Ramification filtration and differential forms

Abstract: Let

L

be a complete discrete valuation field of prime characteristic

p

with finite residue field. Denote by

G a m m a_{L}^{(v)}

the ramification subgroups of

G a m m a_{L} = o p e r a t o r n a m e G a l (L^{s e p} / L)

. We consider the category

o p e r a t o r n a m e {M G a m m a}_{L}^{L i e}

of finite

m a t h b b Z_{p} [G a m m a_{L}]

-modules

H

, satisfying some additional (Lie)-condition on the image of

G a m m a_{L}

in

o p e r a t o r n a m e {A u t}_{m a t h b b Z_{p}} H

. In the paper it is proved that all information about the images of the ramification subgroups

G a m m a_{L}^{(v)}

can be explicitly extracted from some differential forms

O m e g a [N]

on the Fontaine etale

p h i

-module

M (H)

associated with

H

. The forms

O m e g a [N]

are completely determined by a connection

a b l a

on

M (H)

. In the case of fields

L

of mixed characteristic containing a primitive

p

-th root of unity we show that the similar problem for

m a t h b b F_{p} [G a m m a_{L}]

-modules also admits a solution. In this case we use the field-of-norms functor to construct the coresponding

p h i

-module together with the action of a cyclic group of order

p

coming from a cyclic extension of

L

. Then the solution involves the characteristic

p

part (provided by the field-of-norms functor) and the condition for a "good" lift of a generator of the involved cyclic group of order

p

. Apart from the above differential forms the statement of this condition also uses a power series coming from the

p

-adic period of the formal group

m a t h b b G_{m}

.