Explicit reciprocity laws for higher local fields

Reciprocity Laws

Informal notes for a talk at UNCG on reciprocity laws (such as quadratic reciprocity and Shimura-Taniyama). This material is from Fearless Symmetry by Ash and Gross [1] and Galois Representations and Modular Forms by Ribet [2].

Ramification Theory for Higher Dimensional Local Fields

The paper contains a construction of ramification theory for higher dimensional local fields K provided with additional structure given by an increasing sequence of their “subfields of i-dimensional constants”, where 0 i n and n is the dimension of K. It is also announced that a local analogue of the Grothendieck Conjecture still holds: all automorphisms of the absolute Galois group of K, which...

Existence theorem for higher local fields

A field K is called an n-dimensional local field if there is a sequence of fields kn, . . . , k0 satisfying the following conditions: k0 is a finite field, ki is a complete discrete valuation field with residue field ki−1 for i = 1, . . . , n, and kn = K . In [9] we defined a canonical homomorphism from the n th Milnor group Kn(K) (cf. [14]) of an n-dimensional local field K to the Galois group...

Higher dimensional local fields Igor

Suppose that we are given a surface S over a finite field of characteristic p, a curve C ⊂ S , and a point x ∈ C such that both S and C are regular at x. Then one can attach to these data the quotient field of the completion ̂ (ÔS,x)C of the localization at C of the completion ÔS,x of the local ring OS,x of S at x. This is a two-dimensional local field over a finite field, i.e., a complete discr...

Ramification of Higher Local Fields