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Results tagged with automorphic-forms
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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.
13
votes
Estimating volumes in the trace formula
Up to a normalisation factor for the measure you fixed on $G_\gamma(\mathbb A)$, this is the Tamagawa number of $G_\gamma$.
There is indeed a formula for this. It is $$ \frac{ \left| \pi_0\left( Z(\ …
- 149k
16
votes
Accepted
The space of cusp forms for $\mathrm{GL}_2$ over ${\mathbf{F}}_q(T)$
If $D$ has degree $\leq 3$ there won't be any cusp forms. By the Langlands correspondence these correspond to irreducible Galois representations into $GL_2$, unramified away from a degree $3$ divisor, …
- 149k
1
vote
Explicit examples of algebraic Hecke characters with infinite image?
Let $n$ be the maximal order of an element of the class group. Take an ideal, take the $n$th power, take a generator, square it, divide by its conjugate.
- 149k
8
votes
Relation between Fourier coefficients and Satake parameters
There is no coincidence, this is the Weyl character formula for the representation $\operatorname{Sym}^k$ of $GL_3$. The reason that the Langlands dual group comes up is, unsurprisingly, the Satake is …
- 149k
6
votes
Accepted
Adelization for any classical arithmetic subgroup
For a subgroup to have a meaningful lift to the adeles, it is necessary and sufficient for the subgroup to be a congruence subgroup in the sense that for some $N$, the subgroup contains all elements c …
- 149k
11
votes
Accepted
Equivalence between Ramanujan and Selberg conjectures
Let $f$ be an automorphic form corresponding to an automorphic representation $\pi =\otimes_v \pi_v$ of $GL_2(\mathbb A_{\mathbb Q})$.
For an unramified prime $p$, the following are equivalent (for …
- 149k
4
votes
Accepted
Is the universal elliptic curve $\overline M_{1,2}$ a toric stack?
The fibers aren't rational curves when viewed as stacks, because they have four points with extra automorphisms. I think this will be problematic.
The universal family of elliptic curves is the quoti …
- 149k
8
votes
Why is the Langlands dual group always taken over $\mathbb{C}$?
The Satake isomorphism gives a relationship between the convolution algebra of $F$-valued functions on $G (\mathcal O) \backslash G(K) / G(\mathcal O)$ and the ring of conjugacy-invariant polynomial f …
- 149k
9
votes
Accepted
Global Langlands function fields
The abstract of V. Lafforgue's paper https://arxiv.org/abs/1404.6416 says
For any reductive group G over a global function field, we use the cohomology of G-shtukas with multiple modifications and …
- 149k
16
votes
Accepted
What is the matter with Hecke operators?
If $K$ is a local field and $\mathcal O$ is its ring of integers, we say an irreducible representation of $G(K)$ is unramified if it contains a vector invariant under $G(\mathcal O)$. It is known that …
- 149k
10
votes
Elliptic curves and supercuspidal representations of conductor $p^2$
Because $p \geq 5$, the ramification of the Galois representation is tame, hence the action of the inertia group on that Galois representation factors through a cyclic group. For the exact same define …
- 149k
3
votes
How to relate Rankin triple L-function to its Dirichlet series
It's easier to give an elementary formula for this sum directly than in terms of the sum you give, though one can give a formula in terms of this sum by taking a ratio of Euler factors.
Let $q$ be a p …
- 149k
6
votes
About different cohomology theories used to study Shimura varieties
For Eichler-Shimura, I am not a historian, but I have a guess. I think the first papers of Eichler and Shimura handle only the correspondence between modular forms of weight 2 and the Jacobians of (co …
- 149k
15
votes
Accepted
How can I see the relation between shtukas and the Langlands conjecture?
I think shtukas are best understood ahistorically. I would start with the modular curves, but specifically with the (geometric) Eichler-Shimura relation. This says that the Hecke operator at $p$, view …
- 149k
3
votes
On the local factor of Rankin-Selberg L-functions
If we write $$L_p( f\times g, s) =\prod_{i=1}^4 \left(1 - \frac{m_i}{p^s} \right)^{-1}$$ for $p\nmid N$ and $L_p ( f\times g, s) =\prod_{i=1}^2 \left(1 - \frac{m_i}{p^s} \right)^{-1}$ then the Rankin …
- 149k