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Krattenthaler, Christian; Rivoal, Tanguy

On the integrality of the Taylor coefficients of mirror maps. (English) Zbl 1267.11077

Duke Math. J. 151, No. 2, 175-218 (2010).
Summary: We show that the Taylor coefficients of the series \({\mathbf q}(z)=z{\text{ exp}}({\mathbf G}(z)/{\mathbf F}(z))\) are integers, where \({\mathbf F}(z)\) and \({\mathbf G}(z)+{\text{log}}(z){\mathbf F}(z)\) are specific solutions of certain hypergeometric differential equations with maximal unipotent monodromy at \(z=0\). We also address the question of finding the largest integer \(u\) such that the Taylor coefficients of \((z^{-1}{\mathbf q}(z))^{1/u}\) are still integers. As consequences, we are able to prove numerous integrality results for the Taylor coefficients of mirror maps of Calabi-Yau complete intersections in weighted projective spaces, which improve and refine previous results by B. H. Lian and S.-T. Yau and by V. V. Zudilin [Math. Notes 71, No. 5, 604–616 (2002); translation from Mat. Zametki 71, No. 5, 662–676 (2002; Zbl 1043.11060)]. In particular, we prove the general “integrality” conjecture of Zudilin about these mirror maps

Cited in 2 Reviews
Cited in 18 Documents

MSC:

11G42 Arithmetic mirror symmetry
14J33 Mirror symmetry (algebro-geometric aspects)
11F30 Fourier coefficients of automorphic forms
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
12H25 \(p\)-adic differential equations
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
33C20 Generalized hypergeometric series, \({}_pF_q\)

Citations:

Zbl 1043.11060
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\textit{C. Krattenthaler} and \textit{T. Rivoal}, Duke Math. J. 151, No. 2, 175--218 (2010; Zbl 1267.11077)
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Online Encyclopedia of Integer Sequences:

Numbers k for which the numerator of the k-th harmonic number H_k is divisible by the third power of a prime less than k.
Numbers k such that the numerator of the k-th alternating harmonic number H’(k) is divisible by the square of a prime less than k.
Numbers L such that there is a prime p <= L for which v_p(H_L - 1) > 0, where v_p(x) is the p-adic valuation of x and H_L is the L-th harmonic number.
Numbers L such that there is a prime p <= L for which v_p(H_L - 1) > 1, where v_p(x) is the p-adic valuation of x and H_L is the L-th harmonic number.
Numbers L such that there is a prime p <= L for which v_p(H’(L) - 1) > 0, where v_p(x) is the p-adic valuation of x and H’(L) is the L-th alternating harmonic number.

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