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Some Classical Problems in Number Theory via the Theory of K3 Surfaces

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Mathematics in the 21st Century

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 98))

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Abstract

The theory of the elliptic modular function plays an important role in many situations in number theory. The elliptic modular function is obtained as a one-to-one correspondence between the parameter space of the family of elliptic curves (given by the Weierstrass normal form) and its period domain (i.e., the complex upper half plane). The K3 surface is considered to be a two-dimensional counterpart of the elliptic curve. So, if we consider a family of algebraic K3 surfaces with some normal form, we can obtain its modular function. We call it a K3 modular function (see [18, 19], some mathematical physicists call it a mirror map for K3 surfaces).

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Authors and Affiliations

  1. Graduate School of Science, Chiba University, Inage-ku Yayoi-cho 1-33, Chiba, 263-8522, Japan

    Hironori Shiga

  2. Abdus Salam School of Mathematical Sciences, Lahore, Pakistan

    Hironori Shiga

Authors
  1. Hironori Shiga
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Correspondence to Hironori Shiga .

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Editors and Affiliations

  1. Institut des Hautes Études Scientifiques (IHÉS), Bures-sur-Yvette, France

    Pierre Cartier

  2. Abdus Salam School of Mathematical Sciences, Lahore, Pakistan

    A.D.R. Choudary

  3. Université Pierre et Marie Curie (Paris VI),, Paris Cedex 05, France

    Michel Waldschmidt

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Shiga, H. (2015). Some Classical Problems in Number Theory via the Theory of K3 Surfaces. In: Cartier, P., Choudary, A., Waldschmidt, M. (eds) Mathematics in the 21st Century. Springer Proceedings in Mathematics & Statistics, vol 98. Springer, Basel. https://doi.org/10.1007/978-3-0348-0859-0_10

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