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4
votes
0answers
504 views

base change and Langlands' combinatorial exercise

Hi, Is it correct that Langlands' combinatorial exercise (as he terms it in his paper "Shimura varieties and the Selberg trace formula") is to establish base change identities between orbital ...
1
vote
2answers
730 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 ...
6
votes
1answer
267 views

strong approximation and one-dimensional automorphic representations

Hi, Let $D$ be a quaternion algebra over $\mathbf Q$ such that $D\otimes\mathbf R = M_2(\mathbf R)$. Let $\pi = \pi_\infty \otimes_p \pi_p$ be an irreducible automorphic representation of ...
6
votes
2answers
700 views

Decomposition of $L_0^2(GL_2({\mathbb{Q}}) \backslash GL_2(A), \psi)$

Two questions concerning the decomposition of $L_0^2(GL_2({\mathbb{Q}}) \backslash GL_2(A), \psi)$, where $\psi$ is a Hecke character on the adelic ring $A$: It is known that when $\psi$ is trivial ...
3
votes
2answers
2k views

A stupid question about Automorphic forms

Okay, so an automorphic form $f$ on a reductive group $G/ \mathbb{Q}$ and arithmetic subgroup $\Gamma$ is a smooth function satisfying the following conditions: (a) invariance with respect to left ...
13
votes
2answers
759 views

Why isn't meromorphic continuation enough for converse theorems?

This is a very naive question which really does little more than highlight my ignorance of how converse theorems really work. Take an algebraic gadget which should be conjecturally associated to an ...
4
votes
1answer
1k views

Why is the Arthur trace formula so powerful?

Considering the Arthur trace formula, why are the sort of convolution operators, whose "normalized traces" are given in geometric terms and spectral terms, actually able to distinguish all automorphic ...
4
votes
2answers
604 views

Why is the simple trace formula a weaker tool than the Arthur trace formula?

What are some concrete examples of theorems which can be deduced from the Arthur trace formula, which do not follow from the simple trace of Kazdhan and Flicker? (So I do not mean weaker in the sense ...
5
votes
2answers
608 views

Unitary groups over number fields

When defining unitary groups over number fields, one usually takes $F$ to be a totally real number field, $E$ a CM quadratic extension of $F$, and $V$ a hermitian space attached to $E/F$. Then $U(V)$ ...
3
votes
0answers
329 views

Non-vanishing of twists of L functions for GL(4)

Hello, This is a question in the spirit of Nonvanishing of central L-values of quadratic twists? and the application I have in mind is to p-adic L-functions a la Ash-Ginzburg. The question is ...
4
votes
1answer
607 views

Why is the cuspidal spectrum discrete?

Hi, I have a short question concerning the spectral theory of automorphic forms. What is the main property of the unipotent group $N$, which consist of matrices in the form ...
12
votes
3answers
639 views

There is no lattice in PSL(2,R) which contains PSL(2,Z) properly?

How can I see that there is no lattice in $G=\mathrm{PSL}_2( \mathbb{R})$ which contains $\Gamma_1=\mathrm{PSL}_2( \mathbb{Z})$ properly, or equivalently, that $X_1 =\mathrm{PSL}_2(\mathbb{Z}) ...
15
votes
3answers
1k views

Non-vanishing of p-adic L-functions

In Non-vanishing of L-series of modular forms (easy case?) it was answered that for a cuspidal newform $f$ of weight strictly greater than 2, then $L(f,1)$ is non-zero. (Here the $L$-series is ...
7
votes
2answers
958 views

Relation between Theta series and Eisensteinseries

In "Mackey - Unitary Group Representation in Physics, Probability and Number Theory" on page 326, George Mackey mentions a result of Ludwig Siegel, which was later generalized to semi-simple Lie ...
2
votes
2answers
556 views

Are the $\Gamma(N)$ the only normal congruence subgroups of $\mathrm{SL}_2(\mathbb{Z})$?

The answer to the original question is no, see JSE! Are the $\Gamma(N)$ the only normal congruence subgroups of $\mathrm{PSL}_2(\mathbb{Z})$ with no finite subgroups (elliptic elements)? What about ...
10
votes
1answer
523 views

Double coset spaces of reductive groups and integral representations of L-functions

Let $G$ be a reductive group over a number field $k$, with center $Z$. Let $P$ be a parabolic subgroup. Let $H$ be a reductive subgroup of $G$. To what extent can we understand the double coset space ...
12
votes
4answers
1k views

Meromorphic continuation of Eisenstein series

I am interested what kind of (different) proofs of meromorphic continuation Eisenstein series (for general parabolic subgroups) exist in the literature? The only one I understand well, is Bernstein's ...
12
votes
3answers
1k views

Nonvanishing of central L-values of quadratic twists?

Let $\pi$ be a cuspidal automorphic representation of GL(2) over a number field (if you want, assume it's $\mathbb Q$ and $\pi$ comes from a holomorphic modular form). In the case $\pi$ has trivial ...
6
votes
2answers
428 views

Orbits of SL_n acting on matrices of determinant p

Fix a positive integer $n$ and let $S$ be the set of $n$ by $n$ matrices with entries in $\mathbf{Z}_p$ (the $p$-adic integers) whose determinant is $p$. The group $G:=\mathrm{SL}_n(\mathbf{Z}_p)$ ...
9
votes
2answers
834 views

Automorphic forms and Galois representations over imaginary quadratic fields: generalizing Taylor's theorem?

Let $K/\mathbf{Q}$ be an imaginary quadratic field, with $\sigma \in \mathrm{Gal}(K/\mathbf{Q})$ a generator. Suppose $\pi$ is a cuspidal automorphic representation of $GL_2 / K$ with central ...
5
votes
2answers
568 views

Automorphic form encoding the orders of $N$ modulo $p$.

Let $N$ be a nonzero rational number. For every prime number $p$ with $v_p(N)=0$, let $a_p$ denote the index in $\mathbb Z/p\mathbb Z$ of the subgroup generated by $N$ modulo $p$. So we have $a_p=1$ ...
4
votes
0answers
241 views

spectral decomposition for elliptic surfaces?

I'm looking for explicit formulae for the spectral decomposition of $L^2(S)$, where $S$ is an elliptic surface (of complex dimension 2). To be precise, the elliptic surface I'm looking at is the ...
6
votes
3answers
690 views

Local to Global principle for reductive groups

Let $G$ be a reductive group over an algebraic number field $k$. Denote with $k_v$ a local field and with $A$ the ring of its adeles, let $G_k$, $G_{k_v}$ resp. $G_A$ be the group of its $k$- resp. ...
6
votes
2answers
1k views

Automorphic forms on GL(3)

We know that the classical Maass forms on GL(3) are depicted, for instance, in D.Goldfeld's book. I wonder that if there exists "holomorphic" automorphic forms on GL(3) as an analogue of GL(2) case. ...
35
votes
2answers
2k views

Langlands in dimension 2: the Yoshida conjecture

Background: One prominent part of the Langlands program is the conjecture that all motives are automorphic. It is of interest to consider special cases that are more precise, if less sweeping. ...
8
votes
1answer
589 views

Finite dimensional automorphic representations of a definite quaternion with prime discriminant and Hecke action

Before stating the questions that I have, which are very specific and probably not so interesting to someone who has never thought about these things, I need to introduce some notation. Let $p$ be ...
1
vote
2answers
868 views

Arithmetic geometry from a bird's-eye view

Is ist true that Arithmetic Geometry can roughly be separated into two areas: 1) Showing that motivic $L$-functions are automorphic. 2) Calculating special values of these $L$-functions.
30
votes
3answers
2k views

Are there motives which do not, or should not, show up in the cohomology of any Shimura variety?

Let $F$ be a real quadratic field and let $E/F$ be an elliptic curve with conductor 1 (i.e. with good reduction everywhere; these things can and do exist) (perhaps also I should assume E has no CM, ...
26
votes
6answers
3k views

How is representation theory used in modular/automorphic forms?

There is certainly an abundance of advanced books on Galois representations and automorphic forms. What I'm wondering is more simple: What is the basic connection between modular forms and ...
6
votes
1answer
582 views

$L$-functions for $\Theta$-lifts

Let $E/F$ be a quadratic extension of number fields. Let $W$ be a hermitian space over $E$ of dimension $2,$ and let $V$ be a skew-hermitian space of dimension $3$ over $E.$ Consider the associated ...
8
votes
0answers
458 views

Automorphic representations attached to abelian varieties

Let $A$ be an abelian variety defined over $\mathbb{Q}$, of dimension $d$. It is widely expected that there is an automorphic representation $\pi_A$ of $GL(2d)/\mathbb{Q}$ whose L-function agrees ...
10
votes
1answer
749 views

Simple explicit example of local Jacquet-Langlands theorem for inner forms of GL(n), and consequences

This one will be very easy for the experts. Let $F$ be a nonarch local field, let $n\geq1$ be an integer, choose $0\leq d<n$ and let $D$ be the central simple algebra over $F$ with invariant $d/n$ ...
13
votes
3answers
954 views

Constructing coherent sheaves on Shimura varieties.

Let me first run through the setting of my question in an example I understand well; that of modular curves. If $Y_1(N)$ denotes the usual modular curve over the complexes, the quotient of the upper ...
4
votes
1answer
600 views

Rallis inner product formula for U(2,2) and U(3)

Victor Tan has a couple of papers on a regularized Siegel-Weil formula for U(2,2) and U(3). The papers I'm talking about are: "A Regularized Siegel-Weil Formula on U(2,2) and U(3)", Duke, 1998. "An ...
6
votes
2answers
769 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
9
votes
2answers
2k views

p-adic L-functions

For modular forms, it is known that you can construct p-adic L-functions by interpolating (p-power conductor) twists of their associated L-functions at special values. Similarly, Kubota-Leopoldt's ...
6
votes
3answers
1k views

Terminology occuring in automorphic representation and relationship between them

When one tries to read about automorphic representation few terms come up more than others namely, 1.Cuspidal 2.Square Integrable 3.Absolutely Cuspidal 4.Super Cuspidal My understanding about ...
9
votes
2answers
502 views

Is there a canonical notion of “mod-l automorphic representation”?

As the title says. In particular, I am interested in the story for a general reductive group $G$, say defined over $\mathbb{Q}$. I can imagine that mod-$\ell$ (algebraic) automorphic representation ...
4
votes
2answers
430 views

Conductor of monomial forms with trivial nebentypus

Is it true that the conductor of a holomoprhic or a Maass cusp form with trivial nebentypus corresponding to a two-dimensional dihedral representation (over $\mathbb{Q}$ )is non-square-free?
5
votes
1answer
403 views

extending cusp forms

Let $E/F$ be a quadratic extension of number fields, and let $V$ be an $n$-dimensional Hermitian space over $E$. Let $\tilde{G} := GU(V)$ and $G := U(V)$. Suppose that $(\pi, V_{\pi})$ is an ...
40
votes
6answers
3k views

What are the local Langlands conjectures nowadays, for connected reductive groups over a $p$-adic field?

Let me stress that I am only interested in $p$-adic fields in this question, for reasons that will become clear later. Let me also stress that in some sense I am basically assuming that the reader ...
13
votes
2answers
1k views

Overview of automorphic representations for $SL(2)/{\mathbf{Q}}$?

In short: what does Labesse-Langlands say? Slightly more precise: what are the cuspidal automorphic representations of $SL_2(\mathbf{A}_{\mathbf{Q}})$, together with multiplicities? Let's say that I ...
22
votes
5answers
4k views

Where stands functoriality in 2009?

Robert Langlands is famous in number theory for making famous and deep conjectures about very abstract things called automorphic forms, somewhere in the 60s. There's a very interesting article by ...