All Questions
Tagged with arithmetic-groups automorphic-forms
7 questions
2
votes
0
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205
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Two basic questions on congruence subgroups
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$I have two questions related to congruence subgroups.
Let $$\Gamma=\Gamma_0(N)=\Big\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \...
- 1,019
4
votes
1
answer
364
views
Adelization for any classical arithmetic subgroup
In the classical setting, we can define automorphic forms on $\text{SL}_n(\mathbb{R})$ with respect to any lattice $\Gamma$. In fact, for $n \geq 3$, all lattices are arithmetic subgroups.
I have ...
- 767
9
votes
0
answers
259
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Conway big picture for congruence subgroups of $\mathrm{SL}_3(\mathbb{Z})$
I saw in Conway’s paper "Understanding groups like $\Gamma_0(N)$" that the so-called Big Picture can give simple interpretations for important objects in number theory, such as Hecke ...
- 767
1
vote
0
answers
257
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Does the standard arithmetic subgroup of a closed $\mathbb{Q}$-algebraic groups have non-trivial $\mathbb{Q}$-characters?
I am trying to understand the Borel-Harish Chandra theorem about arithmetic subgroups being lattices.
Suppose $G$ is an algebraic group inside $GL_n(\mathbb{C})$ such that it is definable as a zero ...
- 885
3
votes
0
answers
162
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Hecke eigensystem in cohomology vs. compactly supported cohomology
What follows is a question that's probably well-known to experts, but I haven't been able to find a reference.
Let $\mathrm G$ be a connected, semisimple $\mathbb Q$-group. Let $K \subset \mathrm G(\...
- 391
13
votes
1
answer
910
views
Holomorphic cusp forms and cohomology of GL(2,Z)
Let $V_{k}$ denote the complex representation of $\mathrm{GL}(2)$ given by $\mathrm{Sym}^k(V)$, where $V$ is the defining 2-dimensional representation. Assume that $k$ is even. I would like to compute ...
- 40.3k
11
votes
1
answer
565
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What can the theory of automorphic forms for $SL(n,\mathbb{Z})$ say about $SL(n,\mathbb{Z})$?
While reading "Automorphic Forms and L-functions for the Group $GL(n,R)$" by D. Goldfeld, I've got a feeling that linear groups over $\mathbb{R}$ and $\mathbb{Z}$ are considered only as technical ...
- 3,186