Ramification theory in non-abelian local class field theory

Non-Abelian Class Field Theory for Riemann Surfaces

Let T be a Tannakian category with a fiber functor ω : T → VerC, where VerC denotes the category of finite dimensional C-vector spaces. An object t ∈ T is called reducible if there exist non-zero objects x, y ∈ T such that t = x ⊕ y. An object is called irreducible if it is not reducible. If moreover every object x of T can be written uniquely as a sum of irreducible objects x = x1 ⊕ x2 ⊕ . . ....

Ramification Theory for Local Field with Imperfect Residue Field

Definition 1.3. The most natural way to define higher ramification subgroups of the Galois group G is due by Hilbert: g ∈ Ga if and only if vl(gx− x) ≥ a + 1, ∀x ∈ Ol. Indeed, G−1 = G, G0 = I is the inertia subgroup, and G1 = W is the wild inertia subgroup. However, there is a disadvantage of this. Namely, it does not respect quotient and hence it does not give a filtration on the absolute Galo...

Lectures on Field Theory and Ramification Theory

Local Class Field Theory Contents

1 Wedderburn theory and the Brauer group 2 1.1 Algebras and modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 The Brauer group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Splitting fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Crossed products . . . . . . . . . . . . . . . . ...

Ramification theory of schemes over a local field

We introduce the Swan class of an -adic etale sheaf on a variety over a local field. It is a generalization of the classical Swan conductor measuring the wild ramification and is defined as a 0-cycle class supported on the reduction. We establish a Riemann-Roch formula for the Swan class. Let K be a complete discrete valuation field of characteristic 0. We assume that the residue field F is a p...