Extensions of absolute values on two subfields

TL;DR

This paper characterizes absolute values on a field extending two subfields and generalizes Abhyankar's lemma, with applications to counting points on fibered products of curves.

Contribution

It introduces a unified framework for absolute values extending two subfields and generalizes Abhyankar's lemma, simplifying ramification analysis.

Findings

Provides a description of absolute values extending two subfields.

Generalizes Abhyankar's lemma on ramification indices.

Applies results to count points on fibered products of curves.

Abstract

We describe the absolute values on a field which simultaneously extend absolute values on two subfields. We also give a common generalization of many versions of Abhyankar's lemma on ramification indices, which is both widely applicable and easy to state. We then apply these results to count points on the fibered product of two curve morphisms C --> X and D --> X which lie over prescribed points on C and D.