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Darboux evaluations for hypergeometric functions with the projective monodromy \(\hbox {PSL}(2,\mathbb {F}_7)\)

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Abstract

Algebraic hypergeometric functions can be compactly expressed as radical functions on pull-back curves where the monodromy group is simpler, say, a finite cyclic group. These so-called Darboux evaluations have already been considered for algebraic \({}_{2}\text{ F }_{1}\)-functions. This article presents Darboux evaluations for the classical case of \({}_{3}\text{ F }_{2}\)-functions with the projective monodromy group \(\hbox {PSL}(2,\mathbb {F}_7)\). The pullback curves are of genus 0 (in the simplest case) or of genus 1. As an application of the genus 0 evaluations, appealing modular evaluations of the same \({}_{3}\text{ F }_{2}\)-functions are derived.

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Notes

  1. Correspondingly, \((\#\Lambda )/7=24\) is the smallest degree of Darboux coverings that reduce the projective monodromy \(\Lambda \) to a cyclic group. But smaller degree Darboux coverings exist that give pull-back transformations to reducible monodromy representations. In [35], Darboux coverings of degree 21 of the considered Fuchsian equations are given. They transform the projective monodromy \(\Lambda \) to a dihedral group of order 8.

  2. The branching pattern (or the passport [17]) of a Belyi map gives the branching orders of all points in the 3 fibers \(\varphi =0\), \(\varphi =1\), \(\varphi =\infty \). Particularly, \(7^31^3\) prescribes 3 seven-fold points and 3 non-branching points in one fiber, while \(3^8\) prescribes all points to be threefold in other fiber, etc.

  3. We choose a different parametrization than in [7, p. 67] in order to have \(x_7(\tau )=O(q)\) in (4.23) with consistency.

  4. These are essentially the modular forms mentioned right after formula (4.1); see [8, Example 29]. Specifically, \(X=-\eta (\tau )^4\,K_3(\tau )\), \(Y=\eta (\tau )^4\,K_2(\tau )\), \(Z=\eta (\tau )^4\,K_1(\tau )\).

  5. There are these typos in [7]: Formula (4.19) is actually for \(j_7^{-1}\), not for \(j_7\). In formula (4.23), a minus sign is missing before \(\mathsf{y}^2\mathsf{z}/\mathsf{x}^3\). Formula (1.18) should be adjusted by \(40\Phi _4^2\Phi _6 (3\Phi _{14}^2+1008\Phi _4\Phi _6^4+56\Phi _4^2\Phi _6\Phi _{14}-832\Phi _4^4\Phi _6^2-256\Phi _4^7)\). In formula (4.35), the factor \(5\phi ^2-15\phi -7\) should be \(5\phi ^2-14\phi -7\).

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  1. Institute of Applied Mathematics, Vilnius University, Vilnius, Lithuania

    Raimundas Vidunas

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Vidunas, R. Darboux evaluations for hypergeometric functions with the projective monodromy \(\hbox {PSL}(2,\mathbb {F}_7)\). Ramanujan J 53, 85–121 (2020). https://doi.org/10.1007/s11139-019-00229-x

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