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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.

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Today I want to talk about a paper by Dick Gross, where he describes how to attach a motive M of Artin-Tate type to a connected reductive group G over a field k. The motive M and its L-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 k=\mathbb{F}_q is a finite field, then the twisted dual motive M^\vee(1) yields a formula for the order of G(\mathbb{F}_q), as shown by Steinberg.

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

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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 K, consider F=K((t)). This is complete under the t-adic valuation and has residue field K.

(ii) Let F be the p-adic completion of \mathbb{Z}_p((t)). F is a characteristic zero complete DVR in which p is prime, and its residue field is \mathbb{F}_p((t)).

Suppose that X is a two-dimensional scheme of finite type and let x\in X be a closed point and y\subset X an irreducible curve containing x. Let A=\mathcal{O}_{X,x} and let \mathfrak{p} be the height one prime in A which is the local equation of y at x. Consider the following sequence of localisations and completions:

A \rightsquigarrow \hat{A} \rightsquigarrow \hat{A}_{\mathfrak{p}\hat{A}} \rightsquigarrow \widehat{\hat{A}_{\mathfrak{p}\hat{A}}} \rightsquigarrow {\rm Frac}\widehat{\hat{A}_{\mathfrak{p}\hat{A}}}=: F_{x,y}

It follows from the excellence of A that \mathfrak{p}\hat{A} is a radical ideal of the completion \hat{A}. We then localise and complete at \mathfrak{p}\hat{A} and again use excellence to deduce that 0 is a radical ideal in the resulting ring, i.e., \widehat{\hat{A}_{\mathfrak{p}\hat{A}}} is reduced. The total field of fractions F_{x,y} 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 X via the family of two-dimensional local fields (F_{x,y})_{x\in y\subset X}.

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