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.
435
questions
4
votes
1
answer
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Conditional convergence of Artin $L$-functions
Let $k$ be a number field and $V$ a non-trivial irreducible Artin representation over $k$. Consider the associated Artin $L$-function with corresponding Euler product decomposition $L(V,s)= \prod_v ...
1
vote
1
answer
187
views
Classification of L functions and Dirichlet series by poles
I am interested in the connection between particular Dirichlet series' abscissa of convergence and the poles of L-functions.
Let $D(z) = \sum_{n=1}^\infty\frac{a_n}{n^z}$ be a Dirichlet series ...
2
votes
2
answers
211
views
Conditional convergence of exponential sums related to a Hecke modular form
Definition
Consider the Fourier coefficients $\psi(n)$ of the modular form $\eta^4(6\tau)$,
which are defined in terms of $q=\exp(i2\pi\tau)$ by the identity:
$$\eta^4(6\tau) = q \prod_1^\infty (1-q^{...
0
votes
1
answer
132
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A question about the setup of zero density estimates for Dirichlet $L$-functions
For $L(s,\chi)= \sum_{n \geq 1}\frac{\chi(n)}{n^s}$, where $s = \sigma + it$, we define the function $N(\sigma, T, \chi)$ which counts the zeros $\rho = \beta + i\gamma$ for which $L(\rho, \chi) =0$ ...
1
vote
0
answers
124
views
Analytic properties of $L$-functions attached to a compatible system of $\ell$-adic Galois representations
Let $F$ and $E$ be number fields, $G_F$ be the absolute Galois group of $F$, and $S$ be a finite set of primes of $F$. For $\lambda$ a prime of $E$ we denote by $\ell$ its residual characteristic. We ...
2
votes
0
answers
70
views
The logarithmic derivative of a twisted L-function?
Let $F$ be a quadratic number field with class number $h_F = 1$. Let $\zeta_F$ be the Dedekind zeta function, we have
$$ \frac{\zeta_F ' (1+it)}{\zeta_F (1+it)} \ll \frac{\log t}{\log\log t} .$$
(I ...
11
votes
2
answers
616
views
Does the number of roots of the modular form associated to an elliptic curve, on the positive imaginary axis, equal the analytic rank?
Recently I've been playing around with elliptic curves and have seemingly come up with a conjecture that I could not find elsewhere:
Let $E$ be an elliptic curve, and $f(q)$ its associated modular ...
2
votes
0
answers
135
views
Analyticity of unramifed part of Rankin-Selberg $L$-functions on $\Re(s)=1$
I have only a little knowledge about automorphic representations and $L$-functions. Now I am reading the textbook of Goldfeld and Hundley on automorphic representations, and also planning to read the ...
2
votes
1
answer
235
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'$\times$' or '$\otimes$' when writing $L$-functions?
Recently, I came across the Langlands correspondence theorem, there is the following line:
$$L(s,\pi(\sigma) \times \pi(\tau)) = L(s,\sigma \otimes \tau), $$
where $\sigma$ and $\tau$ are ...
1
vote
0
answers
131
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Behavior of Dirichlet L-functions at the edge of the critical strip
Given a Dirichlet L-function $L(\chi, s)$ of a primitive character $\chi$, what is the asymptotic behavior of $L(\chi, 1+it)$ for real $t$? I am looking for as many answers for the same question. This ...
1
vote
0
answers
123
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Reference request: unfolding of Integral representation of an L-function
Are there any text or papers that thoroughly address unfolding of integral representation of an L-function such as D. Ginzburg's On Spin L-function for Orthogonal Groups page 762-763 and page 774 (or ...
14
votes
1
answer
719
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Euler's proof of $\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$
Euler proved
$$\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$$
where the reasoning of the signs thus is prepared, so that of the second may be had as $-$, prime ...
1
vote
0
answers
89
views
Motivation behind a result of Munshi on nonvanishing of L-functions in families of elliptic curves
In this article in Compositio (2011), Munshi proves a mean value result for
$$ \sum_{d} r(d) \Lambda^{(l)}(1/2,f,\chi_d) F(d/Y),$$
where here $f$ is a primitive holomorphic form of level $q$ with ...
0
votes
0
answers
111
views
Are $\zeta'(0)$ and $\beta'(0)$ algebraic numbers?
Let $\zeta$ be the Riemann zeta function and $\beta$ the Dirichlet beta function. We know that $\zeta (0)=-1/2$ and $\beta (0)=1/2$ are algebraic numbers over $\mathbb{Q}$. This led me to the ...
3
votes
1
answer
209
views
Non-vanishing of archimedean integral representations
Let $\psi$ denote a non-trivial additive character of $\mathbb{R}$ and $n$ be a positive integer. Let $(\pi,V)$ and $(\pi',V')$ be two irreducible generic Casselman-Wallach representations of $G_n=\...
5
votes
1
answer
225
views
p-adic L functions from Selmer groups - how canonical are they?
For this question, I am going to be very concrete but very much appreciate broader viewpoints. Let $F$ be a number field and define $F_n = F(\mu_{p^n})$ and let us suppose for simplicity that $\mu_p \...
4
votes
1
answer
201
views
Abscissa of convergence of the $\tau$ Dirichlet series
Define the $\tau$ Dirichlet series $L$ by
$$L(s)=\sum_{n=1}^\infty \frac{\tau (n)}{n^s}$$
where $\tau$ is defined by
$$q\prod_{n=1}^\infty (1-q^n)^{24}=\sum_{n=1}^\infty \tau (n)q^n$$
where $|q|\lt 1$....
0
votes
0
answers
381
views
Spiegel Vermutung: no Siegel zeros iff GRH is equivalent to Goldbach's conjecture
I apologize for using German language in the title, but this question came to my mind after watching the French movie "le théorème de Marguerite" in which the protagonist gets an insight ...
2
votes
0
answers
99
views
Quotient of integral representation of archimedean exterior square L-function
Let $\psi$ denote a non-trivial additive character of $\mathbb{R}$ and $r=2n$ be a positive even integer. Let $(\pi,V)$ denote an irreducible generic admissible Casselmann-Wallach representation of $...
3
votes
1
answer
196
views
Non-Schwartz test functions for the explicit formula for L-functions
The statements of the explicit formula for L-functions that I am aware of require the test function to be a Schwartz function (see, e.g., equation (4.11) in Section 4 of Low lying zeros of families of ...
1
vote
1
answer
206
views
Bound on Von Mangoldt for automorphic L-functions
Following the notation in Iwaniec+Kowalski, let $L(f,s)$ be an L-function. Denote
$$\frac{L'}{L}(f,s)=\sum_{n\ge1} \Lambda_f(n)n^{-s} $$
In terms of the local roots of the Euler product:
$$ \Lambda_f(...
5
votes
1
answer
259
views
Explicit description for action of Weyl element in Whittaker model for GL2
Let $F$ be a non-archimedean local field and let $\pi =\mathscr{B}(\chi, \chi^{-1})$ be a principal series representation of $\mathrm{PGL}_2(F)$ induced from a character $\chi$ of $F^\times$. Let $w = ...
1
vote
0
answers
171
views
Clarifications about the Iwasawa Main Conjecture
I would like to clarify a couple of things regarding the Iwasawa main conjecture.
In the paper where Mazur and Wiles prove the main conjecture, on page 182, it is written that $h_p(\omega^i, T)$ is ...
6
votes
0
answers
221
views
A curious series for $L(2,(\frac{-3}{\cdot}))$
Let
$$K:=L\left(2,\left(\frac{-3}{\cdot}\right)\right)=\sum_{k=1}^\infty\frac{(\frac k3)}{k^2}=\sum_{j=0}^\infty\left(\frac1{(3j+1)^2}-\frac1{(3j+2)^2}\right),$$ where $(\frac k3)$ is the Legendre ...
3
votes
0
answers
260
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 ...
0
votes
0
answers
87
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 ...
2
votes
0
answers
219
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 ...
2
votes
1
answer
217
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 ...
3
votes
1
answer
204
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
$$ ord_{s=i+1-n}L(H^i(X),s)=\dim H^{i+1}_{\mathcal{D}}(X_\mathbb{R},\...
5
votes
1
answer
481
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On the notion of cuspidality
Let $k/\mathbb{Q}$ be a number field and $\mathbb{A}$ its ring of adèles. As usual $\mathbb{A} = \mathbb{A_f} \times \mathbb{A_{\infty}}$.
The standard definition of an automorphic representation $(\...
1
vote
1
answer
184
views
The theta function of an odd Dirichlet character
The theta function $\theta_\chi(t)$ of a Dirichlet character $\chi$ is defined to be $\theta_\chi(t) = \frac{1}{2} \sum\limits_{n=-\infty}^\infty \chi(n) e^{2\pi i n^2 t}$ if $\chi(-1) = 1$ (i.e., $\...
4
votes
0
answers
176
views
Several L-functions but one Galois representation: How to choose
Let $\mathbf{G}$ be a reductive group which enjoys all the nice properties a reducive group can dream of. Fix $(\mathbf{G},X)$ a Shimura datum associated with it and assume that if $K\leq\mathbf{G} $ ...
12
votes
1
answer
533
views
$p$-adic L function of an odd Dirichlet character
Apologies for a naive question (especially for Iwasawa theorists): it is well-known
and trivial to prove that the usual (elementary) construction of $p$-adic L functions
attached to odd Dirichlet ...
6
votes
0
answers
219
views
Correspondence between motives and automorphic representations
What I know:
I understand motives via its realization; in Coates' and Perrin-Riou's paper On $p$-adic L-functions Attached to Motives over $\mathbb{Q}$ (see http://doi.org/10.2969/aspm/01710023), the ...
1
vote
0
answers
131
views
Zeroes of certain $L$-functions on the critical line and GGP conjectures
Global Gan-Gross-Prasad conjecture (on various groups) says that nonvanishing of certain automorphic $L$-function $L(s, \pi)$ (of cuspidal representation $\pi$ of some reductive group $G$) at $s = 1/2$...
4
votes
2
answers
524
views
Explicit formula for Artin L-functions
The classical explicit formula for the Riemann Zeta function states that
$$
\psi(x)=x-\sum_{\rho} \frac{x^{\rho}}{\rho}+O(1),
$$
where $\psi(x)=\sum_{n \leq x} \Lambda(n)$ and the sum is over all non-...
6
votes
1
answer
172
views
What is the conductor of an automorphic representation for $\Gamma_0(q)$ in $GSp(4)$?
Let $\pi$ be a generic cuspidal automorphic representation on $GSp(4)$, with level $\Gamma_0(q)$ (the group of symplectic matrices with lower left block divisible by $q$), i.e.
$$\Gamma_0(q) = \left\{ ...
2
votes
1
answer
307
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 ...
3
votes
0
answers
231
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 $...
2
votes
1
answer
293
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 ...
1
vote
1
answer
246
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\...
5
votes
1
answer
372
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 ...
2
votes
1
answer
151
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 ...
2
votes
0
answers
143
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 ...
3
votes
1
answer
221
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 ...
6
votes
1
answer
503
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 ...
7
votes
0
answers
301
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?
11
votes
2
answers
1k
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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 ...
4
votes
0
answers
184
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 ...
2
votes
0
answers
85
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 $q$,...