Geodesic
Kirchhoff’s theorem for Prym varieties
Forum of Mathematics, Sigma, Tome 10 (2022)

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We prove an analogue of Kirchhoff’s matrix tree theorem for computing the volume of the tropical Prym variety for double covers of metric graphs. We interpret the formula in terms of a semi-canonical decomposition of the tropical Prym variety, via a careful study of the tropical Abel–Prym map. In particular, we show that the map is harmonic, determine its degree at every cell of the decomposition and prove that its global degree is $2^{g-1}$. Along the way, we use the Ihara zeta function to provide a new proof of the analogous result for finite graphs. As a counterpart, the appendix by Sebastian Casalaina-Martin shows that the degree of the algebraic Abel–Prym map is $2^{g-1}$ as well. 
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     author = {Yoav Len and Dmitry Zakharov},
     title = {Kirchhoff{\textquoteright}s theorem for {Prym} varieties},
     journal = {Forum of Mathematics, Sigma},
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     year = {2022},
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Yoav Len; Dmitry Zakharov. Kirchhoff’s theorem for Prym varieties. Forum of Mathematics, Sigma, Tome 10 (2022). doi: 10.1017/fms.2021.75

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