All Questions
34 questions
8
votes
1
answer
356
views
Average bounds on Rankin-Selberg coefficients for modular forms
Let $f$ be a cuspidal Hecke newform of weight $k$ and level $N$, and denote by $a_f(n)$ its $n$-th Fourier coefficient. The newform $f$ is normalized so that $a_f(1) = 1$. As a consequence of Rankin-...
- 5,424
1
vote
0
answers
68
views
Connection between a special integral transform and averages of L-functions
Let $\Gamma = \operatorname{SL}_2(\mathbb{Z})$ and $\mathcal{H}$ be the upper half-plane. For $A>1$, define the truncated Eisenstein series $E_A(z,s)$ as $$E_A(z,s) = \begin{cases} E(z,s), & \...
6
votes
1
answer
642
views
Generalizations of Hamburger's Theorem
(Despite the name, the theorem in question is not a joke nor is it a statement about a delicious food).
An old theorem of Hans Hamburger from 1921, as stated in Marvin Knopp's paper "On Dirichlet ...
- 26.9k
3
votes
1
answer
178
views
The lower bound for the automorphic $L$-function $L(s,\pi)$ at the edge of the critical strip $\Re s=1$
Let $\pi$ be any automorphic Maass form on $\text{GL}_m$ of level $N$, say. Assume that the associated $L$-function $L(s,\pi)$ satisfies some good conditions; for example, it satisfies the functional ...
- 81
5
votes
1
answer
162
views
A question on hybrid subconvexity for individual L-functions
Sorry to disturb. I have a question need some explanations from the experts on the MO-website.
As usual, we let $L(f,s)$ be the corresponding $L$-function associated to the newform $f$ on $SL_2(\...
- 81
2
votes
0
answers
141
views
Analyticity of unramifed part of Rankin-Selberg $L$-functions on $\Re(s)=1$
I have only a little knowledge about automorphic representations and $L$-functions. Now I am reading the textbook of Goldfeld and Hundley on automorphic representations, and also planning to read the ...
- 663
2
votes
1
answer
264
views
'$\times$' or '$\otimes$' when writing $L$-functions?
Recently, I came across the Langlands correspondence theorem, there is the following line:
$$L(s,\pi(\sigma) \times \pi(\tau)) = L(s,\sigma \otimes \tau), $$
where $\sigma$ and $\tau$ are ...
- 185
1
vote
1
answer
329
views
Behaviour of a certain $L$ function at $s=1$
I was going through this paper. Corollary 7.3.4 says the $L$-function $L(s,\pi, \rm{sym}^4)$ is holomorphic except possibly at $s=0,1$ and gives a necessary and sufficient condition for it to have a ...
- 424
4
votes
1
answer
318
views
Watson's triple product for automorphic forms shifted by Maass rising operators
Let $\phi_i$ be a holomorphic Hecke eigencusp form of weight $k_i$ for $\Gamma = \mathrm{SL}_2(\mathbb{Z})$, or a Maass cusp form (we then say that $k_i=0$). We assume they are normalised such that $\...
- 153
1
vote
1
answer
249
views
What's the motivation for the $3$ appearing in Iwaniec and Kowalski's definition of the analytic conductor?
In their book Analytic Number Theory, Iwaniec and Kowalski, on page 95, define the analytic conductor by the following formula:
$\displaystyle{{\frak{q}}_{\infty}(s)=\prod_{j=1}^{d}\left(\vert s+\...
- 6,974
14
votes
1
answer
532
views
Bound for $GL(3)$ symmetric square
Let $\pi$ be an automorphic representation of $GL(3)$ over a number field. Let $a_n$ be the coefficients of $L(s, \pi, \mathrm{sym}^2)$. Do we know if
$$\sum_{n>0} \frac{|a_n|}{n^s}$$
and
$$\sum_{n&...
- 5,424
3
votes
0
answers
216
views
Maass--Selberg for any Eisenstein series on higher rank
Does there exist a Maass--Selberg relation for any Langlands Eisensein series on $\mathrm{GL}(n)$? By any I mean an Eisenstein series which is induced from any standard parabolic with any discrete ...
- 1,692
3
votes
1
answer
1k
views
Are there infinitely many L-rigs?
$\DeclareMathOperator{\Q}{\mathbb{Q}}$Call "L-rig" any class $\mathcal{L}$ of L-functions of automorphic representations of $\operatorname{GL}_{n}(\mathbb{A}_{\Q})$ for some $n$ belonging to ...
- 6,974
8
votes
1
answer
595
views
Does the symmetric square L-function vanish at one?
Take a cuspidal automorphic representation $\pi$ of $GL(3)$ over a number field. My question is quite straightforward and can be related to this one :
Can $L(1, \pi, \mathrm{sym}^2)$ be zero? If ...
- 927
5
votes
0
answers
111
views
Archimedean L-factors for symplectic group
Let $\pi$ be an automorphic representation of $GSp(4)$. Provided a representation $r$ of the Langlands dual group of $GSp(4)$ (namely, the standard or the spinor one), it is possible to define a ...
- 927
5
votes
2
answers
433
views
Asymptotic's for Fourier coefficients of $GL(3)$ Maass forms
Let $f$ be a $GL(3)$ Hecke-Maass cusp form and $A(m,n)$ denote its Fourier coefficients.
Are there any lower bounds known for $\sum_{p\leq x}|A(1,p)|^2$ or $\sum_{n\leq x}|A(1,n)|^2$ ? (we know the ...
- 153
7
votes
1
answer
564
views
Functional equation for general number fields
When it comes to general number fields beyond $\mathbb{Q}$, the litterature is not so abundant in analytic number theory. For instance over $\mathbb{Q}$, for primitve Dirichlet characters modulo $q$, ...
- 927
10
votes
1
answer
1k
views
How does Riemann hypothesis implies estimates?
In Iwaniec, Luo and Sarnak article (precisely (4.23)), it is said that GRH for $L(s, \mathrm{sym}^2(f))$, for a holomorphic cusp newform $f$ of level $N$ and weight $k$, implies
$$\sum_{p \nmid N} \...
- 551
7
votes
2
answers
478
views
Rankin-Selberg integral for GL(3) form with Odd Maass form on GL(2)
Let $F$ be a Hecke-Maass cusp form for $SL_3(\mathbb Z)$.
Let $u$ be a Hecke-Maass cusp form for $SL_2(\mathbb Z)$.
The following integral
$$\mathcal L(s,F\times u)=\int_{{SL}(2,\mathbb{Z})\...
- 3,804
8
votes
2
answers
839
views
On the consistency of the definition of the conductor for automorphic forms
Let $\pi$ be an irreducible admissible representation of $\mathrm{GL}_2(F)$, where $F$ is local non-archimedean. The local conductor associated to $\pi$ can be defined in two usual manners:
By its ...
- 5,424
3
votes
2
answers
337
views
Poles of Rankin-Selberg $L(s,\pi\times\tilde \pi)$?
Let $F$ be a number field and let $\pi$ be a cuspidal automorphic representation of $GL_n(\mathbb{A_F})$.
Do we know that $L(s,\pi\times \tilde\pi)$ has a simple pole at $s=1$?
Do we know that $\...
- 3,804
4
votes
1
answer
391
views
Non-vanishing of L-function of modular form
There is a theorem by Langlands and Shalika (link) that the L-function of a cuspidal automorphic representation does not vanish on the line $\mathrm{Re}( s)=1$ (in their normalization which might be ...
- 7,377
13
votes
0
answers
622
views
No Siegel-Landau zeros for $\mathrm{GL}(n)$
The problem of non-existance of Siegel-Landau zeros seems to be uncharacteristically easier for cuspidal automorphic representations $\pi$ on $\mathrm{GL}(n)$ if $n\geq2$. We have in fact:
There ...
- 17.6k
3
votes
1
answer
259
views
Question about mean square estimate for sums of Dirichlet coefficients of Symmetric Power $L$-functions
I have a question related to Coefficients of Symmetric power $L$-functions and I would be grateful if you could answer it.
Let $\lambda_{Sym^rf}(n)$ be the $n$th Dirichlet coefficient of $L(Sym^rf,s).$...
- 1,257
3
votes
0
answers
163
views
Oscillatory integral moments of $L(\frac{1}{2} + it, f \times f)$
Understanding moments and subconvexity bounds for $L$-functions is a big topic with a lot of activity. I'm currently looking at a related problem, bounding
$$
\int_0^T L\left(\tfrac{1}{2} + it, f \...
- 1,570
9
votes
1
answer
830
views
Lindelof Hypothesis implying Selberg Eigenvalue Conjecture?
The Generalized Lindelof Hypothesis says that for the $L$-function of an automorphic form we have
$$L(1/2+it)\ll Q(t)^{\epsilon}$$
for any $\epsilon>0$ where $Q(t)$ is the conductor of $L(s)$ at $...
- 3,804
3
votes
1
answer
204
views
Lower bound of first moment of $L$-function on $\mathrm{GL}(3)$
Let $\pi$ be an automorphic form on $\mathrm{GL}(3,\mathbb{A}_{\mathbb{Q}})$.
Do we know any case that
\[\int_0^{T} \left|L(\frac{1}{2} + it, \pi)\right| dt \gg T\]
holds unconditionally?
I know the ...
- 3,804
9
votes
0
answers
399
views
Symmetric Fifth Power Lift of GL(2) Automorphic Form
Let $\pi$ be an automorphic representation of $GL(2)/\mathbb{Q}$. For simplicity, you can take it to be a Maass form for $SL(2,\mathbb Z)$. Kim, Shahidi, Gelbart-Jacquet prove that
$$L(s, \pi, Sym^m)$...
- 3,804
14
votes
1
answer
1k
views
Is the adjoint L-function on GL(m) holomorphic?
Let $\pi$ be an automorphic representation on $\mathrm{GL}(m)/\mathbb{Q}$.
Define $$L(s,\pi,\mathrm{Ad}):=\frac{L(s,\pi\times\overline{\pi})}{\zeta(s)}.$$ This is an $L$-function with Euler product of ...
- 253
8
votes
1
answer
643
views
Absolute convergence of Rankin–Selberg series
Let $\pi$ and $\pi'$ be two general automorphic representations on $\operatorname{GL}(n)$ and $\operatorname{GL}(n')$ over $\mathbb{Q}$.
I heard that the Rankin-Selberg $L$-function $L(s,\pi\times\pi')...
- 253
5
votes
1
answer
842
views
Generalization of Watson's triple product
In Watson's thesis (page 51) we can find his beautiful triple product formula. My question is that does there exist a generalization of this formula? By generalization, I mean:
If $\phi_n$'s are ...
- 1,692
6
votes
1
answer
1k
views
subconvexity problem for $GL(3) × GL(2)$ $L$-function without involving in symmetric lift
A question in study of subconvexity topic puzzles me for a long time, which mabe a stupid question for many experts. I really wish someone to help me out, and any advice will be highly appreciated.
...
- 177
5
votes
0
answers
394
views
a generalization of a formula of Shimura
Let $\phi$ be a $GL(2)$ automorphic form with Fourier coefficients $a(n)$ and $a(1)=1$.
Obviously we have $L(s,\phi)=\sum \frac{a(n)}{n^s}$.
Shimura have the following formula
$L(s, Ad\; \phi)=\zeta(...
- 585
1
vote
1
answer
324
views
Off critical line zeros for half integer weight $L$-functions
Let $f(z) = \sum_{n=1}^\infty A(n)n^{\frac{k-1}{2}}e(nz)$ be a modular form of weight $k$ for a half integer $k$. Put
$$L(s,f) = \sum_{n=1}^\infty \frac{A(n)}{n^s} $$
to be the $L$-function.
Further ...
