Tate–Nakayama duality

Statement

Let $F$ be a nonarchimedean local field, let $T$ be an $F$ -torus, and let $\widehat T$ be its complex dual torus with Galois action. Local Tate–Nakayama duality gives a canonical perfect pairing

H^1(F,T)\times\pi_0(\widehat T^{\Gamma_F}) \longrightarrow\mathbb Q/\mathbb Z.

Equivalently,

H^1(F,T)\simeq \operatorname{Hom}\!\left( \pi_0(\widehat T^{\Gamma_F}),\mathbb Q/\mathbb Z \right).

Here $H^1(F,T)$ is Galois cohomology (an abelian group in this torus case), and $\pi_0(\widehat T^{\Gamma_F})$ is the component group of the fixed-point subgroup of the dual torus.

Cohomological origin

For a finite Galois splitting extension $L/F$ , cup product with the local fundamental class relates Tate cohomology of the character lattice to Tate cohomology of $L^\times$ . Passing through the character/cocharacter duality of $T$ yields the displayed pairing. Archimedean and global versions have modified targets and exact sequences.

Use in endoscopy

The pairing turns the set of rational conjugacy classes inside a stable conjugacy class into a finite Fourier space. Characters on that space define kappa orbital integrals and appear in the normalization of endoscopic transfer.

References

John Tate, “The cohomology groups $H^i(G,S)$ ,” in Algebraic Number Theory, Academic Press, 1967, 257–268.
Robert E. Kottwitz, “Stable trace formula: elliptic singular terms,” Mathematische Annalen 275 (1986), 365–399.