Paper: On Euler product and the classification of automorphisc forms I&II

# On Euler product and the classification of automorphisc forms I

This paper proved two main theorems: 
1.  $L(s,\pi\times\pi^{\prime})$
![](https://leanote.com/api/file/getImage?fileId=5d39c579ab644103ac003b0d)
Here I borrow from Cogdell's notes. 
Note that this covers $L(s,\pi)$. 
Thus the L-function gives us an analytic method of testing when two cuspidal representations are isomorphic . 
2. Strong multiplicity one theorem for cuspidal automorphic representations. 
![](https://leanote.com/api/file/getImage?fileId=5d39c579ab644103ac003b0b)
The idea is simple, construct $L_{S}(s,\pi\times \pi^{\prime\vee})$ using 1 and the fact that local L function has no-poles at $s=1$.

# On Euler product and the classification of automorphisc forms II

This paper proved 
1. Any finite set of dinstinct automorphic cuspidal representations is linear independent. 
how to understand this linear independent 
In $n=1$, is reformulation of Dirichlet theorem, Let $\chi_{1},\cdots,\chi_{t}$ be distinct characters of $(\mathbb{Z}/f\mathbb{Z})^{\times}$ and $c_1\cdots, c_t$ be complex numbers. If 
$$
\sum_{1\leq j\leq t}c_j \chi_{j}(p)=0
$$
for all but finitely many prime $p$, then $c_1=\cdots =c_n=0$. 
The prove relys on $L(s,\pi\otimes \pi^{\prime})$ has no pole at $s=1$. 
2. Genralized strong multiplicity one 
![](https://leanote.com/api/file/getImage?fileId=5d39c579ab644103ac003b0a)
![](https://leanote.com/api/file/getImage?fileId=5d39c579ab644103ac003b0e)
Note that the cuspidal representations $\tau_{i}$ need not to be unitary here.

This together with Langlans's Covalis lectures gives a classification of automorphic representations of $GL_n$, the notion of isobaric sum. 
3. Application to strong Artin conjecture 
##Problems 
For $\sigma=\sigma_1\otimes \cdots \otimes \sigma_t$ be an automorphic cuspidal representation of $M(A)$, the Eisenstein series allows us to construct a automorphic representations of some group $G_r(A)$. 
1. When $E(\phi,s,g)$ is zero ? Take the expansions of s, $E(\phi,s,g)=s^kE_k(\phi,s,g)$the first non-vanishing
 ?? 
2. When $I_P^{G}(\sigma)$ is reducible, how to dertermine which one is in the image of Einstein series, is the image of Einstein series irreducible ? For that not appears in the Einstein series, is there another way to realize this ? In other words, is every subquotient of $I_P^{G}(\sigma)$ autormophic ? 
3. How to dertermine the residue spectum? Find the reference
4. The example $E_4$ and $E_6$. 
5. How to generalize to classical groups? Does there exist a Arthur multiplicity formula ? 
 ![](https://leanote.com/api/file/getImage?fileId=5d42f3fcab64410f60000d02)
It fails for classical groups !

#Non-vanishing results 
![](https://leanote.com/api/file/getImage?fileId=5d39c579ab644103ac003b0c)
For $Re(s)>1$, the non-vanishing results follows from the local non-vanishing results plus the absolutely convergent. 
For $Re(s)=1$, it requires the technique of analyzing L-functions via their occurence in the constant terms and fourier coefficients of Eisenstein series. See 
"A non-vashing theorem for zeta functions of GL_n--Jacquet Shalika Invent.math" For $L(s,\pi)$
and 
Shahidi "On nonvashing of L-functions" for $L(s,\pi\otimes \pi^{\prime})$

Partial L function and complete L-function ?

Mixiu

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