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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.

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 $(F_{x,y})_{x\in y\subset X}$.