Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Arithmetic Duality

James S. Milne

TL;DR

This work surveys Tate’s arithmetic dualities—local and global—within Galois and flat cohomology, tracing how dualities for finite modules and abelian varieties yield powerful global consequences. It connects local duality to global exact sequences (nine-term sequences), Étale and Artin–Verdier dualities on curves, and the interplay with the Tate–Shafarevich group and BSD conjectures. The text then extends these dualities to flat cohomology, proving p-part dualities via the Cartier operator and de Rham–Witt theory, culminating in the Artin–Tate conjecture and its p-adic analogs. The historical interludes frame the development alongside key proofs (e.g., Ext-sequences for global duality, Artin–Milne duality) and show how these dualities underpin major results like BSD invariance under isogeny and the connection between Brauer groups, NS-ranks, and L-functions. Overall, the article ties arithmetic duality to both foundational theory and deep conjectures across number fields and function fields, highlighting a cohesive framework for modern arithmetic geometry.

Abstract

In the 1950s and 1960s Tate proved some duality theorems in the Galois cohomology of finite modules and abelian varieties. As for most of Tate's work this has had a profound influence on mathematics with many applications and further developments. In this article, I discuss Tate's theorems and some of these developments.

Arithmetic Duality

TL;DR

This work surveys Tate’s arithmetic dualities—local and global—within Galois and flat cohomology, tracing how dualities for finite modules and abelian varieties yield powerful global consequences. It connects local duality to global exact sequences (nine-term sequences), Étale and Artin–Verdier dualities on curves, and the interplay with the Tate–Shafarevich group and BSD conjectures. The text then extends these dualities to flat cohomology, proving p-part dualities via the Cartier operator and de Rham–Witt theory, culminating in the Artin–Tate conjecture and its p-adic analogs. The historical interludes frame the development alongside key proofs (e.g., Ext-sequences for global duality, Artin–Milne duality) and show how these dualities underpin major results like BSD invariance under isogeny and the connection between Brauer groups, NS-ranks, and L-functions. Overall, the article ties arithmetic duality to both foundational theory and deep conjectures across number fields and function fields, highlighting a cohesive framework for modern arithmetic geometry.

Abstract

In the 1950s and 1960s Tate proved some duality theorems in the Galois cohomology of finite modules and abelian varieties. As for most of Tate's work this has had a profound influence on mathematics with many applications and further developments. In this article, I discuss Tate's theorems and some of these developments.

Paper Structure

This paper contains 31 sections, 14 theorems, 49 equations.

Table of Contents

  1. Local duality (Tate 1957)
  2. The dual abelian variety
  3. Global duality (Tate 1962)
  4. Tate's 1963 proof
  5. Global duality (variant)
  6. Étale duality over a curve
  7. Etale duality $\leftrightarrow$ Tate duality (char $p\neq0$).
  8. Artin--Verdier duality (1964)
  9. Applications to abelian varieties
  10. The pairing on the Tate--Shafarevich group
  11. The Birch--Swinnerton-Dyer (BSD) conjecture
  12. Local flat duality
  13. Finite coefficients
  14. Abelian varieties
  15. Global flat duality
  16. ...and 16 more sections

Key Result

Theorem 1

Let $K$ be an algebraic number field, and $j\colon U\hookrightarrow\mathop{\mathrm{Spec}}\nolimits(\mathcal{O}{}_{K})$ an open subset. For any constructible sheaf $F$ on $U$, the Yoneda pairing is a nondegenerate pairing of finite groups, except possibly on the $2$-torsion when $K$ has a real prime. Here $H_{c}^{r}(U,F)\overset{{\hbox{\tiny def}}}{=}H^{r}(X,j_{!}F)$.

Theorems & Definitions (21)

  • Theorem : Artin--Verdier 1964
  • Theorem : Tate, Cassels for elliptic curves
  • Conjecture : BSD, Tate
  • Theorem : Tate, Cassels for elliptic curves
  • Theorem : Shatz thesis, 1962
  • Theorem : Milne 1970/1972
  • Theorem : Artin--Mazur--Milne duality
  • Conjecture : Tate
  • Conjecture : BSD
  • Conjecture : Artin--Tate
  • ...and 11 more