Skip to main content
Search type Search syntax
Tags [tag]
Exact "words here"
Author user:1234
user:me (yours)
Score score:3 (3+)
score:0 (none)
Answers answers:3 (3+)
answers:0 (none)
isaccepted:yes
hasaccepted:no
inquestion:1234
Views views:250
Code code:"if (foo != bar)"
Sections title:apples
body:"apples oranges"
URL url:"*.example.com"
Saves in:saves
Status closed:yes
duplicate:no
migrated:no
wiki:no
Types is:question
is:answer
Exclude -[tag]
-apples
For more details on advanced search visit our help page
Results tagged with
Search options questions only not deleted user 38145
1 vote
0 answers
91 views

The kernel $K(x,y)$ as an integral over Eisenstein series for $\operatorname{GL}_2$

Let $G = \operatorname{GL}_2$, $f \in C_c^{\infty}(G(\mathbb A)/Z(\mathbb A))$, and $V = L^2( G(\mathbb Q)Z(\mathbb A)\backslash G(\mathbb A))$ (trivial central character). Then the operator $R(f)$ o …
D_S's user avatar
  • 6,180
1 vote
0 answers
87 views

$H(\mathbb A)^0/H(k)$ is homeomorphic to a closed set in $G(\mathbb A)/G(k)$

The setting is $k$ is a number field, $\mathbb A$ is the ring of adeles of $k$, $H$ is a closed subgroup of an algebraic group $G$, both defined over $k$, and $H(\mathbb A)^0$ is the subgroup of $h \in …
D_S's user avatar
  • 6,180
2 votes
2 answers
271 views

Finiteness of the volume of $G(F) \backslash G(\mathbb A)$

Let $G$ be a semisimple algebraic group over a number field $F$ with trivial center. Let $\mathfrak S \subset G(\mathbb A)$ be a Siegel domain (defined in terms of a given maximal split torus and min …
D_S's user avatar
  • 6,180
2 votes
2 answers
208 views

Definition of cusp form in $L^2$ and convergence over $N_{\mathbb Q} \backslash N_{\mathbb A}$

Let $G$ be an adjoint semisimple group over $\mathbb Q$ with parabolic subgroup $P = MN$ in good position relative to a compact subgroup $U= \prod\limits_v K_v$ of $G(\mathbb A)$. Let $L$ be the spac …
D_S's user avatar
  • 6,180
5 votes
1 answer
222 views

Do we have $G(\mathbb A_S) G(k) = G(\mathbb A)$ for sufficiently large $S$?

Let $\mathbb A$ be the ring of adeles of $k$, and $\mathbb A_S = \prod\limits_{v \in S} k_v \prod\limits_{v \not\in S} \mathcal O_v$ for any (large) finite set of places $S$ containing the archimedean …
D_S's user avatar
  • 6,180
9 votes
1 answer
606 views

The correct determinant exponent of the weight $k$-operator for defining Hecke operators/ade...

For $g \in \operatorname{SL}_2(\mathbb R)$, and $\mathbb H$ the upper half plane, and $k\geq 1$ an integer, the weight $k$-operator on functions $f: \mathbb H \rightarrow \mathbb C$ is defined by $$ …
D_S's user avatar
  • 6,180
1 vote
0 answers
166 views

Absolute convergence of the Fourier series of a smooth adelic function

Let $f: \mathbb A/\mathbb Q \rightarrow \mathbb C$ be a smooth function. Smooth means that $f$ is continuous, smooth in the archimedean argument, for every $(x_0,y_0) \in \mathbb A = \mathbb R \time …
D_S's user avatar
  • 6,180
5 votes
1 answer
520 views

Sufficient condition for the absolute convergence of Fourier series of a function on the ade...

What about when $G = \mathbb A_k/k$ for $k$ a number field, and $\mathbb A_k$ the adeles of $k$? …
D_S's user avatar
  • 6,180
2 votes
0 answers
300 views

Question about the Fourier expansion of adelic Eisenstein series for $\operatorname{GL}_2$

My reference is Daniel Bump's book, Automorphic Forms and Representations, Chapter 3.7. Let $k$ be a number field, $G = \operatorname{GL}_2$, $B$ and $T$ the usual Borel subgroup and maximal torus o …
D_S's user avatar
  • 6,180
7 votes
0 answers
289 views

What does it mean for a complex valued function on $G(\mathbb A)$ to be smooth (or smooth of...

Let $\mathbb A$ denote the adeles of $k$, $\mathbb A_f$ the finite adeles, and $k_{\infty} = \prod\limits_{v \mid \infty} k_v$. …
D_S's user avatar
  • 6,180
6 votes
2 answers
346 views

Definition of unitary representation of $\mathbf G(\mathbb A_k)$

Let $k$ be a global field, and let $G = \mathbf G(\mathbb A_k)$ for a connected, reductive group $\mathbf G$ over $k$. In these notes by Jayce Getz and Heekyoung Hahn, a unitary representation of $G$ …
D_S's user avatar
  • 6,180