All Questions
30 questions
43
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 ...
- 3,337
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=\...
- 26k
27
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 $\...
- 1,969
21
votes
1
answer
757
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 ...
- 17.4k
18
votes
0
answers
740
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 $S$...
- 28.7k
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
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 ...
- 363
16
votes
0
answers
878
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 ...
- 643
12
votes
2
answers
2k
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 ...
- 3,309
12
votes
1
answer
1k
views
Motivic L-function vs motivic zeta function
Let $M$ be a pure motive over a field $k$. Roughly speaking, the L-function of $M$ is the product over all primes $p$ of
$$L_p(M,s)=\det(I-Fr_p|_{M_\ell^I} N(p)^{-s})^{-1}$$
where $Fr_p$ is a ...
- 3,799
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 ...
- 17.6k
10
votes
2
answers
2k
views
Main conjecture for elliptic curves
Suppose $E$ is an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at a prime $p$. Then one can define nonnegative integers $ \lambda_{E}^{alg} $, $ \mu_{E}^{alg} $, $ \lambda_{E}^...
- 1,209
9
votes
2
answers
327
views
Is the following the right definition of $L$-functions (on the Galois side)?
This question may be too elementary for this forum, but I have asked it on math stackexchange and didn't get an answer. I have now deleted it so there wouldn't be duplicates... Here is the question as ...
- 91
9
votes
0
answers
262
views
Injectivity of map in Beilinson's conjectures
In Beilinson's conjectures on special values of L-functions, he uses the image of the motivic cohomology of a a regular proper model in the motivic cohomology of the generic fiber to state the ...
- 1,553
7
votes
1
answer
1k
views
Analytic continuation for $L$-functions of elliptic curves
Let $E$ be an elliptic curve over a number field.
When $E$ has no CM and is a $\mathbb Q$-curve (i.e. it is $\overline{\mathbb Q}$-isogenous to all of its conjugates), it is nowadays known that $E$ ...
- 509
7
votes
1
answer
759
views
On Deligne's determinant of motives
This is a question about Deligne's conjecture on special values of L-functions. I have to confess that I've never understood the definition of the determinant which is supposed to give the right ...
- 71
7
votes
1
answer
398
views
Higher-dimensional Artin L-functions
I begin by clarifying that the "higher-dimensional" in my question refers to analogues of Artin L-functions over higher dimensional base schemes than $\mathrm{Spec}(\mathbb{Z})$.
Now for the set-up. ...
- 21.3k
7
votes
1
answer
768
views
Abelian varieties and Selberg class
Hello everyone,
I would like to know whether, assuming Selberg's orthonormality conjecture, it would be possible to establish a "natural" correspondence between abelian varieties and functions ...
- 195
6
votes
1
answer
486
views
Strengthened supercongruences for Ramanujan-type formulas for $1/\pi^k$
The question below is again a follow-up of an old question.
Motivation: Zhi-Wei Sun listed a number of supercongruences attached to Ramanujan-type $1/\pi$ formulas in the arXiv paper which can be ...
- 3,337
5
votes
2
answers
1k
views
Local factors of Hasse-Weil zeta function - what do they have in common?
Let $X$ be a regular scheme, flat and of finite type over $Spec(\mathbb{Z})$ (add "projective" if you want). Then the Hasse-Weil zeta function of $X$ is defined as a product over all prime numbers of ...
- 5,637
5
votes
1
answer
1k
views
Generalizing Eichler-Shimura to higher dimension, again
This question is related to
Intuition behind the Eichler-Shimura relation?
and
L-functions and higher-dimensional Eichler-Shimura relation
Answering the first question above, Matt Emerton gives a ...
- 1,783
4
votes
2
answers
365
views
Main conjecture for elliptic curves invariant under a $\mathbb{Q}$-isogeny
Suppose $E$ is an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at a prime $p$. Then one can define nonnegative integers $ \lambda_{E}^{alg} $, $ \mu_{E}^{alg} $, $ \lambda_{E}^...
- 1,209
4
votes
1
answer
431
views
Automorphicity of L-Factors of Zeta Functions
Associated to a variety over a number field $K$, one has a family of "Hasse-Weil" L-functions, which can be combined (as an alternating product) to give the Hasse--Weil zeta function of the variety, ...
- 643
4
votes
0
answers
232
views
holomorphic continuation of motivic $L$-functions
The question is rather easy to formulate: when is the $L$-function of a pure motive over $\mathbb{Q}$ expected to have a holomorphic (as opposed to simply meromorphic) continuation to the complex ...
- 41
3
votes
0
answers
302
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 ...
- 930
2
votes
2
answers
1k
views
L-functions and algebraic geometry
Robert Langlands commented in a letter to Deligne that perhaps some of the deepest problems of algebraic geometry lie in L-functions. I want to understand the general philosophy and the connection ...
- 2,106
2
votes
2
answers
606
views
Axioms for zeta functions
The Selberg class is an axiomatization of arithmetically significant zeta functions (a.k.a. L-functions) by a few analytic properties (functional equation etc.) However there do exist other zeta ...
- 333
2
votes
1
answer
308
views
convergence of L-functions of curves
Let $C$ be a smooth projective curve over $\mathbb{Q}$. Its associated L-function is defined by
$$
L(C, s)=\prod_{p \text{ prime}} L_p(C, s),
$$
where, if $p$ is a prime of good reduction, $L_p(C,...
- 21
2
votes
0
answers
220
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 ...
user30211
1
vote
0
answers
351
views
Do those manifolds atrached to L-functions give rise naturally to motives? [closed]
Edited after Will Sawin's comment:
Consider the set $\mathcal{M}$ of all automorphic L-functions belonging to the Selberg class. Such a set is closed for the product $.$ and the tensor product $\...
- 6,984