Computing periods of hypersurfaces
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Submission date: 05. Apr. 2018
published in: Mathematics of computation, 88 (2019) 320, p. 2987-3022
DOI number (of the published article): 10.1090/mcom/3430
MSC-Numbers: 32G20, 14C30, 14D07, 14K20, 68W30
Keywords and phrases: Picard--Fuchs equations, Hodge theory, Griffiths--Dwork reduction, periods, algorithms
Link to arXiv:See the arXiv entry of this preprint.
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.