没必要细读，但还是写一个简单介绍。

定位：这是自守表示的入门书，但和真正的前沿（例如covering groups、exceptional groups、GGP conjectures、stablization of trace formulas、applications of relative trace formula）还差得远。

这不算是一本特别好的书，首先非常厚，其次大量计算劝退新人（但这点计算量实际上算不了什么，关键是我们能够按照idea走下去，计算就和百位数加减法一样，理解了之后会知道该算什么，大概需要什么）。这书关键在于GL_2，能够看到很多一般理论的影子，以及一些GL_2时产生的幻觉，在第3，4章的最后会介绍一般的猜想和理论，相当于科普。

实际上可以当成工具书没必要看特别仔细，主要是出于无聊发现没人写这个书评，就补一个粗浅的版本。有时间再写更advanced的书的review，比如会议文集。

第1章主要是是模形式和整体逆定理。第1节提了Riemann zeta函数乃至Dirichlet L function的延拓，这个就是标准的Poisson求和（在非平凡特征时使用Gauss和不为0这一性质插值特征，得到twisted Possion求和）。GL_n也可以这样做但边界项更麻烦（如果加cusp条件可以抵掉），一般群是不知道的，我们猜测有Braverman-Kazhdan-Ngo program。记忆Riemann zeta函数的函数方程最简单的方式是：对好的Schwartz function f，直接计算(\int_{0}^{\infty} f(x) x^s dx)\zeta(s)，展开后拆成(0,1), (1,\infty)两部分，(0,1)用Poisson求和转到(1,\infty)，然后取在Fourier变换下不变的函数f=e^{-\pix^2}计算。习题1.1.7提了另一种得到函数方程的方法即使用mellin transform of theta function，习题1.1.10提了Special value and Stark conjecture 。

之后介绍Eisenstein级数的定义（group theoretic 表示容易看出是模形式），fundamental domain的标准图像，怎么求SL_2(Z)的模形式空间维数（这是特殊情况，可避免Riemann-Roch，最关键的是先用Jacobi三乘积公式得到\Delta在上半平面没有零点，然后做k=2,4,6,8，这时考虑E_h (f/\Delta)^6，\Delta^{h/12}/ E_h，一般权考虑乘\Delta降权），例如怎么证明没有权2的SL_2(Z)模形式。

然后介绍Peterson inner product, 证明Eisenstein is orthongal to cusp (习题1.3.6)。由模性易得函数方程（\int_{0}^{\infty} f(iy)y^s dy/y) ）。

也介绍了Hecke operator，这是很重要的，就随便提一提：

Key 1: use SL_2(Z) \ M_{det=d} / SL_2(Z) finite ， elementary divisor thm ->

SL_2(Z) \ GL_2(Q)^+ / SL_2(Z) ={diags} /~

-> Hecke algebra for \Gamma(1) is commutative, Idea: stable under transposition

Key 2: T_{a} := ( \Gamma a \Gamma = union of \Gamma a_i )= \sum f|_{a_i}

Key 3: T is self adjoint on cusp forms

pf: <f|a,g>=<f,g|a^{-1}>, so it's left and right invariant, hence only depends on double coset, hence

<T_af,g>=\sum <f|ai,g>= deg(a)<f|a,g>

But <f|a,g>=<f,g|a^{-1}>=<f,g|(a^{-1})^t>

But det a (a^{-1})^t is similar to a, hence

<f|a,g>=<f, g|a>

Hence a basis of Eigenforms

Key 4: T_n := (SL_2\det=n/SL_2), but it's

\Gamma(1) (a b / 0 d) (ad=n, a,d>0, b mod d) Ex 1.4.4

Key 5: Hecke eigenform -> Euler product

As T_p^{k+1}=T_pT_p^{k}-RT_{p^{k-1}}

dim S_12=1 ->\Delta must be eigenform

Ex 1.4.5 SL_2(Z) -> SL_2(Z/n) surjective which is related to strong approximation

Ex 1.4.9 general \Gamma_0(N), should care only about new forms

函数方程反过来的逆定理则是对称性的应用，toy model：假如一个幂级数在伸缩下不变即a_nx^n=a_n(tx)^n for some t not root of unity, 那么它显然是常数，证明概要：

SL_2(Z) case: by checking on generators, only need to show z^kf(-1/z)=f, as the difference is holomorphic， only need to prove vanish at iy (y>0), using Mellin inversion formula !

General case: similar idea, again consider difference, show difference is holmorphic, and invariant under a infinite order elliptic matrix hence zero (lemma 1.5.1), more complicated

1.6是大家喜欢的Rankin-Selberg方法，在Galois一侧两个表示的张量积当然还是一个表示，可以定义L function，在自守一侧的类比可以靠Rankin-Selberg方法（注意，三个张量积的类比还是开的）：

Rankin-Selberg -> give function eq, integral with non-holomorphic Eisenstein E(z,s)

Thm 1.6.1

Bessel function

Compute Fourier coefficient

Function eq follows from symmetry of Fourier coefficients, analytic property depends on constant term, entire except s=1 and s=0, and res(z)==1/2 if s=1

Group-theoretic explanation

E(z,s)= \pi^{-s} \Gamma(s) \zeta(2s) \sum_{\Gamma_{\infty} \ \Gamma} Im(yz)^s

Key property (1.6.1): (\Lambda(s) encodes Mellin of \phi_0)

\Lambda(s)= \int_{\Gamma(1)\H} E(z,s)\phi(z) dxdy/y^2 （这个就直接计算）

Hence res_{s=1}\Lambda(s)=1/2\int_{\Gamma(1)\H} \phi(z) dxdy/y^2

Proof: unfold right side, use \phi is autormophic, in the end integral over \Gamma_{\infty} \ H=[0,1] x (0,\infty)

应用: 考虑 \phi(z)=f(z) \bar g(z) y^{k/2}，得到L(fxg,s)的解析延拓和函数方程。

第2章阿基米德位基本没处理，如果想了解可以读Vogan等人的工作。

第3章整体理论可以读3.1，3.3，3.5，3.7，证明延拓性和函数方程（包括转移积分，Eisenstein serieqiu），证明强重数一（Whittaker model存在且唯一，在其他群会遇到non-generic的问题，除特例比如SL_2外没有重数一，引入L-packet）。

第4章关心p-adic field上GL_2的表示论，用到了层论和分布的语言来证重数一（直觉是如果在G轨道或者说fiber上都是平凡的，那么就平凡；在可迁的情况不变分布就是Haar测度的twist，所以能直接算），有很多巧妙的计算，比如spherical Wittaker function，比如Jacquet module dim=2 （for irreducible principle series）。

补一些杂记（没有看书，自己回忆的，所以可能有点小错误）：

3.1

Tate thesis

\zeta(s,\chi,\Phi)=(up to finite factor) L(\chi,s)

Local zeta integral extends and has function e.q \gamma

Pf: prove \zeta(\Phi_1, s, \chi) \zeta (\Phi_2^, 1-s, \chi^{-1} ) is symmetric for \Phi_1, \Phi_2 (0<Res <1)

Choose \Phi^ that vanishes at neighborhood of zero, we see \gamma has a meromorphic extension, we see any \Phi zeta integral has an extension.

Note. For any s_0 in C, we can choose \Phi, s.t the local zeta integral has no zero or pole at s_0 (we can even choose it to be 1 in the p-adic case)

Note. Another proof, regard each side as functionals of \phi in Hom_{F^x}( S(F) \otimes \chi | |^s, C), which is 1-dim.

Can be done for GL_2, A= \int_{(a 0 // 0 1)}, then

dim Hom_A (\pi \otimes ||^s, \mathbb C) <=1

unless possible two excpetional s \in \mathbb C/ 2\pi I log q

Global zeta integral (convergent for Res>1) extends and has function e.q , no \gamma, just \zeta(\Phi,s,\chi)= \zeta(\Phi^,1-s, \chi^{-1}). It's entire unless \chi=|x|^{\lambda}

Pf: split the integral to two parts |x|>1 and |x|<1,

|x|>1 can extend no problem, for |x|<1 transform integral over A^x to A^x/ F^x, applying Poisson summation formula

Get \int_{A^x/F^x, |x|<1} \sum_{a in F } \Phi(ax) \chi(x) |x|^s d^{\times} x

- \Phi(0) \int_{...} \chi(x) |x|^s d^{\times} x

( \Phi(0) \int_{...} \chi(x) |x|^s d^{\times} x is the boundary term, which is \int_{0}^1 \int_{x in A^x/F^x |x|=t} \chi(x) |x|^s d^x x dt/t , which is zero unless \chi=||^{\lambda}, in that case it's Vol(A^1/F^x)/s+{\lambda})

Apply Poisson summation formula

\sum_{a in F } \Phi(ax) = 1/|x| \sum_{a in F } \Phi^(a/x)

Change the variable x -> x^{-1} ,

Separate another boundary term, we see int over {|x|<1} is the same on {|x|>1} with s -> 1-s, \chi -> \chi^{-1}, \Phi -> \Phi^{-1}

Now we have global function e.q for zeta integral, and local function e.q for zeta integral, proved in two different way

But global zeta integral= (finitely many product of local zeta integral) L_S(\chi,s)

Final piece: local L function=gcd of zeta integral when \Phi varies, unramfied can be see directly. Arch if v=real, \chi_v=(sign)^e then

L_v(s,\chi_v):= \pi^{-(s+e)} \Gamma((s+e)/2)

If v=complex, \chi_v=|x|^it (x/|x|^{1/2})^k

L_v := 2 (2\pi)^{s+v+|k|/2} \Gamma(s+v+|k|/2)

We see function eq. for L(\chi,s), with a epsilon factor.

L(s,\chi)=epsilon(s,\chi) L(1-s,\chi)

3.3

Strong approximation

Pf: for GL_n (modulo), use GL_n(A) and GL_n(F) action on lattices transitively, get the maximal compact open case.

For general level, use SL_n(Z) -> SL_n(Z/mZ) is surjective.

Cor. The natural map GL_2(R)^+ -> GL_2(Q) \ GL_2(A) / K_0(N) is surjective, easy to see if g_1=ag_2k, then g_1=a_{\infty}g_2, a in \Gamma_0(N). Hence we see

\Gamma_0(N) \ GL_2(R)^+ = GL_2(Q) \ GL_2(A) / K_0(N)

Cor. (Plus fundamental domain)

GL_n(F) Z(A)\ GL_n(A) has finite volume.

Gelfand, Graev, Piatetski-Shapiro

\phi is Compact operator on cusp forms

Siegel sets

Do a poisson summation formula on the Kernel

Cor. (3.3.2) L_0^2([GL_2],w) is direct sum of irr invariant subspaces.

Tensor product theorem: use Hecke algebra, almost formal

Admissiblity follows from compact operator has finite dimensional eigenspace for nonzero eigenvalue.

Multiplicity one: follows from uniqueness of Whittaker model

Strong multiplicity one: modifying finite places using Krillov model contains C_c^{\infty}(F^x)

3.5

Prop 3.5.2 when spherical, can choose GL_2(o) invariant vector

Pf. Because it's principle series, we can write an explicit function (using Iwasawa decomposition G=BK),

f(bk)=\chi(b)\delta(b)^{1/2},

hence W_v|_{GL_2(o)}=1

Thm 3.5.4

By above, we can define

W_{\zeta}(g)= \prod_{v} W_{v,\zeta_v} (g_v)

It's well defined.

Thm 3.5.5

Existence of Whittaker model for automorphic representations

Pf. For any \phi in V, consider Fourier coefficient

W_{\phi}(g)= \int_{A/F} \phi( (1 x \\ 0 1)g) \phi(-x) dx

Idea. Consider F(x)=\phi( (1 x \\ 0 1)g) , it's F-inv, hence has a Fourier expansion

F(x)= \sum_{a in F} c_{a} \psi(ax)

Then c_{a}= \int_{A/F} F(x) \psi(-ax) dx

if a=0, then \phi cuspidal hence zero, or a not 0, do x -> x/a, use \phi is GL_2(F) inv, a in F Get c_{a}= W_{\phi}(diag{a,1}g)

Hence we see F(0)= \sum_{a in F} c_{a} i.e \phi(g)=\sum_{a} W_{\phi}(diag{a,1}g)

It's easy to check \phi - > W_{\phi}, gives a Whittaker model. (Injective hence not zero)

Cor. Multiplicity one (3.3.6)

Pf. Choose W_v s.t for a.e v W_v|GL_2(O_v)=1 (standard one in the spherical rep)

And for finite many places we don't know iso, we choose W_v(y 0 \\ 0 1) in C_c^{\infty} (F^x_v) by theory of Krillov model, and be the same for i=1,2.

Form \phi_i in V_i, claim \phi_1=\phi_2. Firstly they agree on diag{*,1}, and agree on GL_2(F) under left because automorphic , and agree on K_0 under right by some K_0 compact open, and agree on GL_2(F_{\infty}) by assumption arch places are the same, hence strong approximation gives \phi_1=\phi_2, hence V_1 \cap V_2 not 0, must be the same.

Tate thesis to GL_n

1. Godement-Jacquet based on GL_n inside M_n, and Weil reps

2. Jacquet-Langlands ... based on Whittaker functions

Spherical reps, if unitary then tempered or complementary

First peice: define local L for spherical, get L_S for global L function

Then definie global zeta integral Z(s,\phi)= \int_{A^x/F^x} \phi(diag{y,1}) |y|^{s-1/2} d^xy

Expand \phi as Fourier series, the int can be written over A^x, hence product of local one

Local zeta integral （defined by Krillov model）:

Z(s,W_v)= \int_{F_v^{x}} W_{v}(diag{y,1}) |y|^{s-1/2} d^xy

Local convergent if Res >1/2:

As we have explicit description for Krillov model, we see it's zero if |y| >>0, and when |y| -> 0, it depends on Jacquet module J(V) (if supercuspidal then zero , if principal then \chi_i(y)|y|^{1/2} , note local reps from global one is unitary, hence some conditions on \chi_i, namely, |\chi(y)|=|y|^t , then |t|<1/2)

Arch place, Whittaker is some hypergeometric function

Again, direct computation -> local zeta integral = local L function in good enough case e.g spherical and W_v is the standard one : f(bk)=\chi(b)\delta(b)^{1/2} be the spherical vector, then W_v= an integral of $f$, and we normalize such W_v(1)=1

Then compute W_v(y 1)= q^{-m/2}(a_1^{m+1}-a_2^{m+1})/(a_1-a_2) if m>=0 , 0 if m<=0, where m=ord(y)

Again, we can twist the zeta function by a unitary character \chi, get

Z(s,\phi,\chi)

Global function e.q for zeta is very easy:

Let w_1= (0 1 \\ -1 0), then by auto of \phi

Z(s,\phi,\chi)= \int_{A^x/ F^x} \phi(w_1 diag(x 1)) \chi(x) |x|^{s-1/2} d^x x

= \int_{A^x/ F^x} \phi(diag(1 x)w_1) \chi(x) |x|^{s-1/2} d^x x

Change x -> x^{-1} , and use central char we see

= int_{A^x/ F^x} w_{\phi}(x^{-1})\phi(diag(x 1)w_1) \chi(x^{-1}) |x|^{1/2-s} d^x x

So Z(s,\phi,\chi)= Z(1-s, w_1.\phi, \chi^{-1} w_{\phi}^{-1})

Note. If central char is trivial, then this is very symmetric.

Cor. Global Z has an analytic extension

Local eq , gamma factor

Thm 4.7.5

There get function e.q for global L_S function, note L(s, \pi_v, w_v^{-1}\chi_v)=L(s, \pi_v^{dual}, \zeta^{-1} ) for spherical one

then we define L for every place (g.c.d for testing vector from Krillov model), get a function e.q for global L

L(s,\phi,\chi)

3.7

Eisenstein series revisited

E(z,s)= (group explanation) \pi^{-s} \Gamma(s) \zeta(2s) \sum_{z in \Gamma_{\infty} \ \Gamma} Im(yz)^s

Compute \int_{\Gamma \ H}E(z,s)\phi(z) dxdy/y^2 -> Rankin-Selberg

Compute Fourier coeff of E(z,s)

\chi_i=\zeta_i ||^{s_i}, s_i vary with s_1+s_2 fixed, form V=Ind(\chi_1, \chi_2)

Choose f_{s_1, s_2} in V s.t f|_K is independent of s_i, get a flat section

Get a GL_2(A)-equivariant map

\pi(\chi_1, \chi_2) -> A([GL_2],w)

f -> (g -> \sum_{y in B(F)\GL_2(F)} f(yg))

Constant of Fourier -> intertwining integral

E((1 x \\ 0 1)g,f) has a Fourier expanasion w.rt x

->

E(g,f)=\sum_{a in F} \int_{x in A/F} E((1 x \\ 0 1)g,f)\phi(-ax) dx

A complete representative for B(F)\GL_2(F) is

Id,

w_0 (1 t \\ 0 1) (t in F)

Here w_0 =(0 -1 \\ 1 0)

We see

\int_{x in A/F} E((1 x \\ 0 1)g,f)\phi(-ax) dx= \int_{x in A} f(w_0(1 x \\ 0 1)g) \phi(-ax) dx (a not 0)

And if a=0, there is an additional term

Connection with local whittaker function, interwining integral

Thm 3.7.1

Normalized Eisenstein (by L_S (\zeta_1\zeta_2^{-1}, 2s)) 2s=s_1-s_2+1 has an analytic extension

The only possible pole is at s=1-v/2 (s=1/2-v/2, the residue cancel)

Pf. Deal with constant term, interwining integral, non-constant term, spherical whittaker function

Note. If Eisenstein has a pole at s=s_0, then the residue is a function of g, must be automorphic.

3.7.4 orthogonal to the cusp form (we assume \chi_1\chi_2=w)

Pf. Rankin-Selberg method

E(g,f)\phi(f) is invariant under center by assumption on central char, and GL_2(F) by auto,

B=NT, first int over N (use f is N-inv)

3.7.5 residue is constant

Interwining integral

Function eq. (More carefuly)

Pf. Showing they have same constant term, hence the diffference is cusp, hence orthongal to Eisenstein, get it's zero

第4章

Note. Sheaf theory, C^{\infty}(X) is regarded as a ring with mult given by pointwise mult of functions

Jacquet module of B(\chi_1, \chi_2)

Composition of Intertwining integrals

Intertwining : just int f( w_0(1 x 0 1 )g) , no character

Whittaker : int f( w_0(1 x 0 1 )g) and with \phi(-x)

Use Whittaker model of principal series to compute composition of intertwining integrals

P is important,

THM 4.5.2

B(\chi_1,\chi_2) is isomorphic to another pair, then must be interchange of \chi_1 and \chi_2

Pf. Sheaf theory, using P to get distribution

Spherical

Key. Spherical Hecke alg is commutative

Key. We can construct spherical vector, G=BK

Key. A simple module is determined by its character

T(p)T(p^k)=T(p^{k+1})+ q R(p) T(p^k)

Pf. Decompose K\diag{w,1}K to diag{1,w}K \cup (w b \\ 0 1 )K (b mod p)

-> H_K \cong C[T_1, T_2^{+/-}]

We know composition of M in 4.5, for spherical we can do more, e.g compute M on the sphercial vector

THM 4.6.5 !!

Explicit computation

Idea. Only compute it on good representatives in

N(F)Z(F)\ G(F)/K

i.e a_n=diag{w^n,1}

(and assume conductor of \phi is trivial)

Casselman basis

Spherical function=matrix coefficient of spherical vectors

unitarizabale

Easy if unitary char, complementary, construct it again by int over K, but using intertwining integral (form B(\chi,\chi^{-1}) to B(\chi^{-1}, \chi)) to transform

Difference notations: spherical vector, spherical Whittaker function

local function e.q for GL_2

4.7.3 irreducible

Thm 4.7.2 describe Krillov model

Pf. dimJ<=2, we know J as a T module

V_N=C^{\infty}_c(F^x)

Pf. V_N can't be zero, as J(V) is finite dimensional (if no zero, then V is subrep of principal series, pf by hand for this case). V_N is in C^{\infty}_c(F^x), latter is irreducible as a mirabolic module, hence whole space.

Uniqueness principle

Thm 4.7.4

Pf. If L_1, L_2, restriction to V_N, they are both twisted Haar measures, hence indepedent, so c_1L_1+c_2L_2 facto through J(V), but dim J(V)<=2, hence c_1L_1+c_2L_2=0 for all but possible 2 s.

-> Thm 4.7.5 function e.q

Z(1-s, pi(w_1). \phi, w^{-1}\zeta^{-1})

= (gamma factor) Z(s, \phi, \zeta)

w= (0 1 \\ -1 0)

Note. No Fourier transform !

Pf. Define both sides as functionals on V, with parameter s. Show they are good under diag{y,1} -> \chi(y)^{-1} |y|^{-s+1/2}

4.7.6 Local converse (central char + gamma with all twist)

Pf. Consider Bruhat decomposition, Krillov model, V=V_1 \cap V_2, know the action of B agree. Only need to show \pi(w_1) agree on V

by gp action, only need to show \phi_i=\pi_i(w_1)\phi is the same on the point 1.

F_{\zeta}(n)= \int_{|y|=q^{-n}} [ \phi_1(y)-\phi_2(y) ] \zeta(y) d^xy.

Fourier inversion

-> \sum_{\zeta in O^x} F_{\zeta}(1)= \phi_1(1)-\phi_2(1)

But \sum F_{\zeta}(n)x^n= Z(s,\phi_1,\zeta)-Z(s,\phi_2,\zeta)=0

Prop 4.7.7

\gamma compatible with parabolic induction.

Pf. Application of Weil rep (Jacquet-Langlands)

4.8.1 induction from open subgroup

It's supercuspidal as we can use Mackey formula

Thm 4.8.3 SL_2 <-> O(V)

Schwartz space S(V)

Thm 4.8.5

F_B(v) has a formal Fourier transform

4.8.9 Another realization of Whittaker model (take E=F\oplus F, get principal series)