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Results tagged with automorphic-forms
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user 11919
An automorphic form is a well-behaved function from a topological group $G$ to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup $\Gamma \subset G$ of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups.
3
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About Section 4.2 in 'introduction to the spectral theory of automorphic forms' by Iwaniec
Hejhal uses that $K\in L^2(F\times F)$, which follows from the continuity of $K$ and the compactness of $F$. The latter assumption appears on the first page of Hejhal's book: "Let $F$ denote a compact …
- 105k
10
votes
A stupid question about Automorphic forms
I am no expert in the general theory, but let me share some thoughts. In some sense (c) accounts for the fact that $K$ does not have enough open subgroups (unlike at nonarchimedan places), so that the …
- 105k
6
votes
Any reference on Eisenstein Series for $\Gamma_0(N)$ in $\mathrm{GL}(2)$
For detailed information I recommend "Chapter 7. Eisenstein series" in Miyake: Modular Forms (Springer Verlag, 2006).
- 105k
9
votes
Accepted
Are theta functions cuspidal representations?
"Cusp form" means "cuspidal automorphic form" by definition. So yes, $\theta(z;u)$ is an automorphic form. But it is not a form lying in a cuspidal automorphic representation, because it is not a Heck …
- 105k
4
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Accepted
Do we know absolute bounds for the norm of Satake parameters?
We don't know any upper bound for $|\alpha_{p,j}|$ that is independent of $p$. On the other hand, we do know that each $|\alpha_{p,j}|$ is bounded by $p^{1/2}$, hence if $\pi$ is an automorphic repres …
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3
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Accepted
How does the solenoid structure of $\mathbb{A}/\mathbb{Q}$ lift to $PGL(2, \mathbb{A})/ PGL(...
Yes, the quotient $\mathrm{PGL}_2(\mathbb{Q})\backslash\mathrm{PGL}_2(\mathbb{A})$ and its generalizations for other (reductive) algebraic groups is a complicated object, and this is to a large extent …
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4
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Accepted
Fourier Transform of Eisenstein Series - Sum of Divisors or Ramanujan Sums?
I think all your questions are answered by the following calculation (assume $m\geq 1$ and $\Re(s)>1$):
$$ \sum_{c=1}^\infty\frac{r_m(c)}{c^{2s}} = \sum_{c=1}^\infty\frac{1}{c^{2s}}\sum_{\substack{\te …
- 105k
4
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Accepted
Hecke-module structure implicit in definition of automorphic forms in Borel-Jacquet's Corval...
The definition on the top of page 195 in B-J's article (in Corvallis 1) is ok, because $f\ast\xi$ is well-defined for any smooth $f:G(\mathbf{A})\to\mathbf{C}$ and any $\xi\in H$.
It suffices to ver …
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8
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Accepted
On the pole of local L-function
No, you are not right. The local $L$-function $L(s,\chi_v)=(1-\chi_v(\omega)q^{-s})^{-1}$ has infinitely many poles, namely the solutions of the equation $q^s=\chi_v(\omega)$. All these poles are simp …
- 105k
6
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Accepted
Newform of a cuspidal Automorphic Representation
Yes, the classical version of this adelic newform is the newform in the sense of Atkin-Lehner, and vice versa. See Casselman: On some results of Atkin and Lehner (Math. Ann. 201 (1973), 301-314), espe …
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5
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Accepted
Local component of global irreducible representation of GL_2(A_F)
Not quite. The local component $\pi_v$ is one of four types.
If $\pi_v$ belongs to the tempered principal series, then it is of the form $\rho(\chi_1,\chi_2)$, where $\chi_1$ and $\chi_2$ are arbit …
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7
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Orbits of SL_n acting on matrices of determinant p
To complement Qiaochu Yuan's answer, check out Lemma 9.3.2 in Goldfeld's book Automorphic Forms and L-functions for the group GL(n,R). On the two sides of (9.3.3) you can see the decompositions of the …
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6
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Accepted
Why is the cuspidal spectrum discrete?
This result is due to Gelfand, Graev, Piatetski-Shapiro and has a short proof. I suggest you read Bump: Automorphic forms and representations, Prop. 3.2.3, pp. 285-289.
Let me switch to $G=\mathrm{P …
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4
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Accepted
Jacquet Langlands correspondence
In what sense is the Weyl law different for congruence subgroups and cocompact groups?
At any rate, the Jacquet-Langlands correspondence is not a bijection between the two cuspidal spectra. More pre …
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3
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Automorphic forms and Galois representations over imaginary quadratic fields: generalizing T...
Taylor's theorem is about regular algebraic representations $\pi$. Most representations are not algebraic in any sense (cf. my answer to this question), hence they are not connected to any Galois repr …
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