TY - JOUR

T1 - Kirchhoff's theorem for Prym varieties

AU - Len, Yoav

AU - Zakharov, Dmitry

N1 - Publisher Copyright:
©

PY - 2022/2/16

Y1 - 2022/2/16

N2 - 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. 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 as well.

AB - 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. 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 as well.

KW - 2020 Mathematics Subject Classification 14T05 14H40

UR - http://www.scopus.com/inward/record.url?scp=85125108196&partnerID=8YFLogxK

U2 - 10.1017/fms.2021.75

DO - 10.1017/fms.2021.75

M3 - Article

AN - SCOPUS:85125108196

SN - 2050-5094

VL - 10

JO - Forum of Mathematics, Sigma

JF - Forum of Mathematics, Sigma

M1 - e11 1-54

ER -