Swan Conductor Research Articles

Swan conductors for p-adic differential modules, I: A local construction

We define a numerical invariant, the differential Swan conductor, for certain differential modules on a rigid analytic annulus over a p-adic field. This gives a definition of a conductor for p-adic Galois representations with finite local monodromy over an equal characteristic discretely valued field, which agrees with the usual Swan conductor when the residue field is perfect. We also establish analogues of some key properties of the usual Swan conductor, such as integrality (the Hasse–Arf theorem), and the fact that the graded pieces of the associated ramification filtration on Galois groups are abelian and killed by p.

Irregularity and $p$-adic Swan conductor

Let V be a de-Rham representation of the Galois group of a local field of mixed characteristic (0,p) . We relate the Swan conductor of the associated Weil-Deligne representation to the irregularity of the corresponding p -adic differential equation.

Katz Correspondence for Quasi-Unipotent Overconvergent Isocrystals

Let k be a field and X = Spec (k[t,t−1]). Katz proved that a differential equations with coefficients in k((t−1)) is uniquely extended to a special algebraic differential equation on X when k is of characteristic 0. He also proved that a finite extension of k((t−1)) is uniquely extended to a special covering of X when k is of any characteristic. These theorems are called canonical extension or Katz correspondence. We shall prove a p-adic analogue of canonical extension for quasi-unipotent overconvergent isocrystals. As a consequence, we can show that the local index of a quasi-unipotent overconvergent is equal to its Swan conductor.

Canonical extensions, irregularities, and the Swan conductor

Canonical extensions, irregularities, and the Swan conductor

On Swan conductors for Brauer groups of curves over local fields

For an element w w of the Brauer group of a curve over a local field, we define the “Swan conductor” sw ⁡ ( w ) \operatorname {sw}(w) of w w , which measures the wildness of the ramification of w w . We give a relation between sw ⁡ ( w ) \operatorname {sw}(w) and Swan conductors for Brauer groups of henselian discrete valuation fields defined by Kato.

The local index and the Swan conductor

In this paper we study local indices of systems of p-adic linearly differential equations which arise from p-adic representations of the absolute Galois group of local field of characteristic p with finite monodromy. We show the induction formula of the local index of p-adic differential equations and prove the equality between the local index of differential equations and the Swan conductor of p-adic Galois representations by inductive methods.

Reduced Norm Map of Division Algebras over Complete Discrete Valuation Fields of Certain Type

We study a ramification theory for a division algebra D of the following type: The center of D is a complete discrete valuation field K with the imperfect residue field F of certain type, and the residue algebra of D is commutative and purely inseparable over F. Using Swan conductors of the corresponding element of Br(K), we define ψ-function of D/K, and it describe the action of the reduced norm map on the filtration of D.

On the Swan conductor in positive characteristic

Let X be a proper smooth surface over an algebraically closed field of positive characteristic and U be a complement of a simple normal crossing divisor. For a smooth l -adic sheaf Ƒ on U , Deligne proved a formula calculating the Euler characteristic of Ƒ by local invariants. Kato gave another formula for the Euler characteristic in case where Ƒ is of rank 1, using class field theory. In this paper, we show that local terms of these formulas coincide, admitting certain conjectures for vanishing cycles.