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.

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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 ...
asdfasdgasdfgasfgasdf's user avatar
2 votes
2 answers
208 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^{...
Christopher-Lloyd Simon's user avatar
0 votes
1 answer
124 views

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 $L(\rho, \chi) =0$ ...
Josh's user avatar
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1 vote
0 answers
121 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 ...
LWW's user avatar
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2 votes
0 answers
66 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 ...
Misaka 16559's user avatar
11 votes
2 answers
606 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 ...
KStarGamer's user avatar
2 votes
0 answers
132 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 ...
LWW's user avatar
  • 653
2 votes
1 answer
224 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 ...
Misaka 16559's user avatar
1 vote
0 answers
118 views

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 ...
edward cornfoot's user avatar
1 vote
0 answers
121 views

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 ...
L-JS's user avatar
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14 votes
1 answer
715 views

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 ...
Nomas2's user avatar
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1 vote
0 answers
87 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 ...
Anurag Sahay's user avatar
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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 ...
Nomas2's user avatar
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3 votes
1 answer
202 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=\...
Akash Yadav's user avatar
5 votes
1 answer
221 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 \...
Asvin's user avatar
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4 votes
1 answer
196 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 $|q|\lt 1$....
Nomas2's user avatar
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0 votes
0 answers
360 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 ...
Sylvain JULIEN's user avatar
2 votes
0 answers
98 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 $...
Akash Yadav's user avatar
3 votes
1 answer
191 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 ...
Tristan Phillips's user avatar
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(...
BobBuilder's user avatar
5 votes
1 answer
254 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 = ...
Steph Curry's user avatar
1 vote
0 answers
168 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 ...
Dekimshita's user avatar
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 ...
Zhi-Wei Sun's user avatar
  • 14.4k
3 votes
0 answers
257 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 ...
MAS's user avatar
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0 votes
0 answers
86 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 ...
Ronald J. Zallman's user avatar
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 ...
Ronald J. Zallman's user avatar
2 votes
1 answer
215 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 ...
dekimashita's user avatar
3 votes
1 answer
197 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},\...
curious math guy's user avatar
5 votes
1 answer
478 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 $(\...
Maty Mangoo's user avatar
1 vote
1 answer
172 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., $\...
Taisong Jing's user avatar
4 votes
0 answers
174 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} $ ...
Marsault Chabat's user avatar
12 votes
1 answer
528 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 ...
Henri Cohen's user avatar
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4 votes
0 answers
208 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 ...
Maty Mangoo's user avatar
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 $s = 1/2$...
Seewoo Lee's user avatar
  • 1,901
4 votes
2 answers
515 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-...
Dekimshita's user avatar
6 votes
1 answer
170 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\{ ...
Desiderius Severus's user avatar
2 votes
1 answer
293 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 ...
user501735's user avatar
3 votes
0 answers
224 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 $...
Adithya Chakravarthy's user avatar
2 votes
1 answer
285 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 ...
chbe's user avatar
  • 91
1 vote
1 answer
245 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\...
ShBh's user avatar
  • 271
5 votes
1 answer
342 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 ...
Seewoo Lee's user avatar
  • 1,901
2 votes
1 answer
149 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 ...
Qingzhi Li's user avatar
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 ...
Qingzhi Li's user avatar
3 votes
1 answer
214 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 ...
Zhang's user avatar
  • 81
6 votes
1 answer
499 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 ...
Adithya Chakravarthy's user avatar
7 votes
0 answers
299 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?
user497366's user avatar
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 ...
Anton Hilado's user avatar
  • 3,269
4 votes
0 answers
179 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 ...
David Schwein's user avatar
2 votes
0 answers
81 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 $q$,...
Markus's user avatar
  • 121
10 votes
1 answer
589 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 ...
Adithya Chakravarthy's user avatar

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