8
votes
0answers
159 views

Symmetric Fifth Power Lift of GL(2) Automorphic Form

Let $\pi$ be an automorphic representation of $GL(2)/\mathbb{Q}$. For simplicity, you can take it to be a Maass form for $SL(2,\mathbb Z)$. Kim, Shahidi, Gelbart-Jacquet prove that $$L(s, \pi, ...
7
votes
1answer
243 views

Is the adjoint L-function on GL(m) holomorphic?

Let $\pi$ be an automorphic representation on GL($m$)/$\mathbb{Q}$. Define $$L(s,\pi,Ad):=\frac{L(s,\pi\times\overline{\pi})}{\zeta(s)}.$$ This is an L-function with euler products of degree $m^2-1$. ...
5
votes
1answer
209 views

absolute convergence of Rankin-Selberg series

Let $\pi$ and $\pi'$ be two general automorphic representation on $GL(n)$ and $GL(n')$ over $\mathbb{Q}$. I heard that the rankin-selberg convolution L-function $L(s,\pi\times\pi')$ is absolutely ...
2
votes
2answers
406 views

GL(2) Local Langlands and Artin's L-function

The context I am thinking of mainly is GL(2), and accordingly, the degree 2 Artin L-function. But comments about GL(n) in general are also welcome. In light the local Langlands correspondence, what ...
3
votes
0answers
80 views

Functoriality for triple product GL(2) x GL(2) x GL(2)

Let $f$, $g$ and $h$ be three general automorphic forms on GL(2). Do we know that $L(s, f\times g\times h)$ comes from an automorphic form on GL(8)?
2
votes
0answers
180 views

Automorphicity of L-Factors of Zeta Functions

Associated to a variety over a number field $K$, one has a family of ``Hasse--Weil'' L-functions, which can be combined (as an alternating product) to give the Hasse--Weil zeta function of the ...
3
votes
0answers
100 views

Do local L-functions/epsilon factors vary continuously with the Fell topology?

Edit due to the comment. Consider $G=GL(2)$ over a local field $F$. The Fell topology on the unitary dual of $G(F)$ is seperable. Given a sequence of irreducible unitary representations $(\pi_n)$ of ...
5
votes
0answers
361 views

a generalization of a formula of Shimura

Let $\phi$ be a $GL(2)$ automorphic form with Fourier coefficients $a(n)$ and $a(1)=1$. Obviously we have $L(s,\phi)=\sum \frac{a(n)}{n^s}$. Shimura have the following formula $L(s, Ad\; ...