Kirchhoff’s theorem for Prym varieties

Abstract We prove an analogue of Kirchhoff’s matrix tree theorem for computing the volume tropical Prym variety double covers metric graphs. interpret formula in terms a semi-canonical decomposition variety, via careful study Abel–Prym map. In particular, we show that map is harmonic, determine its degree at every cell and global $2^{g-1}$ . Along way, use Ihara zeta function to provide new proof analogous result finite As counterpart, appendix by Sebastian Casalaina-Martin shows algebraic as well.