Questions tagged [l-functions]
Questions about generalizations of the Riemann Zeta function of arithmetic interest whose definition relies on meromorphic continuation of special kinds of Dirichlet series, such as Dirichlet L-functions, Artin L-functions, elements of the Selberg class, automorphic L-functions, Shimizu L-functions, p-adic L-functions, etc.
411
questions
3
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0
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208
views
What are the unsolved problems in Formal groups and $L$-functions?
In the 1st page of the introduction of Hazewinkel's Formal Groups and Applications book, there are two ways of constructing formal groups (law):
$\bullet$ Given a Lie group $G$, one can define a ...
- 806
0
votes
0
answers
68
views
Relating the multiplicative Fourier transform and the derived characteristic polynomial
(Tuesday, Sept 5:) For a number field $Fˣ$ and a number ring $Oˣ$ it is common to define:
$Z(f,χ) = ʃ_{Fˣ} f(x) χ(x) dˣ x$
$g(ω,ψ) = ʃ_{Oˣ} ω(x) ψ(x) dˣ x$
where $dˣx$ is the multiplicative Haar ...
- 23
2
votes
0
answers
208
views
Zero dimensional varieties and the L-function $1/(1-p^{-n})$
I am interested in positive characteristic varieties which produce an L-function of the form $\frac{1}{1-χ} = \frac{1}{1-p^{-s}} = \sum_{n = 0}^\infty p^{-ns}$. It seems related to the positive ...
- 23
2
votes
1
answer
175
views
Question on automorphic $L$-functions
Let $\pi$ be an automorphic representation of $\textrm{GL}_n$. Associated to $\pi$, we can define the standard $L$-function $L(s, \pi)$.
My question is: what is the difference between $L(s, \pi)$ and ...
- 133
3
votes
1
answer
178
views
Order of vanishing of $L$-function and mixed Hodge-structures
Let $X$ be a smooth and proper scheme over $\mathbb{Q}$ and choose integers $n,i$ such that $n>\frac{i}{2}+1$. Then we have
$$ ord_{s=i+1-n}L(H^i(X),s)=\dim H^{i+1}_{\mathcal{D}}(X_\mathbb{R},\...
- 4,106
5
votes
1
answer
456
views
On the notion of cuspidality
Let $k/\mathbb{Q}$ be a number field and $\mathbb{A}$ its ring of adèles. As usual $\mathbb{A} = \mathbb{A_f} \times \mathbb{A_{\infty}}$.
The standard definition of an automorphic representation $(\...
- 678
1
vote
1
answer
127
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The theta function of an odd Dirichlet character
The theta function $\theta_\chi(t)$ of a Dirichlet character $\chi$ is defined to be $\theta_\chi(t) = \frac{1}{2} \sum\limits_{n=-\infty}^\infty \chi(n) e^{2\pi i n^2 t}$ if $\chi(-1) = 1$ (i.e., $\...
- 536
4
votes
0
answers
152
views
Several L-functions but one Galois representation: How to choose
Let $\mathbf{G}$ be a reductive group which enjoys all the nice properties a reducive group can dream of. Fix $(\mathbf{G},X)$ a Shimura datum associated with it and assume that if $K\leq\mathbf{G} $ ...
- 935
12
votes
1
answer
480
views
$p$-adic L function of an odd Dirichlet character
Apologies for a naive question (especially for Iwasawa theorists): it is well-known
and trivial to prove that the usual (elementary) construction of $p$-adic L functions
attached to odd Dirichlet ...
- 10.7k
3
votes
0
answers
166
views
Correspondence between motives and automorphic representations
What I know:
I understand motives via its realization; in Coates' and Perrin-Riou's paper On $p$-adic L-functions Attached to Motives over $\mathbb{Q}$ (see http://doi.org/10.2969/aspm/01710023), the ...
- 678
1
vote
0
answers
119
views
Zeroes of certain $L$-functions on the critical line and GGP conjectures
Global Gan-Gross-Prasad conjecture (on various groups) says that nonvanishing of certain automorphic $L$-function $L(s, \pi)$ (of cuspidal representation $\pi$ of some reductive group $G$) at $s = 1/2$...
- 1,683
4
votes
2
answers
452
views
Explicit formula for Artin L-functions
The classical explicit formula for the Riemann Zeta function states that
$$
\psi(x)=x-\sum_{\rho} \frac{x^{\rho}}{\rho}+O(1),
$$
where $\psi(x)=\sum_{n \leq x} \Lambda(n)$ and the sum is over all non-...
- 53
6
votes
1
answer
145
views
What is the conductor of an automorphic representation for $\Gamma_0(q)$ in $GSp(4)$?
Let $\pi$ be a generic cuspidal automorphic representation on $GSp(4)$, with level $\Gamma_0(q)$ (the group of symplectic matrices with lower left block divisible by $q$), i.e.
$$\Gamma_0(q) = \left\{ ...
- 5,324
1
vote
1
answer
252
views
What are the best known upper bounds for $\frac{1}{L(s, \chi)}$?
Let $\chi$ be a Dirichlet character and $L(s, \chi)$ be the corresponding Dirichlet L-function. What are the best known bounds for $\frac{1}{L(s, \chi)}$ in the half-plane of convergence?
I'm aware of ...
- 93
3
votes
0
answers
141
views
Analytic continuation of $L$-functions of base changed elliptic curves
Suppose that $E$ is an elliptic curve over $\mathbf{Q}$. Let $K$ be a number field and let $L(E/K, s)$ be the Hasse-Weil $L$-function of $E$ base-changed to $K$. The modularity theorem tells us that $...
- 1,799
2
votes
1
answer
264
views
$p$-adic analogue of modular forms, upper half-plane, and $L$-functions
In the classical picture, there is the (complex) modular form, defined on the (complex) upper half plane, which is related to the (complex) $L$-function via the Mellin transform. As I have recently ...
- 91
1
vote
1
answer
202
views
$p$-adic $L$-functions and congruence of $L$-values
I am reading about $p$-adic $L$-functions and I have one question in mind.
To start with, I will write a proof I've learned of a congruence of $L$-values:
Theorem: Let $p\geq5$ be a prime, $\alpha\...
- 271
5
votes
1
answer
218
views
Waldspurger's formula and toric periods — classical and adelic versions
As far as I know, there are two versions of Waldspurger's formula (classical and adelic), which can be vaguely stated as follows
(Classical version) Let $f$ be a half-integral weight modular form of ...
- 1,683
2
votes
1
answer
132
views
Basic results concerning the intertwining operator in the $\mathrm{SL}_2$ case
I am reading [Ikeda, Tamotsu, On the location of poles of the triple L-functions]. On page 194, the author recalled some known results concerning $\operatorname{SL}_2$. I would like to know any ...
- 111
2
votes
0
answers
137
views
Meaning of the meromorphic continuation of intertwining operators
I am trying to make sure the meaning of the meromorphic continuation of the intertwining operators.
Assume we deal with a non-Archimedean field $F$ and just consider $G= SL_2$, for simplicity. We fix ...
- 111
3
votes
1
answer
197
views
Positivity of partial Dirichlet series for a quadratic character?
Let $\chi\colon(\mathbb{Z}/N\mathbb{Z})^\times\rightarrow\{\pm1\}$ be a primitive quadratic Dirichlet character of conductor $N$. For any integer $m=1,2,\cdots,\infty$, consider the partial Dirichlet ...
- 71
6
votes
1
answer
464
views
Langlands-Shahidi method in classical language
The Langlands-Shahidi method says that the $L$-functions of automorphic representations appear in the constant terms of Eisenstein series. Since those Eisenstein series have analytic continuation and ...
- 1,799
7
votes
0
answers
297
views
Which automorphic L-functions have an integral representation?
Is there a list of which automorphic L-functions are known to have an integral representation?
- 71
11
votes
2
answers
1k
views
What is the Perrin-Riou logarithm (or regulator)?
Recently I've been rewatching some recordings of old talks on L-functions and explicit reciprocity laws (in particular, the series of talks by Loeffler and Zerbes given at this workshop at the CRM in ...
- 3,209
4
votes
0
answers
144
views
What are the modularity conjectures for Artin motives?
Classically, singular cohomology is an important tool for studying topological spaces, in particular, complex varieties. In the mid-twentieth century it was realized that there are many analogues of ...
- 447
2
votes
0
answers
74
views
Second moment of $S(T)$ for Dirichlet L-functions
Let $S(T)$ denote the argument of the Riemann zeta function. Selberg established that $$\int_0^T |S(t)|^2 \text{d}t\sim\frac{T}{2\pi^2}\log \log T.$$ Let now $\chi$ be a Dirichlet character modulo $q$,...
- 121
10
votes
1
answer
474
views
Why $p$-adic measures?
I'm currently learning about the Kubota–Leopoldt $p$-adic $L$-function and I'm noticing that many people view the Kubota–Leopoldt $p$-adic $L$-function as a measure as opposed to a $p$-adic analytic ...
- 1,799
2
votes
2
answers
201
views
Special values of non-cm $L$-functions
For the sake of simplicity, assume $f$ is a non-cm eigenform of weight $k$ on the group $\mathrm{SL}(2, \mathbb{Z})$. Are there any known results or conjectures regarding any special values of the ...
- 485
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votes
4
answers
3k
views
Why do we care about the eigenvalues of the Frobenius map?
The Riemann hypothesis for finite fields can be stated as follows: take a smooth projective variety X of finite type over the finite field $\mathbb{F}_q$ for some $q=p^n$. Then the eigenvalues $\...
- 1,748
4
votes
0
answers
439
views
Question about a paper by Franca and LeClair in analytic number theory
I am reading an article "Transcendental equations satisfied by the individual
zeros of Riemann $\zeta$, Dirichlet and modular
L-functions" by G. Franca and A. LeClair (2015) see here. The ...
- 41
3
votes
0
answers
151
views
Cohomology and p-adic L-functions
The definitions of $p$-adic $L$-functions I know all are given by interpolating the values of "usual" $L$-functions at the negative integers. If I want to define the $p$-adic $L$-function of ...
- 4,106
2
votes
2
answers
286
views
Reference for zero sum estimates of Dirichlet L functions
Let $\chi$ be a primitive character mod $p$ (prime) and $\rho = \beta + i \gamma$ be a non-trivial zero of $L(s, \chi)$.
I am reading a paper by Ihara and Murty where they use following estimate :
$\...
- 31
0
votes
1
answer
170
views
Spacings of Satake parameters under Ramanujan conjecture
I would like to know if, under Ramanujan conjecture, the following three distributions are known or conjectured to match:
the distribution of spacings between Satake parameters of an L-function $F$ ...
- 6,894
5
votes
2
answers
273
views
Additivity of Elliptic Curve Rank over Compositum of Fields
Assume that BSD holds for number fields. Let $E/\mathbf{Q}$ be an elliptic curve. For simplicity, let's assume it has Mordell-Weil rank zero. Let $F_1/\mathbf{Q}$ and $F_2/\mathbf{Q}$ be finite, ...
- 1,362
3
votes
1
answer
669
views
Does asymptotic Goldbach imply GRH?
It seems to me that a proof of $\alpha_{n}=o(n)$ where the quantity $\alpha_{n}$ is defined in About Goldbach's conjecture together with the main result of https://kyushu-u.pure.elsevier.com/en/...
- 6,894
-2
votes
1
answer
210
views
Special value of Hecke $L$ function
Let $E:y^2＝x^3-x/ \Bbb{Q}(i)$ be elliptic curve and $L(E,1)$ be a special value of $L$ function of $E$ at $1$.
Let $L(ψ,1)$ be value at $1$ of Hecke $L$ function with respect to Hecke character $ψ$, ...
- 1,119
3
votes
0
answers
207
views
Proof of $L(E,1)/Ω(E)=1/8$ for elliptic curve $E:y^2=x^3-x/ \Bbb{Q}$?
Let
$E:y^2=x^3-x$ be an elliptic curve over $ \Bbb{Q}$ and
$ω_E=dx/2y=dx/2\sqrt{x^3-x}$.
Then
$$
\begin{split}
\Omega(E)&=\int_{E(\Bbb{R})} ω_E\\
\\
&=2\int\limits_1^{+\infty} dx/\sqrt{x^3-x}...
- 1,119
4
votes
0
answers
408
views
Ramanujan's conjecture on modular forms and Riemann hypothesis
I just watched Kannan Soundararajan's talk on the distributions of valus of zeta and $L$-functions at virtual ICM 2022. In his talk, he introduced a theorem on Ramanujan's ternary form $\phi_{1}: x^{2}...
- 1,683
4
votes
1
answer
268
views
Selberg class definition and Riemann hypothesis
Looking at the Selberg class definition on Wikipedia, under "Comment on definition", there is this paragraph:
"The condition that the real part of $\mu_i$ be non-negative is because ...
- 1,101
4
votes
1
answer
348
views
Calculating the explicit constant – Siegel zeros and class numbers
Let $\chi$ denote the Legendre symbol of conductor $q$. A Siegel zero for the $ L $ series associated to $ \chi $, which we denote by $ L(s,\chi) $ is a real zero $ \sigma $ satisfying $ 1-\frac{c}{\...
- 507
2
votes
0
answers
166
views
Extending the analogy between cyclotomic units and elliptic units
There is a nice analogy between cyclotomic units and elliptic units given as follows:
Cyclotomic units are related to special values of the Riemann Zeta function. This is because the logarithmic ...
- 1,799
4
votes
0
answers
211
views
The link between Satake parameter and Godement-Jacquet L-function of an automorphic representation of $GL_{n}$
Origin of the question: I'm reading the following survey of K. Martin, more generally I'm looking for the "best way" to define L-function associated to an automorphic representation of a ...
- 935
9
votes
1
answer
312
views
Is $\frac{1}{L(1+it)}$ unbounded?
Let $\chi$ be a Dirichlet character and $L(s, \chi)$ be the corresponding L-function. Is $$\frac{1}{L(1+it, \chi)}$$ unbounded for $t \in \mathbb{R}$? I'm aware that this is true if $L=\zeta$, but I'm ...
2
votes
1
answer
124
views
Do Artin L functions have polynomial growth in in the critical strip?
Given an irreducible representation $\rho$ of the Galois group $G$ of a number field $K$ over $\mathbb{Q}$, we have the associated Artin $L$ function which we denote by $L(s, \rho)$. By Brauer ...
- 507
1
vote
0
answers
72
views
Explanation about Lapid-Rallis iductive argument (doubling method)
I am reading Lapid-Rallis "On the local factors of representations of classical groups" and I am completely stuck with the proof of Proposition 3.
In the case $\mathcal V$ is not anisotropic,...
- 11
2
votes
0
answers
166
views
Symmetric square L-function with non square-free level
Let $f$ be a primitive holomorphic cusp form of weight $k$, level $N$ and nebentypus $\chi$, with its $L$-function $L(s,f)=\displaystyle\sum_{n\geq1}\lambda_f(n)n^{-s}$ for $\mathrm{Re}(s)>1$. Let $...
- 21
1
vote
0
answers
160
views
Large values of $L(1,\chi)$ for quadratic Dirichlet characters $\chi$
Granville and Soundararajan, in "Upper Bounds for $L(1, \chi)$", first paragraph,
say it is known that there exist quadratic Dirichlet characters $\chi$ for which $L(1, \chi)$ is about $\log\...
- 1,213
3
votes
0
answers
229
views
Have there been recent developments of Booker's approach to L-functions as distributions?
Andrew Booker introduced a framework to study L-functions through distributions in https://arxiv.org/abs/1308.3067v2. This allowed him and others to get new results about zeros of automorphic L-...
- 6,894
2
votes
2
answers
223
views
Sign of the special value at s=0 of Hecke L-functions
Let $L/K$ be an abelian extension of number fields with Galois group $G$ and let $\chi : G \to \{\pm 1\}$ denote a real linear character of $G$. Denote $L(\chi,s)$ the Artin L-function associated to $\...
- 607
3
votes
0
answers
81
views
Hoffstein–Lockhart for non-congruence subgroups
Let $\Gamma$ be a non-congruence subgroup of $\operatorname{SL}(2,\mathbb{Z})$ of finite index and let $f$ be a holomorphic cuspidal modular form of weight $k$ for the group $\Gamma$. For simplicity, ...
- 507