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Integrality of mirror maps and arithmetic homological mirror symmetry for Greene--Plesser mirrors

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arxiv 2312.01949 v3 pith:YAJE2EMO submitted 2023-12-04 math.SG math.AGmath.NT

Integrality of mirror maps and arithmetic homological mirror symmetry for Greene--Plesser mirrors

Sheel Ganatra , Andrew Hanlon , Jeff Hicks , Daniel Pomerleano , Nick Sheridan This is my paper
classification math.SG math.AGmath.NT
keywords mirrorgreene--plesserhomologicalsymmetryarithmeticcoefficientsintegralitymaps
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We prove the `integrality of Taylor coefficients of mirror maps' conjecture for Greene--Plesser mirror pairs as a natural byproduct of an arithmetic refinement of homological mirror symmetry. We also prove homological mirror symmetry for Greene--Plesser mirror pairs in all characteristics such that the B-side family has good reduction, generalizing work of the fifth author and Smith over the complex numbers. A key technical ingredient is a new versality argument which allows us to work throughout over a Novikov-type ring with integer coefficients.

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