Vol. 10, No. 6, 2016

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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

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.

Anabelian geometry, moduli spaces, abelian varieties, descent obstruction
Mathematical Subject Classification 2010
Primary: 11G35
Secondary: 14G05, 14G35
Received: 14 July 2015
Revised: 26 May 2016
Accepted: 25 June 2016
Published: 30 August 2016
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
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