# Automorphic forms, representations, and L-functions by Armand Borel

By Armand Borel

Half 1 includes sections on Reductive teams, representations, Automorphic kinds and representations)

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By a slight refinement = of the above methods it can be proved that Fa[{X;};EJ] F/i[{X;};e1] implies a: /3. In particular, the polynomial rings Fa[ {X;};er], a: < ww +ww, are pairwise elementarily inequivalent. e. a typical element in R( (G)) has the form Tax", Ta ER, L <>EG whose support {a: E Gira f. O} is a well ordered subset of G. FIELDS OF RATIONAL FUNCTIONS 47 The field R( (G)) is a real closed field. (This for instance follows from the fact that R[[G+]], the subring consisting of all series with support in the set G+, is a maximally complete valuation ring and G is divisible, cf.

Show that the power series rings of the are elementarily equivalent for any n E N. The corresponding holds if K and L are real closed 77 1 -fields. degQ(L) are either both infinite or both 44 CHAPTER 3: ELEMENTARY DEFINABILITY finite and equal. For ordered fields we show that a well-known invariant behaves in a similar way: We recall some basic facts. Let K with an ordering < be an ordered field. For a E K we denote I a I= { a ~f a ~ 0 -a 1f a< 0 If Bis a subset, a< IBI means a < lbl for all b E B.

8 If L is a field such that L[X] = Q[X], then L = Q. = Proof. L[X] Q[X] implies char(L) = 0, hence Q ~ L ~ L[X]. p, defining Qin Q[X] and Qin L[X]. 2) a formula i/J defining Q in Q(X] and L in L(X]. Hence D Q = L if Q(X] L[X]. = It is now easy to give examples of elementarily equivalent fields [{ and L such that K[X] L[X]: Take [{ = Q and L a field elementarily equivalent but not isomorphic to Q (for instance an ultrapower of Q). t TRANSCENDENCY DEGREE OVER Q 37 Transcendency degree over the field Q of rational numbers To obtain further results concerning elementary equivalence of polynomial rings over a field/{ we need an analysis of subsets of/{ that are definable inside K[X].