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One of the motivations of studying the Langlands Conjectures is the question

What are the finite extensions of the rational numbers?

The local Langlands Conjectures give a dictionary between the Galois Theory of local fields and the representation theory of locally compact reductive groups.

The following videos are of the first lecture by Dick Gross in a series of lectures given at Columbia University in Fall 2011 about the Langlands Conjectures where he explains this in more detail.

Today I want to talk about a paper by Dick Gross, where he describes how to attach a motive of Artin-Tate type to a connected reductive group over a field . The motive and its -function are particularly useful in computing some adèlic integrals that occur in the trace formula. Moreover, the motive contains some other encoded information about the group. For example, if is a finite field, then the twisted dual motive yields a formula for the order of , as shown by Steinberg.

If is a local field with characteristic 0, then the -function is finite if and only if Serre’s Euler-Poincaré measure on is non-zero. Also, there is a local functional equation relating the -function of to that of its twisted dual motive.

The following is taken from notes distributed by Matthew Morrow at the Midwest Number Theory Conference for Graduate Students and Recent PhD’s 2010:

A *two-dimensional local field F* is a complete discrete valuation field whose residue field is a local field.

Some examples of two-dimensional local fields are

(i) For any local field , consider . This is complete under the -adic valuation and has residue field .

(ii) Let be the -adic completion of . is a characteristic zero complete DVR in which is prime, and its residue field is .

Suppose that is a two-dimensional scheme of finite type and let be a closed point and an irreducible curve containing . Let and let be the height one prime in which is the local equation of at . Consider the following sequence of localisations and completions:

It follows from the excellence of that is a radical ideal of the completion . We then localise and complete at and again use excellence to deduce that 0 is a radical ideal in the resulting ring, i.e., is reduced. The total field of fractions is therefore isomorphic to a finite direct sum of fields, and each is a two-dimensional local field.

The two-dimensional adelic philosophy, originally due to A. Parshin, is that we should study via the family of two-dimensional local fields .