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.
434
questions
66
votes
17
answers
12k
views
Shortest/Most elegant proof for $L(1,\chi)\neq 0$
Let $\chi$ be a Dirichlet character and $L(1,\chi)$ the associated L-function evaluated at $s=1$. What would be the 'shortest' proof of the non-vanishing of $L(1,\chi)$?
Background: The non-vanishing ...
58
votes
2
answers
4k
views
"Gross-Zagier" formulae outside of number theory
The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer conjecture. It relates the value at $1$ of the ...
41
votes
3
answers
4k
views
Underlying idea for (automorphic) L-function?
Edit: So with a few more months of math under my belt, I recognize some of the issues with this question. I still hope for an answer, so let me say a few things.
Within the Langlands philosophy, L-...
41
votes
1
answer
4k
views
A mysterious connection between Ramanujan-type formulas for $1/\pi^k$ and hypergeometric motives
The question below is the follow-up of this question on MathOverflow.
Motivation: As is stated in the former question, those identities(formula (35)-(44)) of $1/\pi$ attributed to Ramanujan are ...
38
votes
4
answers
6k
views
Modular forms and the Riemann Hypothesis
Is there any statement directly about modular forms that is equivalent to the Riemann Hypothesis for L-functions?
What I'm thinking of is this: under the Mellin transform, the Riemann zeta function $...
36
votes
1
answer
4k
views
Special values of L-functions as periods
If $M$ is a pure motive over $\mathbb{Q}$, one cas define its $L$-function $L(M,s)$ which conjecturaly is a meromorphic function over $\mathbb{C}$ with finitely many poles.
For example, when $M=\...
35
votes
1
answer
2k
views
The modularity theorem as a special case of the Bloch-Kato conjecture
In the homepage for the CRM's special semester this year, I found the interesting statement that the modularity theorem (formerly the Taniyama-Shimura-Weil conjecture) is a special case of the Bloch-...
30
votes
1
answer
3k
views
Iwasawa main conjectures vs Bloch-Kato conjectures
Let $p$ be a prime, $K$ be a number field, $S$ a finite set of finite places of $K$ containing the set $S_p$ of places above $p$ and the places at infinity, $G:=G_{K,S}$ the Galois group of the ...
29
votes
3
answers
2k
views
Intuition for Zagier's theorem for $\zeta_K(2)$
In 1986, Don Zagier generalized Euler's theorem ($\zeta_\mathbb{Q}(2)=\pi ^2 /6$) to an arbitrary number field $K$:
$$\zeta_K(2)=\frac{\pi^{2r+2s}}{\sqrt{|D|}}\times \sum_v c_v A(x_{v,1})...A(x_{v,s})...
28
votes
1
answer
913
views
Relation between Schanuel's theorem and class number equation
(Crossposted on math stack exchange: https://math.stackexchange.com/questions/4040249/relation-between-schanuels-theorem-and-class-number-equation)
It was recently brought to my attention that there ...
28
votes
1
answer
2k
views
Is there a "Langlands philosophy" reason for the fact that the L-function of the Jacobi theta function is (almost) the Riemann zeta function?
The Jacobi theta function $\theta(z) = 1 + 2 \sum_{n = 1}^\infty q^{n^2}$, with $q = e^{\pi i z}$ is a (twisted) modular form with weight $1/2$. It has an associated $L$-function $L(\theta, s) = \sum_{...
26
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 $\...
26
votes
3
answers
1k
views
Universality of zeta- and L-functions
Voronin´s Universality Theorem (for the Riemann zeta-Function) according to Wikipedia: Let $U$ be a compact subset of the "critical half-strip" $\{s\in\mathbb{C}:\frac{1}{2}<Re(s)<1\}$ with ...
23
votes
2
answers
3k
views
Why are Tamagawa numbers equal to Pic/Sha?
For a connected algebraic group $G$ over a global field $K$ with adeles $A$, the Tamagawa number of $G$ is the volume of $G(A)/G(K)$. It is conjectured (and often known) to be rational, namely the ...
22
votes
2
answers
6k
views
Beilinson conjectures
Continuing an amazingly interesting chain of answers about motivic cohomology, I thought I should learn about the Beilinson conjectures, referred there.
I have found some references, and they seem to ...
22
votes
4
answers
4k
views
Special values of $p$-adic $L$-functions.
This is a very naive question really, and perhaps the answer is well-known. In other words, WARNING: a non-expert writes.
My understanding is that nowadays there are conjectures which essentially ...
21
votes
1
answer
1k
views
Hadamard factorization of L-functions
I have already asked this question here in a different form, but really need an answer.
Let $L(s)$ be a "standard" $L$-function, say with Euler product, functional equation, etc...
(Selberg ...
21
votes
1
answer
737
views
What should motives for $L(E,n)$ look like?
Goncharov and Manin showed in this paper that the zeta values $\zeta(n)$ can be realized as periods of framed mixed Tate motives constructed from moduli spaces $\overline{\mathcal{M}}_{0,n+3}$ of ...
19
votes
2
answers
2k
views
Applications of Artin's holomorphy conjecture
I wonder why the Artin conjecture is so important. The only reason I could figure out is that one could use the holomorphy of Artin L-series and Weil's converse theorem to show modularity of two-...
19
votes
1
answer
3k
views
What is a path in K-theory space?
In a comment on Tom Goodwillie's question about relating the Alexander polynomial and the Iwasawa polynomial, Minhyong Kim makes the cryptic but tantalizing statement:
In brief, the current view is ...
19
votes
3
answers
2k
views
Computing (on a computer) the first few (non-trivial) zeros of the zeta function of a number field.
Let $M$ be the splitting field of
x^8 + 3*x^7 + 13*x^6 + 17*x^5 + 45*x^4 + 37*x^3 + 11*x^2 + 112*x + 108
over the rationals. If I've understood some tables ...
19
votes
1
answer
1k
views
constants in Gamma factors in functional equation for zeta functions.
Usually the Riemann zeta function $\zeta(s)$ gets multiplied by a "gamma factor" to give a function $\xi(s)$ satisfying a functional equation $\xi(s)=\xi(1-s)$. If I changed this gamma factor by a non-...
19
votes
1
answer
2k
views
Two-variable p-adic L-functions of elliptic curves
Suppose $K$ is an imaginary quadratic field (with class number 1, for simplicity), $p \ne 2$ a prime split in $K$, and $K_\infty$ the $\mathbb{Z}_p^2$-extension of $K$.
If $E / \mathbb{Q}$ is an ...
18
votes
1
answer
1k
views
Definitions of $\pi_1 \times \pi_2, \pi_1 \boxplus \pi_2, \pi_1 \boxtimes \pi_2$
Let $\pi_i$ be a smooth, admissible (possibly irreducible) representation of $\operatorname{GL}_{n_i}(k)$ for $k$ a $p$-adic field. I have seen the following representations defined in terms of $\...
18
votes
1
answer
1k
views
Distinct simple zeros of Dirichlet L-functions
Given a finite set of distinct primitive Dirichlet characters, $\chi_1, \dots, \chi_r$, is it known that the product of the L-functions, $$L(s):=\prod_{i=1}^r L(s,\chi_i),$$ has a simple zero? It's ...
18
votes
1
answer
540
views
Is special value of Epstein zeta function in 3 variables a period?
Kontsevich-Zagier's article "Periods" contains the following question
Is $\displaystyle \sum_{x,y,z \in \mathbb{Z}}' \frac{1}{(x^2+y^2+z^2)^2}$ an extended period?
($\sum'$ means we do not sum ...
18
votes
1
answer
2k
views
Stark's conjecture and p-adic L-functions
Not long back I asked a question about the existence of p-adic L-functions for number fields that are not totally real; and I was told that when the number field concerned has a nontrivial totally ...
18
votes
0
answers
710
views
Infinite extensions such that every elliptic curve has finite rank
The comments to this answer seem to make the following claim.
Claim. Let $K$ be the maximal abelian extension of $\mathbf Q$ that is unramified away from $p$ (more generally, away from a finite set $S$...
17
votes
2
answers
3k
views
central/critical/special values of L-functions terminology
I have a question about the terminology for special values
of L-functions. Is the following a correct description of
standard usage:
Suppose L(s) is an L-function which satisfies a functional
...
17
votes
0
answers
956
views
Why arithmetic Langlands?
In trying to understand the import of Akshay Venkatesh' most recent work I found myself wondering anew about that old gnawing mystery: why Langlands? Why should arithmetic of polynomial equations over ...
17
votes
0
answers
1k
views
Special values of Artin L-functions
This question might be naive and might carry the heuristic that we are living in the best possible world a little too far. If so, I appreciate being told so.
Background: Stark's conjecture interprets ...
16
votes
3
answers
1k
views
First formulation of the Dedekind and Hasse-Weil conjectures
I'm looking for the original statement of two important conjectures in number theory concerning L-functions. I'm particularly interested in pinning down the year in which they were first formulated:
...
16
votes
1
answer
2k
views
Relation between Weil Conjecture and Langlands Program
Recently I read Gelbart's An Elementary Introduction To The Langlands Program, which explained the origin of the program, and this question came to me. For an elliptic curve over finite field, the ...
16
votes
1
answer
2k
views
Automorphic factorization of Dedekind zeta functions
It is well known that for abelian number fields, the factorization of its Dedekind zeta function goes like this:
$$\zeta_K(s)=\zeta(s)\prod_{\chi \neq 1} L(s,\chi)$$
with the Dirichlet characters ...
16
votes
0
answers
847
views
L-Functions of Varieties, Zeta Functions of Their Models
Let $k$ denote a number field, with algebraic closure $\bar{k}$. Take a smooth, projective variety $X$ over $k$. If $\mathfrak{p}$ is a prime of $k$, and $l$ is a rational prime different to the ...
15
votes
5
answers
2k
views
$|L'(1,\chi)/L(1,\chi)|$
Let $\chi$ be a primitive Dirichlet character $\mod q$, $q>1$. Is there a neat, simple way to give a good bound on $L'(1,\chi)/L(1,\chi)$?
Assuming no zeroes $s=\sigma+it$ of $L(s,\chi)$ satisfy $\...
15
votes
1
answer
942
views
How do functional equations for zeta functions arise from the structure of a homology group?
I have read in various disparate sources that certain zeta functions satisfy functional equations as a consequence of some structure on some homology group. Here is an example of a quote in this ...
15
votes
2
answers
3k
views
p-adic L-functions
For modular forms, it is known that you can construct p-adic L-functions by interpolating (p-power conductor) twists of their associated L-functions at special values. Similarly, Kubota-Leopoldt's p-...
15
votes
1
answer
1k
views
Jacquet's approach to Rankin--Selberg L-functions
In his book "Automorphic Forms on GL(2), II", Springer Lecture Notes vol. 278, Jacquet defines the Rankin--Selberg L-function of $\pi_1 \times \pi_2$, where $\pi_i$ are automorphic representations of $...
14
votes
3
answers
5k
views
Why are they called L-functions?
I was hoping to see this pop up on the recent big list question about etymology or terms and symbols. Since it has not, and I can't find an answer, I will ask:
What is the reason for the $L$ in $L$-...
14
votes
1
answer
936
views
Euler factors of L-function at bad primes
This is of course a very-well known problem, but still let me ask the questions my way. Let $L(s)$ be a "motivic" $L$-function, whatever that means: in particular, it has an Euler product (including ...
14
votes
1
answer
2k
views
Primer on Eisenstein series
My apologies if this question is a duplicate. I seached, and the closest I could locate is this question, which has very intriguing and intractable (for me) responses.
In my continuing journey of ...
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$$
where the reasoning of the signs thus is prepared, so that of the second may be had as $-$, prime ...
14
votes
3
answers
2k
views
Nonvanishing of central L-values of quadratic twists?
Let $\pi$ be a cuspidal automorphic representation of GL(2) over a number field (if you want, assume it's $\mathbb Q$ and $\pi$ comes from a holomorphic modular form).
In the case $\pi$ has trivial ...
14
votes
1
answer
861
views
BSD conjecture for rank 1 elliptic curves
Let $E/\mathbb{Q}$ be an elliptic curve. The weak Birch and Swinnerton-Dyer conjecture predicts that
$$\text{ord}_{s=1}L(E, s)=\text{rank} E(\mathbb{Q}).$$
Thanks to the work of Gross-Zagier and ...
14
votes
2
answers
969
views
What are zeta functions good for?
I know a couple of answers to the above question:
They can be used for point counting over finite fields/estimating the distribution of primes in characteristic 0.
There are various conjectures/...
14
votes
1
answer
1k
views
Is the adjoint L-function on GL(m) holomorphic?
Let $\pi$ be an automorphic representation on $\mathrm{GL}(m)/\mathbb{Q}$.
Define $$L(s,\pi,\mathrm{Ad}):=\frac{L(s,\pi\times\overline{\pi})}{\zeta(s)}.$$ This is an $L$-function with Euler product of ...
14
votes
1
answer
498
views
Bound for $GL(3)$ symmetric square
Let $\pi$ be an automorphic representation of $GL(3)$ over a number field. Let $a_n$ be the coefficients of $L(s, \pi, \mathrm{sym}^2)$. Do we know if
$$\sum_{n>0} \frac{|a_n|}{n^s}$$
and
$$\sum_{n&...
14
votes
1
answer
961
views
P-adic L-functions of nonabelian twists of elliptic curves
Let $E$ be an elliptic curve and $\rho$ an Artin representation of $\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$. Then there is a "twisted L-function" $L(E, \rho, s)$, corresponding to the ...
14
votes
0
answers
553
views
Moments of derivatives of $L$-functions
I'd like to know why it is important to know the moments of the derivatives of $L$-functions.
The moments of $L$-functions are related to the Lindelöf Hypothesis, but what about the moments of the ...