Questions tagged [l-functions]
L-functions are meromorphic functions on $\mathbb C$ that are used extensively in number theory.
117
questions
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21 views
Estimation of certain Euler product using a symmetric square $L$-function
Let
$$
L(s)=\sum_{n=1}^{\infty}a(n)n^{-s}
$$
be a modular $L$-function of conductor $N$, and let
$$
F_d(s)=\sum_{n=1}^{\infty} \frac{a(d_0 n^2)}{(d_0 n^2)^s} \prod_{p|4Nnd}\left(1+\frac{1}{p}\right)^...
1
vote
0answers
26 views
Absolute value of roots of bad Euler factors.
Suppose $V$ is a smooth projective variety over number field $\mathbb{Q}$. For every prime integer $p$ and positive integer $j$, let
\begin{equation}
Z_p(H^j(V), T):=\det(1-{\rm Frob}_p\cdot T|H^j_{et}...
1
vote
0answers
68 views
Topics for Masters thesis around Birch-Swinnerton-Dyer conjecture [closed]
I am currently looking for a Masters thesis subject in number theory. My favourite subjects are algebraic number theory and cohomologies (I only studied De Rham cohomology).
I've been lately reading ...
-1
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1answer
36 views
Is the reciprocal of a Satake parameter a Satake parameter?
Let $F$ be an automorphic L-function for $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$. Is it known whether for all but finitely many primes $p$ the set of Satake parameters of $F$ at $p$ is ...
2
votes
0answers
24 views
Do you know about the textbook of Selberg class of Dirichlet series?
I have read the Atle Selberg's thesis named "Old and new conjectures and results about a class of Dirichlet series". At the end of this thesis, he wrote
"A more complete account with proofs is ...
0
votes
0answers
24 views
Probability for an L-function to be RS-primitive
Assuming an L-function is any element of the intersection $\mathcal{L}$ of the Selberg class $\mathcal{S}$ and the class of automorphic L-functions $\mathcal{A}$, define the notion of Galois class of ...
0
votes
1answer
66 views
Minimum and maximum of a partial Euler product?
Question: If if $n\in\mathbb{N}$ and $s\in \mathbb{C},$ say $s=\sigma+t\sqrt{-1},$ then Dirichlet Beta function is defined to be
$$
\beta(s)=\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^s};
$$
which for ...
1
vote
0answers
39 views
$p$- adic $L$- Function of a Dirichlet character at positive integers.
Let $p$ be a prime number and $d$ a natural number with $p\nmid d$. Furthermore let $\chi$ be a Dirichlet character with conductor $d$ and $G=Gal(\mathbb{Q}(\mu_{p^{\infty}})/\mathbb{Q})$.
In the ...
0
votes
0answers
11 views
Derivative of dirchlet L function at s=1
I'm trying to derive the chowla-selberg formula and it involves the derivative of the dirchlet L function at s=1. I've tried to manipulate the series in several ways, but there's only one that seems ...
4
votes
0answers
108 views
Relationship between $\theta$ functions and number fields.
I'm trying to have a clear picture of the relationship of theta functions and $L$-functions, and the geometric objects they relate to.
Firstly, I know that $\theta$-functions arise as sections of ...
1
vote
1answer
50 views
Gamma Function Integral Identity
On p.11 of D. Bump's "Automorphic Forms and Representations" he uses the following identity in a proof of the functional equation of a Dirichlet $L$-function:
$$ \int_0^\infty e^{-\pi tn^2}t^{(s+\...
0
votes
1answer
22 views
Expression of the dirichlet coefficient of an L-series in terms of the Satake parameters
Last year, I had found a pdf where the expression of the Dirichlet coefficient $\lambda_{\pi}(p^{\nu})$ in terms of the Satake parameters $\alpha_{p,i}(\pi)$ was given. Unfortunately I don't remember ...
3
votes
1answer
61 views
Asymptotics for average of Fourier coefficients of cusp form
Iwaniec Topics in Classical Automorphic Forms, after introducing the Rankin-Selberg convolution $L$-function
$$L(f \otimes \bar{f}, s) = \sum_{n = 1}^\infty \frac{|a(n)|^s}{n^s}$$
of a weight $k$ ...
0
votes
0answers
17 views
The argument of L-functions
For Riemann Zeta function, we can define $\pi S(t)=\arg\zeta(\frac{1}{2}+it)$ where the argument is the variation from $+\infty+it$ to $\frac{1}{2}+it$. I read few results related to $S(t)$ such as $S(...
3
votes
1answer
51 views
Proof of Chebotarev Density Theorem without Artin Reciprocity
I'm wondering if there is a proof of the Chebotarev density theorem that does not require the use of any big results in class field theory, such as Artin Reciprocity.
As I understand it, the main ...
1
vote
0answers
37 views
Do the properties defining the Selberg class imply the distribution of real parts of non trivial zeros of an L-function is strongly unimodal?
Selberg defined what is now known as the Selberg class as a class of L-functions fulfilling for essential properties, which are analyticity, Euler product, functional equation and Ramanujan-Patersson ...
0
votes
2answers
42 views
Need reference for proof of functional equation for generalized L-functions
When proving functional equation for Riemann zeta function one starts at the definition of gamma function
$$\Gamma(s) = \int_0^{\infty} x^{s-1} e^x\mathrm dx\tag1$$
After a few steps we arrive at
$$ ...
2
votes
2answers
78 views
Test vector for local zeta integral with ramified character
Suppose $\pi$ is an unramified principal series representation of ${\rm GL}_2(F)$, where $F$ is a non-archimedean local field with integers $\mathfrak{o}$. Let $W$ be a function in its Whittaker model....
3
votes
0answers
44 views
Langlands L functions for groups over finite fields.
In some reading on automorphic/Langlands-related papers I have seen some authors refer to the finite field analogues of Langlands objects, such as admissible representations, L factors but a simple ...
2
votes
0answers
35 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.
(1) 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 ...
4
votes
0answers
116 views
Determining elliptic curve analytic rank even/odd
For an elliptic curve over Q that is defined with large coefficients, it can take mathematical software (such as Sage) a long to time calculate the analytic rank. However, it seems to quickly know if ...
-1
votes
1answer
29 views
Permutation group of Satake parameters
Let $L(s)=\prod_{p}L_{p}(s)$ the Euler product of an L-function in the relevant right half-plane. As $ L_{p}(s)=\prod_{j=1}^{d}(1-\alpha_{j}(p)p^{-s} )^{-1}$, the permutation group $ G_{p}$ of the ...
1
vote
2answers
132 views
Evaluating Dirichlet $L$-functions at $s=1$
I'm trying to find references on general methods for evaluating Dirichlet $L$-functions at $s=1$, but it's proving a little harder to google than I'd hoped. Specifically I'm looking for any books or ...
4
votes
1answer
46 views
Series of reciprocal of integers
This is a question I asked myself today...
$ $
Do you know if it is possible to build a strictly-increasing sequence $(u_n)_{n\in\mathbb{N}^\star}$ of positive integers such that $\displaystyle\sum_{...
0
votes
0answers
30 views
Is $ \pi\mapsto(s\mapsto L(s,\pi)) $ bijective?
Let $ \pi $ be an automorphic representation of $ \operatorname{GL_{n}}(\mathbb{A}_{\mathbb{Q}}) $ and $ L(s,\pi) $ the associated L-function. Is the map $ \pi\mapsto L(s,\pi) $ bijective ? In ...
0
votes
0answers
14 views
self-intertwining operators of an automorphic representation
I stumbled a few days ago on the definition of an intertwining operator in https://www.encyclopediaofmath.org/index.php/Intertwining_operator
Considering the case where $ \pi_{1}=\pi_{2}=\pi$, $ ...
0
votes
1answer
23 views
Modular parametrization from equality of $L$-functions
In the literature, an elliptic curve $E/\mathbb{Q}$ is defined to be modular in two different ways
1) if there exists a nonconstant morphism $X_0(N) \to E$,
2) if there exists a modular form $f$ ...
4
votes
1answer
77 views
Is the conductor of an L-function F the absolute value of the discriminant ofsome number field related to F?
In the theory of automorphic forms, ramified primes of an L-function divide the so-called conductor thereof. On the other hand, one can define for a number field $ K $ an integral invariant $ \...
0
votes
1answer
57 views
Uniqueness of $L$-series of cusp forms
For a cusp form $f$, one gets an $L$-series by taking the Mellin transform as we have
$$ \tilde{f}(s) = (2\pi)^{-s} \Gamma(s) L(s,f). $$
My question is: is this operation injective? It seems to me ...
4
votes
1answer
100 views
Convexity Bound of Rankin-Selberg L-Function
Let $f,g$ be primitive modularforms of arbitrary levels $N_1,N_2$, trivial nebentypus and same weight $k$. Let $L(f\otimes g,s)=\zeta(2s)\sum_{n\geq1}\frac{\lambda_f(n)\lambda_g(n)}{n^s}$ be the ...
1
vote
1answer
108 views
Rankin-Selberg convolution : normalization issues
I was told several times on MO that if $ F : s\mapsto\sum_{n>0}\frac{a_{n}}{n^{s}}$ and $ G : s\mapsto\sum_{n>0}\frac{b_{n}}{n^{s}} $ for $ \Re(s)>1 $ are L-functions, then provided the ...
1
vote
0answers
53 views
The $\heartsuit$ operator on $\mathcal{L}^2(SL_2(\mathbb{Z})\backslash \mathbb{H})$
In Goldfeld's text Automorphic forms and L-functions for GL(n,R), for a fixed prime $p$ the operator $\heartsuit \colon \mathcal{L}^2(SL_2(\mathbb{Z})\backslash \mathbb{H})\to\mathcal{L}^2_{cusp}(SL_2(...
10
votes
1answer
388 views
Roadmap to Iwasawa Theory
I haven’t found any posts on this, so I figured I would ask. Beyond learning basic algebra (rings, groups, fields) and complex analysis, what must one study if they want to start learning a good ...
2
votes
0answers
86 views
Upper bound on zero multiplicity of Dirichlet $L$-functions
I was wondering whether there is (a known) upper bound of the order of the non-trivial zeros of Dirichlet $L$-functions. For a zero $s$ of the Riemann zeta function we have the estimate $C\cdot\log(\...
2
votes
0answers
96 views
Simplest nontrivial example of an L-function yielding information about a Diophantine equation
I got excited while reading Langlands' essay, REPRESENTATION THEORY: ITS RISE AND ITS ROLE IN NUMBER THEORY, because he appears to provide concrete motivation for the study of L-functions:
We have ...
2
votes
0answers
43 views
Artin $L$-functions and abelianization
Let $L/K$ be a finite Galois extension of global fields, with $G=\mathrm{Gal}(L/K)$. Let $H \leq G$ and $\chi:H \to \mathbb{C}$ a non-trivial irreducible character of $H$. Then we can define the (...
11
votes
3answers
222 views
Why does the Dedekind zeta function of a number field have a pole at $s=1$?
The analytic class number formula tells us that the Dedekind zeta function $\zeta_K$ of a number field $K$ has a pole at $s=1$ with residue $$\frac{2^{r_1}(2\pi)^{r_2}\text{Reg}_Kh_K}{w_K\sqrt{|\...
4
votes
1answer
94 views
Why does the determinant come in for Artin L-Functions?
Let $L/K$ be a Galois extension of number fields. For $\mathfrak p$ a prime of $K$, unramified in $L$, the Frobenius elements $\sigma_{\mathfrak P}$ for $\mathfrak P \mid \mathfrak p$ are conjugate, ...
1
vote
0answers
60 views
Trying to understand generic nature of L-functions
I’m not a “professional” mathematician but amateur and curious in maths - more like a hobbyist. So this is a little bit advanced topic for me and am trying to visualize/grasp understanding of how L- ...
2
votes
1answer
97 views
Analytic continuation of twisted Hecke $L$-function
Let $K$ be a (real quadratic) number field and $\chi$ be a Hecke character on it. We can think about Hecke $L$-function
$$
L(\chi, s):= \sum_{0\neq \mathfrak{a}\subseteq \mathcal{O}_{K}} \frac{\chi(\...
3
votes
0answers
101 views
Understanding a result of Serre about zeros of $x^3 - x - 1$ in $\mathbb{F}_p$
I'm trying to understand a result of Serre which relates the number of zeros in the finite field $\mathbb{F}_p$ of $f(x) = x^3 - x - 1$ to a modular form. The result can be found in the section 5.2 of ...
1
vote
0answers
27 views
Reference for $L$-functions of curves
I am looking for a reference that explains as easily and as completely as possible how the $L-$function of a curve $C$ (non-singular, projective, geometrically irreducible, defined over $\mathbb{Q}$, ...
0
votes
0answers
57 views
Washington's Formula for $p$-adic $L$ function
I am reading through Washington's construction of $p$-adic $L$ functions in chapter $5$ of his book Cyclotomic Fields. Instead of $p$-adically interpolating as Kubota and Leopoldt originally did, ...
0
votes
1answer
280 views
Is the Generalized Riemann Hypothesis thought to be true?
The Riemann Zeta Function is the analytic continuation of the following function:
$$\zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s}$$
The Riemann Hypothesis states that the zeros of this in the critical ...
-1
votes
1answer
121 views
How to interpret the symbol $L( \frac{1}{2}, \pi \times \chi)$?
I am trying to interpret the symbol $L( \frac{1}{2}, \pi \times \chi)$ where $\chi = \mathbb{A}^\times / \mathbb{Q}^\times$ and $\pi$ is a cuspidal representation of $GL_2( \mathbb{A})$ (where $\...
2
votes
0answers
58 views
Study L-function associated to elliptic curves
I'm studying Iwasawa theory, and now I'm reading Greenberg's article about the application of Iwasawa theory of elliptic curves, but I have some problems on the first chapter, because it's about Hasse ...
0
votes
0answers
63 views
uniform convergence of $L(s,\chi)$ for $\Re(s) ≥ 1 + \delta$ " due to absolute convergence for $\Re(s)>1$?
On page 6 of this link, lemma 2.4 shows $L(s,\chi)$ is absolutely convergent for $\Re(s)>1$.
I understand the proof. However,
they also add: "The above proof also shows that for any $\delta > ...
1
vote
1answer
98 views
Euler product for $ \eta(z)^2 \eta(2z) \eta(4z) \eta(8z)^2 $
I was looking up a modular forms online: $S_3^{new}\big(\chi_8(3, \cdot)\big) $ it can be written as an Eta product:
$$f(z) = \eta(z)^2 \eta(2z) \eta(4z) \eta(8z)^2
= q \prod_{n=1}^\infty (1 - q^n)...
3
votes
1answer
147 views
Generalization of a $\det$ series for $\zeta_{\mathbb{Q}(\zeta_k)}$
With $\mathbb{Q}(\zeta_k)$ a cyclotomic field, $\chi_1,\ldots,\chi_{\phi(k)}$ the Dirichlet characters modulo $k$ and $\tilde{\chi}_1,\ldots,\tilde{\chi}_{\phi(k)}$ the underlying primitive Dirichlet ...
0
votes
1answer
218 views
Bounds for Dirichlet L-functions
In the half plane $\sigma$=Re(s) > 1 , one can find bounds for the Riemann zeta function $\zeta$(s) using either its convergent series or product formula.$\,$ From the Dirichlet series we get the ...
