay (2010), as well as an elaboration on the methods of Galois descent introduced by Bertrand (2009, 2011) in the case of abelian varieties.

Keywords

differential Galois theory, algebraic independence over function fields, semiabelian schemes, Manin maps

Mathematical Subject Classification 2010

Primary: 03C60, 11J95, 12H05, 14K05, 34M15

Milestones

Received: 15 February 2015

Revised: 9 July 2015

Accepted: 3 August 2015

Published: 9 February 2016