## 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)].

### MSC:

 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

LMFDB
Full Text:

### References:

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