Lecture 3: Fourier transforms and Poisson summation

Transcription

1 Mah 726: L-funcions and modular forms Fall 2 Lecure 3: Fourier ransforms and Poisson summaion Insrucor: Henri Darmon Noes wrien by: Luca Candelori In he las lecure we showed how o derive he funcional equaion of he iemann ζ funcion, by leing Λ(s) :=π s/2 Γ(s/2)ζ(s) and hen by showing There are wo key seps in his proof: Λ(s) =Λ( s). (I) Express Λ(s) asamellinransformλ(s) =M(ω)(s), where ω() = e πn2 n= and >,oobainaninegralrepresenaionofλ(s). (II) Exploi he ideniy where θ = x θ(x), x > () x θ(x) := e πn2x =2ω(x)+ The firs sep was proved in he previous lecure, whereas his lecure will be concerned wih he proof of he ideniy (). We firs need o recall some noions from Fourier analysis. Le f : C be an inegrable (i.e. L 2 )funcion. Definiion. The Fourier ransform of f is he funcion f : C given by f(s) = e 2πis f() d. emark 2. The noion of a Fourier ransform makes sense for any locally compac opological group G. IfG is he space of characers χ : G S,henheFourierransformcan be seen as a map L 2 (G) L 2 ( G) by sending f f(χ) = χ()f() d. When G =, all G he characers of G are of he form χ s () =e 2πis for s G.

2 Alhough we can define Fourier ransforms for any funcion f L 2 (), we would like o resric our aenion o a special class of inegrable funcions on which he process of aking Fourier ransforms can be ieraed. Definiion 3(Schwarzfuncion). f is a Schwarz funcion if f is smooh (i.e. C )and if i is of rapid decay (i.e. f(x) x N as x for all N). A Schwarz funcion is clearly inegrable, so we can ake is Fourier ransform. Lemma 4. The Fourier ransform preserves he space of Schwarz funcions. Moreover: (a) ˆf = f( ). (b) f g (s) = f(s) g(s) where f g () = f(x)g( x) dx is he convoluion of f and g. (c) f(λ)(s) = λ f s λ for any λ. In paricular, we will use propery (c), which we prove below. Proof of (c). f(λ)(s) = e 2πis f(λ) d = e 2πisu/λ f(u) du λ = λ f s λ (use u = λ) We will also make use of he following imporan heorem. Theorem 5(Poisson summaion formula). Le f : C be a Schwarz funcion. Then f(n). f(n) = Proof. Consider he funcion F (x) = f(x + n). This is a periodic funcion of period, herefore we can ake is Fourier series expansion: F (x) = a n e 2πinx 2

3 where a n = Therefore: F (x)e 2πinx dx = = = m Z = m Z f(x + m)e 2πinx m Z f(x + m)e 2πinx dx f(x + m)e 2πin(x+m) d(x + m) f()e 2πin d = f(n). f(n)e 2πinx f(x + n) =F (x) = and he resul follows by evaluaing a x =. Going back o he proof of ideniy (), consider now he Gaussian e π2. Proposiion 6. The Schwarz funcion g() =e π2 is is own Fourier ransform. Proof. g(s) = e 2πixs e πx2 dx = e π(x2 +2ixs) dx = e π((x+is)2 +s 2) dx (complee he square) = e πs2 e π(x+is)2 dx = e πs2 e πz2 dz z=is+ We claim ha he inegral is+ e πz2 dz, which is over he line is + parallel o he real line in he complex plane, is he same as e πx2 dx. Thisfollowsbyinegrainge πz2 along he sides of recangles of base 2M on he real axis and heigh s: heinegralalonghewhole perimeer is zero by Cauchy s Theorem, bu he inegral on he verical sides ends o zero as M (see Figure ). Therefore e πz2 dz = e πx2 dx. is+ 3

4 Figure : The inegral of e πz2 along he verical lines ends o as M. To conclude he proof, we need o show ha e πx2 dx =. Buhisfollowsfrom: e πx2 dx =2 e πx2 dx =2 e πx2 dx e πy2 dy =2 π =2 2 π =2 2 r= π/2 θ= 2π e πr2 2π = e πr2 rdθdr Corollary 7. Le f (x) =e πx2. Then f (s) = e πs2 /. Proof. Noe ha f (x) =f ( x). By par (c) of Lemma 4 we mus have: f (s) = f ( x)(s) = f s. Bu f (x) ishegaussianofproposiion6,herefore f (s) =f (s) =e πs2. We are now ready o prove he ideniy (): θ = x θ(x). x 4

5 Proof of ideniy (). Le f (x) =e πx2 and apply Poisson summaion: f (n) θ() = f (n) = = = θ e πn2 / In paricular, he proof of his ideniy concludes he proof of he funcional equaion of he ζ funcion. emark 8. We have defined θ() asafuncionof > bu here is nohing o preven us from exending is domain o he complex righ half-plane {z C : [z] > }. As i is cusomary in his game, we acually roae he domain by 9 degrees and define: θ(z) :=θ( iz) = e πin2 z where z is now a variable in he upper half-plane H = {z C : [z] > }. Thefuncion θ(z) hen saisfies he wo remarkable ransformaion properies: θ(z +2)= θ(z) (by definiion) θ z = iz θ(z) (by ()) corresponding, respecively, o he Möbius ranformaions z z +2andz /z acing on H. Theseransformaionpropeiesareypicalofmodular forms. Inparicular,wesay ha θ(z) isamodular form of weigh /2 on he group a Γ(2) := c b d SL 2 (Z) :b, c mod 2 and a, d mod 2 of deerminan ineger marices which reduce o he ideniy modulo 2. We will see in he near fuure ha he funcion: Λ(s) = ω(i) s d has a funcional equaion analogous o he iemann ζ funcion, and a similar consrucion will apply o general modular forms as well. 5

6 Going back o our general philosophy ha L-funcions corresponds o Galois represenaions, recall ha he iemann ζ funcion is he L-funcion associaed o he rivial Galois represenaion: ρ riv : G Q C and i can herefore be regarded as he simples L-funcion. Nex, we will analyze hose L-funcions ha come from -dimensional represenaions χ : G Q C. By he class field heory of Q (i.e. Kronecker-Weber Theorem) hese represenaions correspond o Dirichle characers. The aached L-funcions are called Dirichle L-funcions. Jus as we did wih he iemann ζ funcion, we will ry and undersand he poles, zeroes and criical values of hese new ypes of L-funcions. 6