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The $$n$$th root of the mirror map. (English) Zbl 1048.11049
Yui, Noriko (ed.) et al., Calabi-Yau varieties and mirror symmetry. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3355-3/hbk). Fields Inst. Commun. 38, 195-199 (2003).
The authors consider the differential equation $(\Theta^{N-1}-Nz(N\Theta +1)\cdots (N\Theta +N-1))f(z)=0$ where $$\Theta =z\dfrac{d}{dz}$$, $$N$$ is an odd prime number. Let $$f_N$$, $$g_N$$ be the power series solutions, with the asymptotic form $$f_N(z)=1+O(z)$$, $$g_N(z)=f_N(z)\log z+G_N(z)$$, $$G_N(z)=O(z)$$. It is proved that all the coefficients of the power series $$\exp \left( \frac{G_N}{Nf_N}\right)$$ are integers. This result implies the integrality property of the $$N$$th root of the mirror map of a Calabi-Yau manifold and improves an earlier result [B. H. Lian and S. T. Yau, Lectures in algebra and geometry. Proceedings of the international conference on algebra and geometry, National Taiwan University, Taipei, Taiwan, 1995, 215–227 (1998; Zbl 0998.12009)] about the integrality of $$\exp \left( \frac{G_N}{f_N}\right)$$.
The technique is based on Dwork’s theory of $$p$$-adic hypergeometric functions [B. Dwork, Ann. Sci. Éc. Norm. Supér., IV. Ser. 6, 295–315 (1973; Zbl 0309.14020)].
For the entire collection see [Zbl 1022.00014].

##### MSC:
 11G25 Varieties over finite and local fields 34M15 Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain 12H25 $$p$$-adic differential equations 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 32Q25 Calabi-Yau theory (complex-analytic aspects) 33C20 Generalized hypergeometric series, $${}_pF_q$$