Simpler form of the trace formula for GL 2 (A)

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1 Simpler form of the trace formula for GL 2 (A Last updated: May 8, 204. Introduction Retain the notations from earlier talks on the trace formula and Jacquet Langlands (especially Iurie s talk and Zhiwei s talk. In particular, G GL 2 oer a number field F. In Iurie s talk, we proed the trace formula for G(A [GJ79, Theorem 6.33]: tr ρ cusp (ϕ + tr ρ sp (ϕ ol( G(F \ G(Aϕ(e + ϕ(x γxdx G(F \ G(A γ elliptic { } + f.p. s ϕ(k ( a k a s dkd a ( A K 2 ol(f \A 0 ϕ(g ( α g log H(wgdg (2 A(A\G(A α + tr(m( iym (iyπ iy (ϕdy (3 4π 4 tr(m(0π 0(ϕ. (4 Here, A is the diagonal torus, and A 0 is the norm- idèles. The meaning of H, π s, M(s and will be recalled in the subsequent sections. Our goal in this talk is to simplify this to the form used in Zhiwei s talk to proe Jacquet Langlands: Theorem.. [GJ79, Theorem 7.2] Assume that ϕ ϕ and that there exist two distinct places, 2 (finite or infinite such that A i \G i ϕ i (g ( α gdg 0 for all α, i.e. the regular hyperbolic orbital integrals of ϕ i anish. Then tr ρ cusp (ϕ + tr ρ sp (ϕ ol( G(F \ G(Aϕ(e + ϕ(x γxdx. G(F \ G(A γ elliptic We will in fact show that each of the terms ( through (4 anishes indiidually.

2 . References As with the other talks on Jacquet Langlands, the main reference is Gelbart and Jacquet s Corallis article [GJ79]. This talk corresponds to Section 7. Daid Whitehouse has online lecture notes on this material [Whi] containing more details. Some of the calculations here hae been lifted from these sources with only notational changes. 2 Orbital terms 2. (2 Hyperbolic term Up to a constant, this is the sum oer α of ϕ(g ( α g log H(wgdg. A(A\G(A We show that this anishes. Recall the height function H : G(A R >0, which was the analogue of y in the SL 2 (R case. This was defined as H(g a b, where g ( a c b k. This can be computed locally; with the analogous definition of H : G(F R >0, we hae H(g H (g. Hence the expression aboe is (abbreiating the arguments of ϕ and H ( ϕ log H u dg [ ( ] ϕ log H u dg A(A\G(A u u A(A\G(A ( ϕ u log H u dg u ϕ dg, u A u\g u A \G and eery product in the last line inoles at least one of or Tate s theory of local Zeta integrals We will recognize the unipotent term ( as a zeta integral and use results from Tate s thesis, which we now recall. We will then express ( as a limit of hyperbolic orbital integrals. Let F be a local field, χ : F C a continuous character, and f a Schwartz Bruhat function on F. In his thesis, Tate studied the local zeta integral (or local zeta function Z(f, χ, s : f(t t s χ(td t, F where d t dt t. For a suitably normalized Haar measure dt, Tate proed a functional equation for this integral and the global L-function using Poisson summation for harmonic analysis on locally compact abelian groups. We will use the following results from this theory: 2 u

3 Z(f, χ, s extends to a meromorphic function on C. The ratio θ(f, χ, s : Z(f,χ,s L(s,χ is entire. (Here, L(s, χ is the local L-factor. For F a number field, f f, and χ χ, define the global analogue Z(f, χ, s Z(f, χ, s, and similarly for L(s, χ and θ(f, χ, s. Recall that L(s, χ is meromorphic on C with at most a simple pole at s. We omit χ from the notation when it is triial. 2.3 ( Unipotent term Define F ϕ : A C by F ϕ (a K ϕ(k ( a kdk, which is easily seen to be Schwartz Bruhat. The unipotent term is f.p. s Z(F ϕ, s f.p. s {θ(f ϕ, sl(s, F }. Writing we get θ(f ϕ, s θ(f ϕ, + θ (F ϕ, (s + L(s, F λ s + λ 0 + λ (s +, f.p. s {θ(f ϕ, sl(s, F } λ θ (F ϕ, + λ 0 θ(f ϕ,. For ϕ ϕ, we hae θ(f ϕ, θ(f ϕ, with the analogous definition of F ϕ, so θ (F ϕ, θ (F ϕ, θ(f ϕu,. u So it suffices to show that θ(f ϕ, anishes at two distinct places. Up to the L-factor, this is ϕ (k ( a k a dkd a ϕ (k ( a ( ( a k a dkd a. This is a limit of regular hyperbolic orbital integrals: for β, β β ϕ (g ( β gdg β A \G β ϕ (k ( a ( β ( a kdkda F β β ϕ (k ( β a(β kdkda F K ϕ (k ( β aβ kdkda F K ϕ (k ( a ( β β ( a kdkda, and now take β. F 3

4 3 Spectral terms For terms (3 and (4, we need to go back and deelop some facts about the representations (π s, H(s of G(A and intertwining operators M(s among them. This material is in [GJ79, Section 4, A C]. 3. Local intertwining operators and meromorphic continuation of M(s Recall that for each for s C we defined the Hilbert space H(s of (equialence classes of functions f : G(A C satisfying f(( αau βa x g ω(a u s+ 2 f(g for all α, β F, x A, a A, u, F + and an L 2 condition. Here, F + is the positie reals embedded in the infinite places. The representation of G(A on H(s by right translation was denoted by π s. The point was that, by Mellin inersion, L 2 (ω, N(AP (FZ(A\GL 2 (A (functions on the boundary, from Akshay s talk and Iurie s talk is a direct integral of H(s. In order to analyze the continuous spectrum, we defined intertwining operators M(s : H(s H( s and used the fact that this has a meromorphic continuation to C and satisfies a functional equation. Here, we used an identification H(s H(0 ( triializing the fiber bundle, in Gelbart and Jacquet s language to be able to say what it means for s M(s to be meromorphic. The representation π s is far from irreducible. To facilitate the analysis, refine this as follows. Note that the diagonal subgroup A 0 A 0 G(A acts on H(s by left translation. This factors through the compact quotient F \A 0 F \A 0 and commutes with the action of G(A by right translation, so (π s, H(s breaks up as a Hilbert direct sum of subrepresentations of G(A according to characters of F \A 0 F \A 0. Since we know how F + F + and Z(A act on H(s by left translation, the characters that occur in this Hilbert direct sum may equialently be iewed as a pair of quasicharacters η (µ, ν of F \A satisfying some conditions. The normalization in [GJ79, Section 4, A] is as follows: let H(η be the space of (equialence classes of functions f : G(A C satisfying f(( a x b g µ(aν(b a /2 f(g for all a, b A, x A, b again with an L 2 condition, and denote the representation of G(A on H(η by right translation by π η. Then π s η π η, where η (µ, ν runs oer pairs of quasicharacters of F \A 0 satisfying µν ω and µ(a ν(a a s for a F +. 4

5 Note that (π η, H(η is a principal series representation; the factor a /2 b is there so that this is the usual normalization. In particular, we know when they are irreducible. We will use this later. We outline the proof in [GJ79, Section 4, B C] of the meromorphic continuation and the functional equation of M(s. For η (µ, ν, write η (ν, µ. Then it is a calculation to check that M(s restricts to M(η : H(η H( η. For η (µ, ν a pair of quasi-characters on F, define the local analogues (π η, H(η and M(η : H(η H( η in the same way. The key computation is to recognize the image of a function in H(η under M(η as a local zeta integral. From there, it follows from Tate s theory that M(η can be normalized using suitable local L-factors to an entire operator, i.e. M(η is some meromorphic function times an entire operator R(η. One checks that their infinite tensor product is defined and so gies a meromorphic continuation of M(η: M(η m(η R(η (5 with m(η a meromorphic function and R(η an entire operator. The functional equations R( η R(η I, M( ηm(η I, M( sm(s I ultimately derie from those of the local zeta integral. 3.2 (4 By the aboe, π 0 η S π η, where S is the set of η (µ, ν with µν ω and µ(a ν(a for a F +. Since M(0 takes π (µ,ν into π (ν,µ and (χ, χ S if and only if χ 2 ω, tr(m(0π 0 (ϕ tr(m((χ, χπ (χ,χ (ϕ. χ:χ 2 ω We know from the study of principal series representations that such π (χ,χ are irreducible, so M((χ, χ acts by some scalar c χ. Thus tr(m((µ, µπ (µ,µ (ϕ c χ tr(π (χ,χ (ϕ c χ tr(π(χ,χ (ϕ. We claim that, for each χ and, 2, we hae tr(π (χ,χ(ϕ 0. To ealuate this trace, we will write π (χ,χ(ϕ as an integral operator and integrate the kernel oer the diagonal, and see that this is an integral of a hyperbolic orbital integral. So let f H((χ, χ. Then (π (χ,χ(ϕ f (x ϕ (yf (xydy ϕ (x yf (ydy G(F G(F ϕ (x ( a F F ( b kf (( a ( b kd adbdk ( ϕ (x ( a ( b k a /2 χ (ad adb f (kdk. F 5

6 Thus π (χ,χ (ϕ is gien by the kernel K(x, y ( ϕ (x ( a ( b ydb a /2 χ (ad a F ( ϕ (x ( b ( a ( a b ydb χ (a F a /2 d a, where the last equality uses the change of ariable b goes to a a b b b a ( a ( b ( a ( b. Hence ( b b a 3.3 (3 tr(π (χ,χ(ϕ K(x, xdx ( A \G ϕ(g ( a gdg and the identity a χ (a a /2 d a. Write π iy π η. Using the functional equation M( ηm(η I, the integrand in (3 becomes the sum oer η of tr(m(η M (ηπ η (ϕ. We show that this anishes. Taking the logarithmic deriatie of the meromorphic continuation (5 of M(η, M(η M (η m (η m(η I + R u (η u R u(η u ( I. u u So tr(m(η M (ηπ η (ϕ m (η m(η tr(π η (ϕ + u tr(r u (η u R u(η u π ηu (ϕ tr(π η (ϕ, u and this anishes since, just as in the term (4, tr(π η (ϕ 0 for, 2. References [GJ79] S. Gelbart and H. Jacquet. Forms of GL(2 from the analytic point of iew. In Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Uni., Corallis, Ore., 977, Part, Proc. Sympos. Pure Math., XXXIII, pages Amer. Math. Soc., Proidence, R.I., 979. [Whi] D. Whitehouse. An introduction to the trace formula. Lecture notes