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K3 surfaces and elliptic fibrations in number theory Elkies, Noam D.


We outline several number-theoretical contexts where K3 surfaces and elliptic fibrations arise naturally: Diophantine equations, Euclidean and hyperbolic quadratic forms, elliptic and Shimura modular curves and higher-dimensional analogues, record ranks for elliptic curves and related tasks, and complex reflection groups and their invariants. Several of these contexts call for explicit formulas for surfaces are known to exist only by transcendental means (Torelli theorem for K3 surfaces). One of these formulas also yields a family of elliptically fibered Calabi-Yau threefolds with Mordell-Weil rank 10.

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