Strong arithmetic mirror symmetry and toric isogenies. (English) Zbl 1447.11072
Malmendier, Andreas (ed.) et al., Higher genus curves in mathematical physics and arithmetic geometry. AMS special session, Seattle, WA, USA, January 8, 2016. Proceedings. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 703, 117-129 (2018).
Summary: We say a mirror pair of Calabi-Yau varieties exhibits strong arithmetic mirror symmetry if the number of points on each variety over a finite field is equivalent, modulo the order of that field. We search for strong mirror symmetry in pencils of toric hypersurfaces generated using polar dual pairs of reflexive polytopes. We characterize the pencils of elliptic curves where strong arithmetic mirror symmetry arises, and provide experimental evidence that the phenomenon generalizes to higher dimensions. We also provide experimental evidence that pencils of K3 surfaces with the same Picard-Fuchs equation have related point counts.
For the entire collection see [Zbl 1388.14009].
For the entire collection see [Zbl 1388.14009].
MSC:
| 11G42 | Arithmetic mirror symmetry |
| 14H52 | Elliptic curves |
| 14J28 | \(K3\) surfaces and Enriques surfaces |
Software:
SageMathReferences:
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