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Mathematics of Computation

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Computing periods of hypersurfaces

Author: Emre Can Sertöz
Journal: Math. Comp. 88 (2019), 2987-3022
MSC (2010): Primary 32G20, 14C30, 68W30, 14D07, 14K20
Published electronically: April 10, 2019
MathSciNet review: 3985484
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Abstract: We give an algorithm to compute the periods of smooth projective hypersurfaces of any dimension. This is an improvement over existing algorithms which could only compute the periods of plane curves. Our algorithm reduces the evaluation of period integrals to an initial value problem for ordinary differential equations of Picard–Fuchs type. In this way, the periods can be computed to extreme precision in order to study their arithmetic properties. The initial conditions are obtained by an exact determination of the cohomology pairing on Fermat hypersurfaces with respect to a natural basis.

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Additional Information

Emre Can Sertöz
Affiliation: Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany

Keywords: Picard–Fuchs equations, Hodge theory, Griffiths–Dwork reduction, periods, algorithms
Received by editor(s): April 14, 2018
Received by editor(s) in revised form: December 7, 2018
Published electronically: April 10, 2019
Article copyright: © Copyright 2019 American Mathematical Society