All Questions
Tagged with automorphic-forms algebraic-number-theory
42 questions
9
votes
1
answer
680
views
Roadmap to Carayol-Deligne-Langlands
Having begun self-study of Fermat's Last Theorem a few years ago, I have only recently begun to understand and appreciate the theorem of Carayol-Deligne-Langlands on local-global compatibility for ...
- 283
1
vote
0
answers
136
views
Notion of "Hodge bundle" for abelian type Shimura varieties
For a Siegel type Shimura datum $(\text{GSp}_{2g}, \mathcal{H}^{\pm})$ and level $K$, we construct the Shimura variety $S_{g,K} := \text{Sh}_K(\text{GSp}_{2g},\mathcal{H}^{\pm})$. We have a universal ...
- 159
3
votes
1
answer
199
views
Counting local representations for $\mathrm{GL}_2$
$\DeclareMathOperator\GL{GL}$Some context.
In number theory, it is natural to study distribution questions for the family of elliptic curves over $\mathbb{Q}$ (or any fixed number field for that ...
- 685
3
votes
0
answers
79
views
Logarithm map for groups defined over adelic ring
I've been reading the book Eisenstein series and automorphic representations and I am struggling to understand the definition of a logarithm map $H:G(\mathbb{A})\rightarrow \mathfrak{h}(\mathbb{R})$ (...
- 323
3
votes
1
answer
98
views
Reference Request: Possible generalizations of the stability of $\gamma$-factors
$\DeclareMathOperator\GL{GL}$
Let $F$ be a nonarchimedean local field. Suppose $\pi, \sigma$ are irreducible admissible representations of $\GL_{n}(F)$ and $\GL_{m}(F)$ respectively, with $n \geq m$. ...
- 639
8
votes
1
answer
567
views
Symmetric power lift of modular forms
Let $f_1$ and $f_2$ be two cuspforms of weights $k_1$ and $k_2$ and nebentypus $\epsilon_1$ and $\epsilon_2$ respectively such that $f_1 \neq f_2 \otimes \chi$ for some Dirichlet character $\chi$ of ...
- 424
3
votes
0
answers
117
views
Reference Request: Local decomposition of GGP period integrals of cuspidal forms on unitary groups
Setup: Let $E/F$ be a CM-extension of global number fields. Let $(V,\phi)$ be an Hermitian space of dimension $n$ over $E$. Let $(V^{\flat}, \phi^{\flat})$ be a subspace of $V$ of dimension $n-1$ on ...
- 639
2
votes
1
answer
264
views
'$\times$' or '$\otimes$' when writing $L$-functions?
Recently, I came across the Langlands correspondence theorem, there is the following line:
$$L(s,\pi(\sigma) \times \pi(\tau)) = L(s,\sigma \otimes \tau), $$
where $\sigma$ and $\tau$ are ...
- 185
6
votes
1
answer
574
views
Automorphic representation of GL(1)
These might be very silly questions, but somehow I am not able to understand it or I might have misunderstood something.
I am reading automorphic forms from this book.
What I have understood till now:
...
- 424
9
votes
0
answers
233
views
A detail in I. Piatetski-Shapiro and S. Rallis's "Doubling paper": computing the integral on negligible orbits
I'm currently reading the paper "L-functions for the classical groups" by I. Piatetski-Shapiro and S. Rallis, where they introduced the doubling method over classical groups.
I'm confused at ...
- 639
3
votes
0
answers
234
views
Basic question on the Langlands conjectures for $GL_n$ over global field of positive characteristic
My field is far from the Langlands conjectures. I am just trying to understand some basic ideas.
At the moment I am interested in a global field $K$ of positive characteristic and the group $G=GL_n$. ...
- 21.8k
5
votes
1
answer
365
views
modularity lifting theorems for non-compact unitary groups
I am reading David Geraghty's paper, 'Modularity lifting theorems for ordinary Galois representations'(https://link.springer.com/article/10.1007/s00208-018-1742-4) and I have a related question, which,...
- 63
3
votes
0
answers
144
views
Ash–Stevens for Hilbert modular forms
In the theory of mod-$p$ modular forms, I learned a while ago about an interesting result that I think is technically due to Serre and Tate, though the proof was first published by Jochnowitz in ...
- 241
1
vote
0
answers
151
views
Action of $T_p$ on automorphic forms, and error in Gelbart's "Automorphic forms on adele groups"?
Let $f\in\mathcal{S}_k(\Gamma_0(N),\chi)$ be a cuspidal modular form, and $\phi_f\in\mathcal{A}_0(\text{GL}_2(\mathbb{Q})\backslash\text{GL}_2(\mathbb{A}_\mathbb{Q}),\omega)$ be its corresponding ...
- 111
8
votes
1
answer
385
views
Classical and adelic automorphic forms from SL(n) to GL(n) over number fields
It's a long post but I felt like I needed to provide some context to my problem. The explicit questions start at the bold font questions below.
In the classical world, it seems that one is usually ...
- 767
2
votes
0
answers
205
views
Two basic questions on congruence subgroups
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$I have two questions related to congruence subgroups.
Let $$\Gamma=\Gamma_0(N)=\Big\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \...
- 1,019
1
vote
0
answers
115
views
Powers of automorphic Eisenstein series
Let $G$ be a reductive group defined over $\mathbb{Q}$. Let $P$ be a standard parabolic subgroup of $G$ with Levi decomposition
$$P = MN.$$
We denote by $R_{disc,M}$ the discrete spectrum of $M$. Let $...
- 103
5
votes
0
answers
213
views
Truncation and weighted orbital integrals in hyperbolic term of trace formula for $\mathrm{GL}(2)$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PGL{PGL}$I am looking at Gelbart--Jacquet's article in the first Corvallis volume (the article entitled Forms of $\GL(2)$ from an analytic point of ...
- 241
4
votes
1
answer
417
views
What are the known number-theoretic functions, that are related to "the number of ideals of norm $n$, that belong to the class $[c]$"?
Let $L$ be a number field, $\mathcal{O}_L$ its ring of integers, and $\mathcal{Cl(O}_L)$ its ideal class group. Let's fix an arbitrary class $[c] \in \mathcal{Cl(O}_L)$. By $r(n)=r([c], n)$, I mean ...
7
votes
0
answers
122
views
Theta Function Associated to Kummer Lattice
This is something which I feel must be out in the literature somewhere, but I have been unable to find anything.
If we let $\text{Km}(A)$ be the Kummer $K3$ surface associated to an abelian surface $A$...
- 1,701
8
votes
2
answers
744
views
A question related to Hilbert modular form
This is a question related to Hilbert modular forms.
Let $\mathbb{K}=\mathbb{Q}(\sqrt D)$ be an imaginary quadratic field with discriminant $D<0$ and $\zeta (\text{mod } m)$ a Hecke character such ...
- 424
3
votes
1
answer
277
views
Level vs. conductor of a supercuspidal representation
What is the relation between level and conductor of a supercuspidal representation of $\operatorname{GL}_2(\mathbb{Q}_p)$ for some prime $p$?
Proposition 3.4 in Loeffler and Weinstein - On the ...
- 424
1
vote
1
answer
274
views
Analogous theorem for Hilbert modular forms
I have studied modular forms and saw a correspondence like a newform correspond to a automorphic representation of $\mathrm{GL}_n(\mathbb{A_Q})$. Does any similar result holds for Hilbert modular ...
- 424
6
votes
1
answer
497
views
Conductor of Principal series representation
Let $\mathbb{F}$ be a local field and let $\pi$ be a principal series representation of $GL_2(\mathbb{F})$ that is $\pi=Ind_B^{GL_2}(\chi_1\otimes\chi_2)$ for two characters $\chi_1$ and $\chi_2$ of ...
- 424
11
votes
1
answer
490
views
conductor formula
Let $\pi_p$ be an irreducible representation of $GL_2(\mathbb{Q}_p)$. Assume $\pi_p$ is ramified,hence it will have a positive conductor. Consider $sym^3(\pi_p)$ which is a representation of $GL_4(\...
- 424
6
votes
1
answer
315
views
Proving automorphy of the Galois representations of number fields without considering the residual representation
All the papers proving automorphy of the representations of Galois groups of number fields that I have come across seem to first reduce the representation modulo a prime, prove the automorphy of the ...
user145520
6
votes
1
answer
937
views
Langlands Reciprocity and Fermat's Last Theorem
Question:
Can Langlands Reciprocity be used to prove Fermat's Last Theorem?
Background
A few years ago I was reading a book on the Langlands Program and the introduction provided a list of ...
- 69
9
votes
0
answers
419
views
Does local Langlands say anything about the isomorphism class of the absolute Galois group?
I have heard some people claim that the Local Langlands Correspondence over $\mathbb{Q}_p$ (when it is known) is a deep theorem about representations of the absolute Galois group of $\mathbb{Q}_p$. My ...
- 91
2
votes
0
answers
196
views
Trying to understand why Eisenstein series is well defined
I am struggling to see why Eisenstein series is well defined, and I would greatly appreciate clarification.
Let
$$
E(x, \lambda) = \sum_{\delta \in P(\mathbb{Q}) \backslash G(\mathbb{Q}) }
e^{\...
- 3,625
5
votes
0
answers
314
views
Hodge-Tate weights of cohomological cuspidal automorphic representation
Let $\Pi$ be an algebraic cuspidal automorphic representation for $GL_{n}/\mathbb{Q}$ cohomological with respect to a dominant integral weight $\mu \in X^{*}(T)$ ($T \subset GL_{n}$ being the standard ...
- 183
5
votes
0
answers
147
views
For Hida theory on $GU(2,2)$ can $p$ be inert in the imaginary quadratic field $K$?
I am familiar with the theory of Hida families of modular forms, so Hida theory on $GL_2$, but I am not familiar with Hida theory on any other algebraic groups. My question concerns Hida families of ...
- 323
4
votes
0
answers
249
views
Orbits of arithmetic subgroups intersection a compact set
Let us suppose we have $G$ a connected reductive group over a number field $F$. Consider $G(\mathbb{A})$ the group over the adeles and $G(\mathbb{Q})$ embedded discretely. For $\gamma \in G(\mathbb{Q})...
10
votes
0
answers
391
views
Residue of Eisenstein Series on GL(n)
Reference: Mœglin, C.; Waldspurger, J.-L.: Le spectre residuel de GL(n)
On GL($n$), the result of Waldspurger shows that if an automorphic representation $\pi$ is non-cuspidal and in the discrete ...
- 3,804
9
votes
1
answer
920
views
Newform of a cuspidal Automorphic Representation
I was going through these notes https://www.dpmms.cam.ac.uk/~ty245/2008_AGR_Fall/2008_agr_week1.pdf . There, Theorem 9.2 states that: If $\pi ^{\infty}$ is a cuspidal automorphic representation of $\...
- 493
22
votes
1
answer
3k
views
What is the $p$-adic Langlands conjecture for $\mathbf{GL}_1$?
In the Boston conference on Fermat's Last Theorem (Summer 1995), Barry Mazur said (around 15m into the video) about class field theory that
If you are a number-theorist and you want to cheer ...
- 18.7k
15
votes
3
answers
1k
views
Philosophy behind cohomological representations
For a given real reductive Lie group $G$, we have the notion of a representation being cohomological using the Lie algebra cohomology. In particular we know that the discrete series representations of ...
23
votes
2
answers
2k
views
Even Galois representations "mod p"
Consider an irreducible $\mathrm{mod}$ $p$ representation:
$$\rho: \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_2(\bar{\mathbb{F}}_p)$$
If $\rho$ is odd, it was conjectured by Serre in ...
- 17.6k
3
votes
1
answer
368
views
References about identities of Gauss sum
I am reading the paper. In the end of page 10, there are the following identities of Gauss sum.
\begin{align}
& h(b) h(a+b) = q^b h(b) h(a), \\
& h(b) g(a+b) = q^b h(b) g(a), \\
& g(a+b) h(...
- 6,201
5
votes
0
answers
266
views
On Langlands Pairing and transfer factors
In the paper "On the definition of transfer factors" Langlands and Shelstad define a certain number of factors $\Delta_{I}$, $\Delta_{II}$,$\Delta_{III,1}$,$\Delta_{III,2}$, which are roots of unity.
...
- 3,472
11
votes
2
answers
4k
views
Explicit examples of algebraic Hecke characters with infinite image?
Jerry Shurman has a lovely set of notes explaining the classical definition of Hecke characters, the idelic definition of Hecke characters, their relationship, and the classification of algebraic ...
- 7,062
5
votes
1
answer
630
views
Special value of $L$-function
Let $p$ be a prime number. Let $f$ be a newform of weight 2 on $Γ_0(p)$, and $E_f$ denote the associated newform quotient of $J_0 (N)$ over $\mathbb{Q}$. Is there a way to express the
algebraic part ...
- 471
2
votes
2
answers
1k
views
Decomposition of Artin L functions
The Dedekind zeta function of an abelian extension $E$ of $\mathbb{Q}$ factors as a product of Artin L function $L(s, \chi)$, where the product runs over all irreducible representations $\chi$ of $Gal(...
- 11.2k