Integrality of certain exponential series.

*(English)*Zbl 0998.12009
Kang, Ming-Chang (ed.), Lectures in algebra and geometry. Proceedings of the international conference on algebra and geometry, National Taiwan University, Taipei, Taiwan, December 26-30, 1995. Cambridge, MA: International Press. 215-227 (1998).

From the text: The authors study exponential series which arise from solutions to generalized hypergeometric differential equations. Their technique uses primarily Dwork’s theory of \(p\)-adic hypergeometric functions and some estimates involving the \(p\)-adic Gamma functions. As a corollary, they deduce some congruence conditions on the coefficients of those hypergeometric series.

In particular, they prove the following: Theorem. Let \(N\) be an odd prime number. Let \(f_N\), \(g_N\) be the (unique) power series solutions, with the asymptotic form \(f_N(z)=1+O(z)\), \(g_N(z)=f_N(z)\log z+O(z)\), to the equation \[ (\Theta^{N-1}-Nz(N\Theta+1)\cdots(N\Theta+N-1))f(z)=0. \] Then all the coefficients of the power series \(\text{exp}(\frac{g_N}{f_N})\) are rational integers.

One of the motivations for the theorem comes from the old problem of studying exponential series, especially those related to hypergeometric series. An entirely different motivation for their theorem comes from mirror symmetry and is exposed in the paper.

For the entire collection see [Zbl 0920.00026].

In particular, they prove the following: Theorem. Let \(N\) be an odd prime number. Let \(f_N\), \(g_N\) be the (unique) power series solutions, with the asymptotic form \(f_N(z)=1+O(z)\), \(g_N(z)=f_N(z)\log z+O(z)\), to the equation \[ (\Theta^{N-1}-Nz(N\Theta+1)\cdots(N\Theta+N-1))f(z)=0. \] Then all the coefficients of the power series \(\text{exp}(\frac{g_N}{f_N})\) are rational integers.

One of the motivations for the theorem comes from the old problem of studying exponential series, especially those related to hypergeometric series. An entirely different motivation for their theorem comes from mirror symmetry and is exposed in the paper.

For the entire collection see [Zbl 0920.00026].

##### MSC:

12H25 | \(p\)-adic differential equations |

14J81 | Relationships between surfaces, higher-dimensional varieties, and physics |

14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |

32G20 | Period matrices, variation of Hodge structure; degenerations |

33C20 | Generalized hypergeometric series, \({}_pF_q\) |

81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |

##### Keywords:

exponential series; solutions to generalized hypergeometric differential equations; Dwork’s theory of \(p\)-adic hypergeometric functions; \(p\)-adic Gamma functions; congruence conditions; mirror symmetry
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\textit{B. H. Lian} and \textit{S.-T. Yau}, in: Lectures in algebra and geometry. Proceedings of the international conference on algebra and geometry, National Taiwan University, Taipei, Taiwan, December 26--30, 1995. Cambridge, MA: International Press. 215--227 (1998; Zbl 0998.12009)

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