In this talk, I will present some results on the class field theory of smooth projective curves over a local field where one allows arbitrary ramification along a proper closed subset. We shall derive these results using some new results on the class field theory of 2-local fields and a duality theorem. This is based on a joint work with Subhadip Majumder.
The well-known Kronecker-Weber theorem affirms that every finite abelian extension of the field ℚ of rational numbers belongs to some cyclotomic extension ℚ(t|tn=1). In his 12th problem D.Hilbert asked how to generalize this theorem for other global fields. In this talk, we give the exposition of the actual state of this problem together with the connection with Carlitz-Drinfeld-Anderson modules. Recall that Anderson module M is a (left) module over non-commutative ring R=Cp[T,τ], Tτ=τT, τa=ap τ, where Cp is a some field of characteristic p greater than 0, such that M is free finite generated over subrings Cp[T] and Cp[τ].
I will speak about results contained in my article "G-torseurs en théorie de Hodge p-adique" linked to local class field theory. I will in particular explain the computation of the Brauer group of the curve and why its fundamental class is the one from local class field theory.
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