All Questions
25 questions
2
votes
0
answers
94
views
Galois representations attached to discrete automorphic representations
Let $F$ be a totally real field. Let $G$ be a (split) connected reductive group over $F$. Let $\pi$ be an irreducible automorphic representation of $G$.
Recall in the work of Buzzard and Gee "The ...
- 6,622
6
votes
1
answer
536
views
How to see that Eisenstein series are eigenfunctions of the laplacian?
Let $\Gamma$ be a discrete subgroup of $PSL_2(\mathbb{R})$ of finite type. Let $c_1,\ldots,c_h\in\mathbb{R}\cup\{\infty\}$ be a set of representatives of the $\Gamma$-equivalence classes of cusps. For ...
- 7,527
3
votes
0
answers
118
views
A question on the averages of Kloosterman sums
Sorry to disturb. Recently, I encountered a puzzle on the sums involving two Kloosterman sums. That is,
For any $h, q_1,q_2\in \mathbb{N}$ with $(q_1,q_2)=1$ and $Q>1$, how two get a bound
$$\sum_{...
- 563
2
votes
1
answer
164
views
What is the sum operation on torsors induced by Weil uniformization?
Let $k$ be an algebraically closed field, $G$ a reductive group, and $C$ a curve. The algebraic version of the Weil uniformization theorem (see e.g. arXiv:1511.06271v2) says that groupoid of $G$-...
- 1,969
4
votes
1
answer
344
views
Tannakian fundamental group of automorphic representations
Let $\mathcal{C}_{\mathrm{aut}}(G, F)$ be the category of automorphic representations of a connected reductive group $G$ over a number field $F$.
If this is a Tannakian category, it has an associated ...
- 87
4
votes
1
answer
211
views
anti-holomorphic Hilbert modular forms as global sections
The classical definition of Hilbert modular cuspforms as given in say, Hida's "On $p$-adic Hecke algebras for $\mathrm{GL}_2$ over totally real fields", defines them as holomorphic functions ...
- 302
23
votes
2
answers
3k
views
How can I see the relation between shtukas and the Langlands conjecture?
The following bullet points represent the very peak of my understanding of the resolution of the Langlands program for function fields. Disclaimer: I don't know what I'm writing about.
Drinfeld ...
- 317
2
votes
0
answers
105
views
Possible Context for this "Siegel-like" Modular Form Construction?
The following construction of something very nearly a Siegel modular form of degree 2 arose in my research. I'm outside the worlds of automorphic forms and number theory, so I'm wondering if it ...
- 1,701
6
votes
2
answers
1k
views
About different cohomology theories used to study Shimura varieties
The classical theory of Eichler-Shimura realizes the space of cusp forms of certain weights and levels in the parabolic cohomology of modular curves, which is the image of the cohomology with compact ...
- 159
4
votes
0
answers
206
views
Higher dimensional generalization of an identity between traces of Hecke operators and number of elliptic curves over finite fields?
In http://www.math.ubc.ca/~behrend/ladic.pdf, the author uses his generalization of Lefschetz trace formula to smooth algebraic stacks to prove an interesting identity (Proposition 6.4.11.):
$\sum_{k}...
- 6,622
3
votes
0
answers
166
views
Automorphy Factor from Vector Bundles on Compact Dual
So I'm coming from an algebraic geometry perspective and I'm trying to carefully piece together the story of interpreting automorphic forms as sections of vector bundles on Shimura varieties. I think ...
- 1,701
0
votes
0
answers
116
views
Gauss lemma for a complete Noetherian domain
Suppose that $R$ is a Noetherian complete domain over a field $K$.
Suppose that a monic polynomial $f(X) \in R[X]$ (i.e., the highest degree $X^e$ in $f$ has the coefficient $1$), satisfies the ...
- 563
26
votes
1
answer
959
views
What automorphic forms are expected to occur in the zeta function of moduli space of curves?
Assume $g \geq 1$ and $n \geq 0$, the moduli stack ${\mathcal {M}}_{g,n}$ classifies families of smooth projective curves of genus $g$ with $n$ marked points , together with their isomorphisms. It has ...
- 6,622
11
votes
1
answer
660
views
Finiteness or infiniteness for Galois representations with unusual Hodge numbers
Say a representation $\operatorname{Gal}(\mathbb Q) \to GL_n(\overline{\mathbb Q}_\ell)$ has big monodromy if the Zariski closure of the image of $\operatorname{Gal}(\mathbb Q) $ contains $SO_n$ or $...
- 148k
5
votes
0
answers
201
views
The Geometry of Jacobi Forms and their Asymptotic Expansions
A Jacobi form of weight $k$ and index $m$ is a meromorphic function $\varphi: \mathbb{H} \times \mathbb{C} \to \mathbb{C}$ satisfying
$$\varphi\bigg(\frac{a \tau + b}{c \tau + d}, \frac{z}{c \tau + d}...
- 1,701
2
votes
1
answer
212
views
When will the value of automorphic function $f(x)$ satisify an algebraic equation?
When will the value of automorphic function $f(x)$ satisfy an algebraic equation? Or what is the value of $x$ such that the value of automorphic function $f(x)$ is algebraic?
If the question is too ...
- 3,094
4
votes
1
answer
698
views
Reference for the proof of Langlands conjecture for $GL_n$ over function fields
Is there any reference written in English for the proof of Langlands conjecture for $GL_n$ over function fields?
- 781
6
votes
1
answer
1k
views
Relationship between motivic Galois groups and Langlands program [duplicate]
I would like to know if there is any relationship between the motivic Galois groups and the Langlands program.
Many thanks.
- 1,720
12
votes
1
answer
572
views
Non-algebraic Hecke characters
Algebraic Hecke characters are ubiquitous in modern number theory. They are in 1-1 correspondence with one dimensional complex Galois representations, and in some precise sense they are the building ...
- 17.6k
12
votes
1
answer
895
views
Is the universal elliptic curve $\overline M_{1,2}$ a toric stack?
It is well-known that the compactification $\overline M_{1,1}$ of the moduli space of elliptic curves over $\mathbb C$ is a weighted projective line with
weights $4$ and $6$. As far as I can tell, ...
- 5,186
11
votes
1
answer
786
views
A frustrating cohomology class on the moduli of abelian surfaces
Here's a very frustrating question that I have been stuck on for some time. I believe that my question could fit in a general framework of what happens when you restrict $L^2$-cohomology classes on a ...
- 40.2k
1
vote
6
answers
1k
views
List of structure theorems for vector valued Siegel modular forms (esp. of genus 2)
What are Siegel modular forms?
We start with defining their common domains $\mathbb{H}_g$ as the set of symmetric $g \times g$ matrices with positive definite imaginary parts.
The symplectic group $...
19
votes
2
answers
5k
views
New Geometric Methods in Number Theory and Automorphic Forms
The MSRI is organising a programme with the above title from Aug 11, 2014 to Dec 12, 2014. Here is a short description from their website :
The branches of number theory most
directly related to ...
- 18.7k
7
votes
1
answer
1k
views
An interesting double coset in the theory of automorphic forms
Does anyone have some idea to describe the double coset $P(F)\backslash G(F)/H(F)$ , say using Weyl group elements ? Here $G=GL_n\times GL_{n-1}$ is defined over a number field $F$ , $H=GL_{n-1}$ ...
- 809
9
votes
2
answers
1k
views
modularity of algebraic varieties
Hello,
Are there any examples of varieties which are not Shimura varieties or abelian varieties
and whose L-functions have been shown to be a product of automorphic L-functions?
Thanks.
N
- 2,842