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Point counting on K3 surfaces and an application concerning real and complex multiplication

Part of: Zeta and $L$-functions: analytic theory Arithmetic algebraic geometry Computational number theory Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  26 August 2016

Andreas-Stephan Elsenhans  and
Jörg Jahnel
Andreas-Stephan Elsenhans
Affiliation:
Mathematisches Institut, Universität Paderborn, Warburger Straße 100, D-33098 Paderborn, Germany email elsenhan@math.uni-paderborn.de
Jörg Jahnel
Affiliation:
Department Mathematik, Universität Siegen, Walter-Flex-Straße 3, D-57068 Siegen, Germany email jahnel@mathematik.uni-siegen.de

Abstract

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We report on our project to find explicit examples of K3 surfaces having real or complex multiplication. Our strategy is to search through the arithmetic consequences of RM and CM. In order to do this, an efficient method is needed for point counting on surfaces defined over finite fields. For this, we describe algorithms that are $p$ -adic in nature.

MSC classification

Primary: 14J28: $K3$ surfaces and Enriques surfaces
Secondary: 11G15: Complex multiplication and moduli of abelian varieties 11Y16: Algorithms; complexity 11M38: Zeta and $L$-functions in characteristic $p$

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