Let m\documentclass[12pt]{minimal}
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\begin{document}$${\mathsf {m}}$$\end{document} be any conical (or smooth) metric of finite volume on the Riemann sphere CP1\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {C}}P^1$$\end{document}. On a compact Riemann surface X of genus g consider a meromorphic function f:X→CP1\documentclass[12pt]{minimal}
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\begin{document}$$f: X\rightarrow {{\mathbb {C}}}P^1$$\end{document} such that all poles and critical points of f are simple and no critical value of f coincides with a conical singularity of m\documentclass[12pt]{minimal}
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\begin{document}$${\mathsf {m}}$$\end{document} or {∞}\documentclass[12pt]{minimal}
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\begin{document}$$\{\infty \}$$\end{document}. The pullback f∗m\documentclass[12pt]{minimal}
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\begin{document}$$f^*{\mathsf {m}}$$\end{document} of m\documentclass[12pt]{minimal}
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\begin{document}$${\mathsf {m}}$$\end{document} under f has conical singularities of angles 4π\documentclass[12pt]{minimal}
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\begin{document}$$4\pi $$\end{document} at the critical points of f and other conical singularities that are the preimages of those of m\documentclass[12pt]{minimal}
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\begin{document}$${\mathsf {m}}$$\end{document}. We study the ζ\documentclass[12pt]{minimal}
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\begin{document}$$\zeta $$\end{document}-regularized determinant Det′ΔF\documentclass[12pt]{minimal}
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\begin{document}$${\text {Det}}^\prime \Delta _F$$\end{document} of the (Friedrichs extension of) Laplace–Beltrami operator on (X,f∗m)\documentclass[12pt]{minimal}
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\begin{document}$$(X,f^*{\mathsf {m}})$$\end{document} as a functional on the moduli space of pairs (X, f) and obtain an explicit formula for Det′ΔF\documentclass[12pt]{minimal}
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\begin{document}$${\text {Det}}^\prime \Delta _F$$\end{document}.
