All Questions
Tagged with analytic-number-theory rt.representation-theory
27 questions
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Whether or not the root number of GL$_3\times$GL$_2$ $L$-function $L(s, F \otimes g)$ contains the coefficients $\lambda_g(n)$ of $g$?
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$Let $p$ and $q$ be two distinct primes. Let $$\Gamma_0(p)= \left\{ g\in \GL_3(\mathbb{Z}):g \equiv \left(\begin{matrix} \ast &\ast&\...
- 41
3
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1
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228
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On the local factor of Rankin-Selberg L-functions
I have a puzzle on the local factors of Rankin-Selberg $L$-functions. Consider two newforms on $\text{GL}_2$. Let $f$ be a newform of square-free level $N$, and $g$ a newform of trivial level. As ...
- 353
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70
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What is the form of the incomplete Eisenstein series on PGL_2(C)?
Let $F$ be an imaginary quadratic number field. Let $G = \mathrm{PGL}_2(\mathbb{C}) $ and $\varGamma = \mathrm{PSL}_2(\mathcal{O}_F)$. We have the Iwasawa decomposition $G = NAK$ where $K = \mathrm{...
- 185
4
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1
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299
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Non-vanishing of archimedean integral representations
Let $\psi$ denote a non-trivial additive character of $\mathbb{R}$ and $n$ be a positive integer. Let $(\pi,V)$ and $(\pi',V')$ be two irreducible generic Casselman-Wallach representations of $G_n=\...
- 145
1
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186
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Consult a question about subconvexity bounds for symmetric-square L-functions in an Arxiv-eprint due to P. D. Nelson
Sorry to disturb, the experts here. Recently, I read a paper of Nelson ("Subconvex equidistribution of cusp forms: reduction to Eisenstein observables--"https://arxiv.org/pdf/1702.02908.pdf)....
- 563
4
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1
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291
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The Langlands parameters of the symmetric cube lifts of cusp forms
I have a question concerning the Langlands parameters of the symmetric cube lifts. Lets $f$ be a $GL_2$-cusp form, and $\operatorname{sym}^2f$ the symmetric-square lift of it. Assume that $\alpha_1,\...
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1
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Global irreducible admissible representations analogue
Let ${A}_{\mathbb{Q}}$ denote the adele over the rational numbers. Then it is known that cuspidal modular forms of level $N$ correspond to some unitary automorphic representation of $\operatorname{GL}...
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109
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Whether or not the Maass form for $\Gamma _0(N)$ on $GL(3)$ covers the classical symmetric lift of a newform on $GL(2)$?
I have a blur which needs some help from the experts here, and may look naive for some experts. Recently I read Zhou's paper "The Voronoi formula on $GL{(3)}$ with ramification" (https://...
- 563
10
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1
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440
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A question on the period integral of Rankin-Selberg $L$-function
$\DeclareMathOperator\GL{GL}$Let $\Pi$ and $\pi$ be irreducible automorphic representations of $\GL_{n+1}(\mathbb{A}_F)$ and $\GL_n(\mathbb{A}_F)$ respectively, where $n \geq 2$, $F$ is a number field ...
- 421
4
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135
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Values at 1 of symmetric power L-functions of Maass cusp forms
I have a blur that whether one has $L(1,\text{sym}^2f)\ll \log^A q$ for some $A>0$? Here $f$ is assumed to be a Maass cusp form of square-free level $q$. If any experts here know something about ...
- 563
4
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204
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$\DeclareMathOperator\sym{sym}$Does $L(s, \sym^2 f \times \sym^2 g)$ have a pole at $s=1$?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\sym{sym}$I encountered a question on the poles of $\GL_3\times \GL_3$ $L$-function, which needs the knowledge of the experts ...
- 563
1
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1
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323
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How to relate Rankin triple L-function to its Dirichlet series
I have a very tricky question which may look naive to many experts here.
Let $f$ be a newform of level prime $P$, and $g,h$ two newforms of level 1, respectively. These three forms $f,g,h$ are all of ...
- 563
4
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1
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336
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Infinite, finitely generated linear group has indices of subgroups divisible by infinitely many primes
This question was previously asked at Math.SE, but didn't receive much attention.
Let $G$ be an infinite finitely generated group and suppose that $G$ is linear, say $G \leq \operatorname{GL}_n(K)$ ...
- 2,821
8
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2
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744
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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
6
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1
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361
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Decay of matrix coefficients of non-tempered representation
A theorem of Cowling--Haagerup--Howe gives an effective decay rate of the matrix coefficients of a tempered representation $\pi$ of a semi-simple algebraic $G$ in terms of Harish-Chandra $\Xi$ ...
- 1,692
4
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0
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215
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L-functions of tempered automorphic representations
Let $G$ be a reductive group over the adele ring $\mathbb A_F$ of a number field $F$. Let also $r$ be a complex finite dimensional representation of the $L$-group $r:{^LG}\to GL(V)$.
It is generally ...
- 3,799
3
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0
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128
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Representation of a matrix ring
Years ago, I read a paper about how to re-write any $n\times n$ matrix $X$ over the ring $\mathbb Z_N$ (where $N=pq$ for two primes $p$ and $q$) as a product of sequences of two simpler matrices $S$ ...
- 475
3
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1
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213
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What role do $(\mathfrak{g},K)$- modules play in the construction of automorphic vector bundles
Looking at the connection between modular forms as sections and automorphic representations it is to me somewhat clear why automorphic representations are (demanded to be) admissible $G(\mathbb{A}^\...
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1
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274
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Generalisation of the Liouville function as irreducible representations for the semigroup ($\mathbb{N},\cdot)$?
This is a duplicate of a question I have asked at here at math stack exchange, but I thought it could be also here of interest.
When looking at the [Liouville function] (https://en.wikipedia.org/wiki/...
2
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1
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169
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Bound of higher rank spherical Whittaker function
I am not much familiar with the literature about upper bound of the spherical Whittaker function on higher rank real groups. Any reference or answer to the following question would be appreciated.
Let ...
- 1,692
3
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1
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181
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Reference for a formula of Kloosterman sum (in connection with Jacobi symbol) and its generalizations
The following is from wikipedia:
The lifting formulas below, however, are often as good as an explicit evaluation. If $gcd(a,p) = 1$ one also has the important transformation:
$$S(a,a;p) = \sum_{m=0}^...
- 253
6
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1
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678
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Root number of the Rankin-Selberg convolution of two newforms
Let $p$ and $q$ be two distinct primes. Let $f\in \mathcal{S}_k^{\ast}(pq,\psi)$ be a holomorphic newform of level $pq$, nebentypus $\psi$, and weight $k$, where $\psi = \chi_p \chi_{0(q)}$, with $\...
- 63
12
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1
answer
2k
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Artin conjecture on L-functions
Artin conjecture on Artin $L$-functions asserts that the Artin $L$-function $L(\rho,s)$ of a non-trivial irreducible representation $\rho$ of the Galois group $\Gamma$ of a number field admits ...
user95222
3
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0
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272
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Spectral decomposition on GL(n)
If $\Delta_1, \ldots, \Delta_{n-1}$ are a basis of the ring of commuting bi-$SL(n,R)$-invariant differential operators, $L_0^2=L_0^2(SL(n,Z)\backslash SL(n,R))$ is the space of cuspidal automorphic ...
- 151
8
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1
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1k
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Multiplicity one theorem
I am reading Dorian Goldfeld's book Automorphic forms and L functions for the groups GL(n,R) (http://www.cambridge.org/us/academic/subjects/mathematics/number-theory/automorphic-forms-and-l-functions-...
17
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2
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1k
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Which L-functions are not "Langlands-Shahidi L-functions"?
The Langlands-Shahidi method, among other things, obtains certain L-functions from the constant term of Eisenstein series attached to so-called $(G,M)$ pairs, where $G$ is a reductive group, $M$ a ...
- 3,799
2
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1
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673
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Different cuspidal automorphic representations with same representations at infinity
Let us fix a representation $\pi_\infty$ of GL(n,$\mathbb R$).
Let us fix a character $\chi$ of K, where K is a compact subgroup of $GL(n,\mathbb A_{finite})$.
$$K=\Pi_{v<\infty}K_v$$
$K_v$ is $...
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