Archimedean Compactifications and the Asymptotics of Hodge-Theoretic Torsion on Moduli Spaces of Polarized Abelian Varieties

This paper investigates the asymptotic behavior of Hodge-theoretic analytic torsion on the moduli space of principally polarized abelian varieties, Ag, as one approaches the boundary of its minimal toroidal compactification. We study families of degenerating abelian varieties over a punctured disk, corresponding to paths approaching a cusp in Ag. The central object of study is the Ray-Singer analytic torsion associated with the Hodge bundle, endowed with its natural Hodge metric. By leveraging techniques from spectral geometry, the theory of theta functions, and Mumford’s construction of degenerating abelian varieties, we derive a precise asymptotic formula for this torsion. Our main result expresses the leading-order and sub-leading logarithmic terms of the torsion in terms of fundamental arithmetic and geometric invariants, including the Faltings height of the singular fiber and special values of automorphic forms. This formula can be interpreted as a boundary contribution term for a version of the arithmetic Grothendieck-Riemann-Roch theorem on non-compact arithmetic varieties. The findings establish a new quantitative bridge between the complex analytic geometry of moduli spaces and the arithmetic of abelian varieties, offering insights into the geometry of Archimedean compactifications.