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8 questions
6
votes
1
answer
500
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Cusp forms have an orthonormal basis of eigenfunctions for all Hecke operators
I am reading Langlands' pape Euler Products and have a few questions. Let $G$ be a split adjoint semisimple group over $\mathbb Q$. If $p$ is a place of $\mathbb Q$, finite or infinite, let $G_{\...
- 105k
5
votes
1
answer
716
views
Relation between Hecke operators and coefficient of L-functions
This question has its seed in this one by Gory, which found an enlightening answer but one of the comments kept me wondering. I am beginning to discover Hecke operators, and there appears to be an ...
- 6,039
3
votes
1
answer
373
views
Hecke eigenvalue at p and at p^k
I am interested in the relationship between the Hecke eigenvalue at $p$ and at $p^k$ for $k \geq 2$ in the unramified and ramified situation for modular/Maass forms.
More precisely, I know from a ...
- 105k
12
votes
0
answers
285
views
Modularity of endomorphism algebras
This question is about comparing Hecke algebras and endomorphism algebras.
Let $\mathbf{A}_f$ be the ring of finite adèles of $\mathbf{Q}$ and let $K$ be a compact open subgroup of $\mathrm{GL}_2(\...
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1
vote
0
answers
171
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Order of individual Fourier coefficient of a Maass form
Let $D$ be a definite quaternion division algebra over $\mathbb{Q}$ and $\mathcal{O}$ be an Eichler order of $D$. Let $F$ be a Maass form in $L^2(PGL_2(\mathcal{O})\backslash PGL_2(D\otimes\mathbb{R}))...
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2
votes
1
answer
491
views
Newform and Galois representation (Shimura-Deligne Reciprocity Law)
Shimura-Deligne Reciprocity Law implies that to a newform $f \colon =f(z) \in S^{\mathrm{new}}_k(\Gamma_0(N))$ of weight $k$, one can associate Galois representation $\rho_{f,\lambda} \colon {\mathrm{...
- 105k
3
votes
2
answers
403
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Gelfand pair and double coset decomposition
Let $F$ be a non-Archimedean local field with ring of integers $O$, $\pi$ be a uniformizer. Let $\tilde{G}$ be a connected algebraic group over $F$ and splits over $F$, fix a split maximal torus $\...
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4
votes
3
answers
1k
views
Definition of Hecke operators
I am confused about the definition of Hecke operators. It will be great if someone provides some references.
Shimura's 'Arithmetic Theory of Automorphic forms' says: Let $\Gamma$ be acting in the ...
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