Also other papers on Artin-Verdier duality. Artin-V duality for etale cohomology of global fields apparently explains Tate’s results on duality in Galois cohomology, which are important in class field theory. See Haberland: Galois cohomology of number fields, for the general number field case. Mazur proved it for totally imaginary quadratic fields. Deninger: function fields, also connections with Poincare duality.
Milne book
Scholbach: Various things in thesis
arXiv:0910.3759 Étale duality for constructible sheaves on arithmetic schemes from arXiv Front: math.AG by Uwe Jannsen, Shuji Saito, Kanetomo Sato In this note we relate three topics for arithmetic schemes: a general duality for étale constructible torsion sheaves, an étale homology theory, and a Gersten-Bloch-Ogus-Kato complex. The results in this paper have been used in other papers of the authors ([JS], [Sa], [SaH] in the list of references).
arXiv:1102.1302 Geometry of Numbers from arXiv Front: math.AG by Lin Weng We develop a global cohomology theory for number fields by offering topological cohomology groups, an arithmetical duality, a Riemann-Roch type theorem, and two types of vanishing theorem. As applications, we study moduli spaces of semi-stable lattices, and introduce non-abelian zeta functions for number fields.
arXiv:0911.4781 Algebraic K-theory of the fraction field of topological K-theory from arXiv Front: math.KT by Christian Ausoni, John Rognes We compute the algebraic K-theory modulo p and v_1 of the S-algebra ell/p = k(1), using topological cyclic homology. We use this to compute the homotopy cofiber of a transfer map K(L/p) –> K(L_p), which we interpret as the algebraic K-theory of the “fraction field” of the p-complete Adams summand of topological K-theory. The results suggest that there is an arithmetic duality theorem for this fraction field, much like Tate–Poitou duality for p-adic fields.
nLab page on Arithmetic duality