Anabelian geometry and descent obstructions on moduli spaces

Patrikis, Stefan; Voloch, José; Zarhin, Yuri

doi:10.2140/ant.2016.10.1191

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Anabelian geometry and descent obstructions on moduli spaces

Stefan Patrikis, José Felipe Voloch and Yuri G. Zarhin

Vol. 10 (2016), No. 6, 1191–1219

DOI: 10.2140/ant.2016.10.1191

Abstract

We study the section conjecture of anabelian geometry and the sufficiency of the finite descent obstruction to the Hasse principle for the moduli spaces of principally polarized abelian varieties and of curves over number fields. For the former we show that the section conjecture fails and the finite descent obstruction holds for a general class of adelic points, assuming several well-known conjectures. This is done by relating the problem to a local-global principle for Galois representations. For the latter, we show how the sufficiency of the finite descent obstruction implies the same for all hyperbolic curves.

Keywords

Anabelian geometry, moduli spaces, abelian varieties, descent obstruction

Mathematical Subject Classification 2010

Primary: 11G35

Secondary: 14G05, 14G35

Milestones

Received: 14 July 2015

Revised: 26 May 2016

Accepted: 25 June 2016

Published: 30 August 2016

Authors

Stefan Patrikis
Department of Mathematics University of Utah 155 S 1400 E Salt Lake City, UT 84112 United States

José Felipe Voloch
School of Mathematics and Statistics University of Canterbury Private Bag 4800 Christchurch 8140 %8052 New Zealand voloch@math.utexas.eduDepartment of Mathematics University of Texas Austin, TX 78712 United States

Yuri G. Zarhin
Department of Mathematics Pennsylvania State University University Park, PA 16802 United States

Vol. 10, No. 6, 2016