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Pdf Notes on Automorphic Forms and the Arthur-Selberg Trace Formula Automorphic forms and the Arthur-Selberg trace formula MA842 BU Spring 2018 Salim Ali Altuğ October 23, 2018 1 Automorphic forms /GL2 (possibly GLn) Lecture 1 18/1/2018 These are notes for Ali Altuğ’s course MA842 at BU Spring 2018. The course webpage is http://math.bu.edu/people/saaltug/2018_1/2018_ 1_sem.html. Course overview: This course will be focused on the two papers Eisenstein Series and the Selberg Trace Formula I by D. Zagier and Eisenstein series and the Selberg Trace Formula II by H. Jacquet and D. Zagier. Although the titles of the papers sound like one is a prerequisite of the other it actually is not the case, the main difference is the language of the papers (the first is written in classical language whereas the second is written in adelically). We will spend most of our time with the second paper, which is adelic. 1.1 Goal Jacquet and Zagier, Eisenstein series and the Selberg Trace Formula II (1980’s). Part I is a paper of Zagier from 1970 in purely classical language. Part II is in adelic language (and somewhat incomplete). Arthur-Selberg conjecture Relative trace formula −−−−−! trace formula the Arthur-Selberg side is used in Langlands functoriality and the Relative is used in arithmetic applications. 1.2 Motivation What does this paper do? “It rederives the Selberg trace formula for GL2 by a regularised process.” Note 1.1 • Selberg trace formula only for GL2 • Arthur-Selberg more general The Selberg trace formula generalises the more classical Poisson summation formula. Poisson summation. Theorem 1.2 Poisson summation. Let f : R R ! then Poisson summation says Õ Õ f n f ξ ¹ º ƹ º n Z ξ Z 2 2 where ¹ 1 1 f ξ f x e xξ dx. ƹ º 2π ¹ º ¹ º −∞ Notation: e x e2πix. To make this¹ lookº more general we make the following notational choices. G R; Γ Z Õ Õ f γ f ξ ¹ º ƹ º γ Γ# ξ G Γ 2 2¹ / º_ where • Γ# conjugacy classes of Γ ( Γ in this case since Γ is abelian). • G Γ dual of G Γ. ¹ / º_ / Selberg. G GL2 R ; Γ GL2 Z Õ ¹ º Õ ¹ º “ ” ··· ··· γ Γ# π “ G Γ ” 2 2 ¹ / º_ relating conjugacy classes on the left to automorphic forms on the right. Arthur and Selberg prove the trace formula by a sharp cut off, Jacquet and Zagier derive this using a regularisation. 1.3 Motivating example Õ1 1 ζ s ¹ º ns n1 converges absolutely for s > 1. Theorem 1.3 Riemann. ζ s has analytic continuation up to s > 0 with a simple pole at s 1 residue 1.¹ i.e.º <¹ º 1 ζ s + φ s ¹ º s 1 ¹ º − where φ s is holomorphic for s > 0. ¹ º <¹ º 1. Step 1: observe ¹ 1 1 s t− dt (for s > 1) s 1 <¹ º − 1 ¹ n+1 Õ1 s t− dt n1 n 2 Step 2: this implies ¹ n+1 1 Õ1 s s ζ s + n− t− dt ¹ º s 1 − − n1 n ¹ n+1 ¹ n+1 1 Õ1 s s + n− t− dt s 1 − − n1 n n we denote each of the terms in the right hand sum as φn s ¹ º ¹ n+1 s s φn s n− t− dt ¹ º n − Step 3: s s φn s sup n− t− j ¹ ºj ≤ n t n+1 j − j ≤ ≤ s s sup j j j j s +1 s +1 n t n+1 t<¹ º ≤ n<¹ º ≤ ≤ by applying the mean value theorem. Í Í So 1n1 φn converges absolutely. Hence φ 1n1 φn is holomorphic One can push this idea to get analytic continuation to all of C, one strip at a time. This is an analogue of the sharp cut off method mentioned above. It’s fairly elementary but somewhat unmotivated and doesn’t give any deep information (like the functional equation). 2. Introduce Õ πn2 t θ t e− ; t > 0 ¹ º n Z 2 + Í πn2 t note that θ t 1 2 1n1 e− . Idea: Mellin¹ º transform and properties of θ to derive properties of ζ. Γ s ¹ 2 1 1 πn2 t s 2 dt e− t / s 2 π / ns 0 t property of θ: 1 1 θ t θ ¹ º pt t Step 1: proof of this property is the Poisson summation formula • πx2 f x e− f ξ f ξ ¹ º ) ƹ º ¹ º • 1 ξ g x f ptx g ξ f ¹ º ¹ º ) ƹ º pt Æ pt Step 2: Would like to write something like ¹ 1 s 2 dt “ θ t t / ” 0 ¹ º t This integral makes no sense 3 • As t , θ 1 thus ! 1 ∼ ¹ 1 s 2 dt θ t t / < A ¹ º t 1 ¹ 1 s 2 dt t / < () A t 1 s < 0 () <¹ º • As t 0 consider ξ 1 so ξ and ! t ! 1 1 1 p θ t θ ξθξ ¹ º pt t p p 1 θ t ξθ ξ ξ ) ¹ º ¹ º ∼ pt so θ t 1 ¹ º ∼ pt ¹ A s 2 dt θ t t / < ) 0 ¹ º t 1 ¹ A s 1 2 dt t¹ − )/ < () 0 t 1 s > 1 () <¹ º so no values of s will make sense for this improper integral. Refined idea: Consider ¹ 1 ¹ 1 s 2 dt 1 s 2 dt I s θ t t / + θ t 1 t / ¹ º 0 ¹ ¹ º − pt º t 1 ¹ ¹ º − º t upshot: I s is well-defined and holomorphic for all s C. Final step:¹ º Compute the above to see 2 2 2 2 s I s + + Γ ζ s ¹ º s 1 s πs 2 2 ¹ º − / which implies 1. ζ s has analytic continuation to s C, with only a simple pole at s 1 with¹ º residue 1. 2 2. I s I 1 s , ¹ º ¹ − º this follows from the property of θ so if we let Γ s Λ s 2 ζ s , s 2 ¹ º π / ¹ º then Λ s Λ 1 s . ¹ º ¹ − º 1.4 Modular forms Functions on the upper half plane, H z C : z > 0 . f 2 =¹ º g Historically elliptic integrals lead to elliptic functions, modular forms and elliptic curves. 4 Note 1.4 When one is interested in functions on Λ where is some object and Λ is some discrete group. Take f a functionO/ on and averageO over Λ to get O Õ f λz . ¹ º λ Λ 2 If you’re lucky this converges, this is good. Elliptic functions. Weierstrass, take Λ !1Z + !2Z a lattice and define 1 Õ 1 1 }Λ z + + . ¹ º z2 z ! 2 !2 ! Λr 0 2 f g ¹ − º Jacobi, (Elliptic integrals) consider ¹ φ dt p ; κ 0 0 1 t2 1 κt2 ≥ ¹ − º¹ − º related by: 2 3 }0 z 4}Λ z 60G2 Λ }Λ z 140G3 Λ ¹ Λ¹ ºº ¹ º − ¹ º ¹ º − ¹ º Õ 2k Gk Λ λ− ¹ º λ Λr 0 2 f g or Õ 1 Gk τ ¹ º mτ + n 2k m;n Z2r 0 ¹ º2 f g ¹ º the weight 2k holomorphic Eisenstein series. Fact 1.5 Let ¹ 1 ds u p 3 y 4s 60G2s 140G3 − − then y }Λ u . ¹ º 1.5 Euclidean Harmonic analysis Lecture 2 23/1/2018 We’ll take a roundabout route to automorphic forms. Today: Classical harmonic analysis on Rn. Classical harmonic analysis on H. The aim (in general) is to express a certain class of functions (i.e. 2) in terms of building block (harmonics). L In classical analysis the harmonics are known (e nx ), then the question becomes how these things fit together. In number theory¹ º the harmonics are extremely mysterious. We are looking at far more complicated geometries, quotient spaces etc. and arithmetic information comes in. Example 1.6 R, f : R C, being periodic in 2 S1 leads to a fourier expansion ! L ¹ º Õ f x an e nx . ¹ º ¹ º n Z 2 5 1.5.1 R2 We have a slightly different perspective. R2 G G via translations (i.e. right regular representation of G will be G 2 G ). I.e. g x x + g. L ¹ º · Remark 1.7 This makes R2 a homogeneous space. R2 with standard metric ds2 dx2 + dy2 is a flat space κ 0. To the metric we have the associated Laplacian (Laplace-Beltrami operator, ) r · r @2 @2 ∆ + @x2 @y2 we are interested in this as it is essentially the only operator, we will define automorphic forms to be eigenfunctions for this operator. Note 1.8 The exponential functions φu v x; y e ux + v y ; ¹ º ¹ º 2 2 2 are eigenfunctions of ∆ with eigenvalue λu v 4π u + v i.e. ; − ¹ º ∆ + λu v φu v 0. ¹ ; º ; These are a complete set of harmonics for 2 R2 . The proof is via fourier L ¹ º inversion. ¹ ¹ f x; y fÆ u; v φu;v x; y du dv ¹ º R2 ¹ º ¹ º where ¹ ¹ ¯ fÆ u; v f u; v φu;v x; y dy dx. ¹ º R2 ¹ º ¹ º A little twist. We could have established the spectral resolution ( of ∆) by considering invariant integral operators. Using the spectral theorem if we can find easier to diagonalise operators that commute we can find the eigenspaces for those to cut down the eigenspace. Recall: an integral operator is ¹ L f x K x; y f y dy ¹ º¹ º ¹ º ¹ º invariant means L g f gL f g G ¹ º ¹ º 2 in our case g f x f g + x .
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