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A hypergeometric version of the modularity of rigid Calabi-Yau manifolds. (English) Zbl 1456.11073
This paper considers the fourteen one-parameter families of Calabi-Yau threefolds whose periods are expressed in terms of hypergeometric functions. For these fourteen families, periods are solutions of hypergeometric equations with parameter \((r, 1-r, t, 1-t)\), where \begin{multline*} (r,t)=\Big(\frac{1}{2},\frac{1}{2}\Big),\Big(\frac{1}{2},\frac{1}{3}\Big),\Big(\frac{1}{2},\frac{1}{4}\Big), \Big(\frac{1}{2},\frac{1}{6}\Big),\Big(\frac{1}{3}\Big),\Big(\frac{1}{3},\frac{1}{4}\Big),\Big(\frac{1}{3},\frac{1}{6}\Big),\\ \Big(\frac{1}{4},\frac{1}{4}\Big),\Big(\frac{1}{4},\frac{1}{6}\Big),\Big(\frac{1}{6},\frac{1}{6}\Big),\Big(\frac{1}{5},\frac{2}{5}\Big), \Big(\frac{1}{8},\frac{3}{8}\Big),\Big(\frac{1}{10},\frac{3}{10}\Big),\Big(\frac{1}{12},\frac{5}{12}\Big). \end{multline*} At a conifold point, any of these Calabi-Yau threefolds becomes rigid, and the \(p\)-th coefficient \(a(p)\) of the corresponding modular form of weight \(4\) can be recovered from the truncated partial sums of the corresponding hypergeometric series modulo a higher power of \(p\), where \(p\) is any good prime \(>5\).
This paper discusses relationships between the critical values of the \(L\)-series of the modular form and the values of a related basis of solutions to the hypergeometric differential equation. It is numerically observed that the critical \(L\)-values are \(\mathbb{Q}\)-proportional to the hypergeometric values \(F_1(1), F_2(1), F_3(1)\), where \(F_j(z)\) are solutions of the hypergeometric equation for the hypergeometric function \(F_0(z)=_4F_3(z)\) with parameters \((r, 1-r, t, 1-t)\). This confirms the prediction of Golyshev concerning gamma structures [V. Golyshev and A. Mellit, J. Geom. Phys. 78, 12–18 (2014; Zbl 1284.33001)].

11F33 Congruences for modular and \(p\)-adic modular forms
11T24 Other character sums and Gauss sums
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14J33 Mirror symmetry (algebro-geometric aspects)
33C20 Generalized hypergeometric series, \({}_pF_q\)
Zbl 1284.33001
Full Text: DOI arXiv
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