Introduction to Local and Global Euler Characteristic Formulas

TL;DR

This paper introduces the local and global Euler characteristic formulas in Galois cohomology, providing detailed proofs based on classical ideas from Tate, Hida, and Milne, aimed at number theory researchers.

Contribution

It offers a detailed, accessible proof of Tate's Euler characteristic formulas, connecting foundational concepts in Galois cohomology with modern expositions.

Findings

Clarifies the proof of Euler characteristic formulas

Connects classical and modern approaches in Galois cohomology

Serves as an educational resource for number theorists

Abstract

This is a note of talks I gave at the number theory seminar at Tsinghua University in Fall 2011. We will introduce the local and global Euler characteristic formulas given by John Tate(1962) for Galois cohomology. We will give a detailed proof based on the idea in Hida's book[1,ch4.4.5] and Milne's book[2,ch1.5].