Bad reduction of genus $2$ curves with CM jacobian varieties

TL;DR

This paper demonstrates that genus 2 curves with CM jacobians over number fields typically exhibit stable bad reduction at some prime, using height calculations and L-function analysis.

Contribution

It introduces a novel approach to relate the stable bad reduction of genus 2 curves with CM jacobians to height formulas and L-function derivatives, combining multiple advanced techniques.

Findings

Genus 2 curves with CM jacobians usually have bad reduction at some prime.

The height of the jacobian can be computed via two different formulas linking to L-functions.

The contribution at finite places reflects the stable bad reduction of the curve.

Abstract

We show that a genus $2$ curve over a number field whose jacobian has complex multiplication will usually have stable bad reduction at some prime. We prove this by computing the Faltings height of the jacobian in two different ways. First, we use a formula by Colmez and Obus specific to the CM case and valid when the CM field is an abelian extension of the rationals. This formula links the height and the logarithmic derivatives of an $L$-function. The second formula involves a decomposition of the height into local terms based on a hyperelliptic model. We use results of Igusa, Liu, and Saito to show that the contribution at the finite places in our decomposition measures the stable bad reduction of the curve and subconvexity bounds by Michel and Venkatesh together with an equidistribution result of Zhang to handle the infinite places.