Canonical heights, periods and the Hurwitz zeta function

TL;DR

This paper introduces a canonical height for log pairs over number fields, linking it to K-stability, periods, and special functions, with explicit computations for certain curves and applications to Shimura curves.

Contribution

It defines a new canonical height related to K-stability, expresses it via periods, and provides explicit formulas for specific algebraic curves, connecting arithmetic geometry with special functions.

Findings

Canonical height finite iff the complexification is K-semistable.

Explicit height formulas for certain log surfaces and Fermat curves.

Procedure to analyze Shimura curves' integral models using height formulas.

Abstract

Let (X,D) be a projective log pair over the ring of integers of a number field such that the log canonical line bundle K_(X,D) or its dual -K_(X,D) is relatively ample. We introduce a canonical height of K_(X,D) (and -K(X,D)) which is finite precisely when the complexifications of K_(X,D) (and -K(X,D)) are K-semistable. When the complexifications are K-polystable, the canonical height is the height of K_(X,D) (and -K(X,D)) wrt any volume-normalized K\"ahler-Einstein metric on the complexifications of K_(X,D) (and -K(X,D)) The canonical height is shown to have a number of useful variational properties. Moreover, it may be expressed as a limit of periods on the N-fold products of the complexifications of X, as N tends to infinity. In particular, using this limit formula, the canonical height for the arithmetic log surfaces (P_1,D) over the integers, where D has at most three components, is computed explicitly in terms of the Hurwitz zeta function and its derivative at s=-1. Combining this explicit formula with previous height formulas for quaternionic Shimura curves yields a procedure for extracting information about the canonical integral models of some Shimura curves, such as wild ramification. Furthermore, explicit formulas for the canonical height of twisted Fermat curves are obtained, implying explicit Parshin type bounds for the Arakelov metric.