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In part the motivation comes from applications, such as physics. Some of the more recent interest in Calabi-Yau varieties e.g. was triggered by the discovery of mirror symmetry by string theorists. Certain classes of families of diophantine polynomials describe simple types of Calabi-Yau spaces in toric varieties and provide a fairly large number of quite different types of diophantine equations. During the first post-mirror symmetry decade this interest came mostly from classical algebraic geometers, but over the past few years some number theorists have become interested in Calabi-Yau spaces as well. Questions like the The question of automorphy of Calabi-Yau type motives is an example of a concrete questionproblem. This problem is of interest already in dimension two, i.e. for families of K3 surfaces described e.g. by hypersurfaces in weighted projective spaces or toric varieties. In the case of CY threefolds this problem has played an important role in the work of Clozel, Harris, Shepherd-Barron, Taylor on the Sato-Tate conjecture. This involves a 1-parameter family of quintics in projective 4-space ${\mathbb P}^4$.

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In part the motivation comes from applications, such as physics. Some of the more recent interest in Calabi-Yau varieties e.g. was triggered by the discovery of mirror symmetry by string theorists. Certain classes of families of diophantine polynomials describe simple types of Calabi-Yau spaces in toric varieties and provide a fairly large number of quite different types of diophantine equations. During the first post-mirror symmetry decade this interest came mostly from classical algebraic geometers, but over the past few years some number theorists have become interested as well. Questions like the automorphy of Calabi-Yau type motives is an example of a concrete question. This problem is of interest already in dimension two, i.e. for families of K3 surfaces described e.g. by hypersurfaces in weighted projective spaces or toric varieties.