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Results tagged with l-functions
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user 43108
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.
30
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 $K$:
$$\zeta_K(2)=\frac{\pi^{2r+2s}}{\sqrt{|D|}}\times \sum_v c_v A(x_{v,1})...A(x_{v,s} …
- 17.6k
19
votes
Underlying idea for (automorphic) L-function?
This is not exactly an answer, but should illustrate how tentative the opinion of experts on L-functions is, when it comes to explain what L-function really are. Two quotes from the first volume on nu …
- 17.6k
16
votes
3
answers
1k
views
First formulation of the Dedekind and Hasse-Weil conjectures
I'm looking for the original statement of two important conjectures in number theory concerning L-functions. I'm particularly interested in pinning down the year in which they were first formulated:
…
- 17.6k
16
votes
1
answer
2k
views
Automorphic factorization of Dedekind zeta functions
It is well known that for abelian number fields, the factorization of its Dedekind zeta function goes like this:
$$\zeta_K(s)=\zeta(s)\prod_{\chi \neq 1} L(s,\chi)$$
with the Dirichlet characters di …
- 17.6k
13
votes
0
answers
284
views
Propagation of modularity and the Artin conjecture
The (still incomplete) solution of the Artin conjecture on dimension $\leq2$ has been a massive research effort that has spanned (knowingly or not) around a century.
A very natural question is, what …
- 17.6k
13
votes
0
answers
620
views
No Siegel-Landau zeros for $\mathrm{GL}(n)$
The problem of non-existance of Siegel-Landau zeros seems to be uncharacteristically easier for cuspidal automorphic representations $\pi$ on $\mathrm{GL}(n)$ if $n\geq2$. We have in fact:
There a …
- 17.6k
11
votes
3
answers
1k
views
References for general Hasse-Weil zeta function
Most research on the Hasse-Weil zeta function focuses on some particular type of algebraic variety, and general surveys usually deal mostly with the better understood elliptic curve case.
I am lookin …
- 17.6k
9
votes
1
answer
890
views
Tate's thesis for Artin L-functions
As far as I know, Tate's thesis has been successfully applied in two fronts:
Hecke L-functions, by Tate and Iwasawa (and Teichmüller, Witt, Schmid)
Automorphic L-functions, by Jacquet, Shalika, Shap …
- 17.6k
8
votes
Accepted
Artin conjecture on L-functions
This is the status as far as I know. For dimension $\leq 2$ it is up to date. For higher dimensional representations I'm sure it is very incomplete, so feel free to edit or comment.
Dimension 1. Known …
- 17.6k
8
votes
Analytic continuation for $L$-functions of elliptic curves
As said in the comment above, the answer to your first question is no. Hecke L-functions are expected to factor as a product of irreducible cuspidal automorphic representations of $\mathrm{GL}_n(\math …
- 17.6k
7
votes
Irrationality of Dedekind zeta values
As with all motivic L-functions, irrationality of special values breaks into two rather different problems:
Critical values. In the case of Dedekind zeta functions, those are the even positive intege …
- 17.6k
7
votes
Is special value of Epstein zeta function in 3 variables a period?
Nice question. I think this is what's going on:
If $Q$ is a (positive definite) quadratic form, then its Epstein zeta function $Z_Q(s)$ can be expressed as a linear combination of L-functions of cusp …
- 17.6k
6
votes
Accepted
Weil Conjectures Analog for Multivariate Zeta Functions
The Weil conjectures have to do with local zeta-function over finite fields.
The Riemann zeta function and the multiple zeta functions are defined over $\mathbb{Q}$
So Weil conjectures are simply th …
- 17.6k
6
votes
Accepted
Corvallis 1979 proceedings
Given the importance of the proceedings, I leave a link to the copy at Library Genesis:
A. Borel and W. Casselman (Editors) Automorphic Forms, Representations and L-functions (1979)
- 17.6k
6
votes
Reference for the odd dihedral case of Artin's conjecture
There's a beautiful history behind this.
Basically, Artin and Hecke were working on different sides of this "dihedral modularity conjecture" at the same time (the 20s) and at the same place (Hamburg) …
- 17.6k