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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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