Communications in Number Theory and Physics

Volume 2 (2008)

Number 3

Frobenius polynomials for Calabi–Yau equations

Pages: 537 – 561

DOI: http://dx.doi.org/10.4310/CNTP.2008.v2.n3.a2

Authors

Kira Samol (Institut für Mathematik, Johannes Gutenberg-Universität, Mainz, Germany)

Duco van Straten (Institut für Mathematik, Johannes Gutenberg-Universität, Mainz, Germany)

Abstract

We describe a variation of Dwork’s unit-root method todetermine the degree 4 Frobenius polynomial for members ofa 1-modulus Calabi–Yau family over $\P^1$ in terms of theholomorphic period near a point of maximal unipotentmonodromy. The method is illustrated on a couple ofexamples from the list \cite{AESZ}. For singular points, wefind that the Frobenius polynomial splits in a product oftwo linear factors and a quadratic part $1-a_pT+p^3T^2$. Weidentify weight 4 modular forms which reproduce the $a_p$as Fourier coefficients.

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