All Questions
Tagged with motives nt.number-theory
75 questions
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
45
votes
2
answers
3k
views
What are the possible motivic Galois groups over $\mathbb Q$?
Let $E$ be a motive over $\mathbb Q$. (I should precise, that by a motive I mean
here a pure motive over $\mathbb Q$, with coefficients in $\mathbb Q$, that I see here as a conjectural object which ...
- 26k
19
votes
2
answers
2k
views
What is the relationship between these two notions of "period"?
The motivation for this question is to understand a recent theorem of Francis Brown which implies that all periods of mixed Tate motives over $\mathbb{Z}$ lie in $\mathcal{Z}[\frac{1}{2\pi i}]$, where ...
- 9,061
12
votes
2
answers
2k
views
Status of Beilinson conjectures?
(I hesitate to post that question here, but I received on answer on FB:)
Does anyone know how the current status of work on them is? And how the possible generalizations etc. which one thinks ...
3
votes
2
answers
375
views
critical values of motives
Hi friends,
I have some questions concerning the critical values of motives, in the sense of Deligne. I will only look at motives of the form $h^i(X)$ where $X$ is a smooth projective algebraic ...
- 31
5
votes
0
answers
654
views
The Shafarevich Conjecture and motivic Langlands stacks.
Hi, I recently learned about an amazing conjecture of Shafarevich
(proved by Faltings) about the finiteness of the number
of curves of a fixed genus with good reduction outside a
finite number of ...
6
votes
1
answer
688
views
Followup questions about the relationship between modular forms and motives
It occurs more and more that I ask a question on math stackexchange and then realize that it is more appropriate to mathoverflow. Hopefully this reflects well on myself... In any case, I copy here ...
- 61
11
votes
0
answers
1k
views
Linear algebra of elliptic curves over p-adic fields
Unfortunately, the following question is somewhat ill-posed. However, I hope to make what I'm looking for sufficiently clear.
Given two elliptic curves (or Abelian varieties) over $\mathbb{C}$, one ...
- 1,537
5
votes
0
answers
834
views
Motivic Galois group and Shimura varieties
Hi,
Suppose that one has a Shimura variety $Sh(G,X)$ where $(G,X)$ is the corresponding Shimura datum and suppose that it can be interpreted as a moduli space of motives (e.g. PEL type Shimura ...
- 647
15
votes
1
answer
769
views
Crystalline realization of mixed Tate motives
Deligne and Goncharov, in their article of 2005, mention that the crystalline realization functor has yet to be worked out. What's the current state of the literature on this? And how big of an issue ...
- 313
6
votes
2
answers
2k
views
motive of a modular form
What is the idea behind a "motive of a modular form"? (I know what a motive is and what a Weil cohomology is. I want to know how to get the motive [what is the idea of Scholl?], and why this is ...
user19475
10
votes
1
answer
2k
views
How does the conjectural Langlands group fit into the Tannakian point of view?
I've read that one way to formulate the Langlands program is the following:
Let $\mathcal{L}_ {\mathbb{Q}}$ be the conjectural Langlands group. Then the category of semi-simple (continuous) ...
14
votes
2
answers
2k
views
How would a motivic proof of the Riemann hypothesis over finite fields go?
It is well known that Grothendieck had a different idea than Deligne about how one should go about proving the Riemann hypothesis for finite fields. However, since Grothendieck's desired proof never ...
- 6,348
3
votes
1
answer
777
views
Is the "L-function of the complex cohomology" of a motive equal to the L-function of its l-adic realization?
Let's say I have a motive in $\mathcal{M}_{num}(K)$ ($K$ a number field). For each prime $l$ there is a realization of this motive in terms of etale cohomology with coefficients in $\mathbb{Q}_l$. ...
- 6,348
10
votes
1
answer
1k
views
Motives from the fundamental group made nilpotent
I am reading the fascinating paper of Deligne on "le groupe fondamental de la droite projective moins trois points", and other stuffs related to anabelian geometry. This suggested the following ...
- 26k
141
votes
0
answers
13k
views
Grothendieck-Teichmüller conjecture
(1) In "Esquisse d'un programme", Grothendieck conjectures
Grothendieck-Teichmüller conjecture: the morphism
$$
G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T})
$$
is an isomorphism.
Here $...
- 7,527
7
votes
1
answer
796
views
Crystalline realizations of Artin motives
What are the crystalline realizations of Artin motives?
In a paper by Kisin and Wortmann, "A note on Artin motives" (google it and you'll find it immediately), they define a suitable category of ...
- 1,841
8
votes
3
answers
3k
views
Why is the zeta function of a variety over a finite field not a polynomial? (question about motives)
I've been doing some light(?) reading on motives and the standard conjectures in an attempt to put various things that I tangentially know in perspective.
The question is this: the Weil conjectures ...
- 5,929
5
votes
2
answers
671
views
Automorphic form encoding the orders of $N$ modulo $p$.
Let $N$ be a nonzero rational number. For every prime number $p$ with $v_p(N)=0$, let $a_p$ denote the index in $\mathbb Z/p\mathbb Z$ of the subgroup generated by $N$ modulo $p$. So we have $a_p=1$ ...
- 4,015
18
votes
1
answer
6k
views
Deligne's proof of Ramanujan's conjecture
I am trying to understand Deligne's proof of the Ramanujan conjecture and more generally how one associates geometric objects (ultimately, motives) to modular forms.
As the first step, which I ...
- 1,783
35
votes
4
answers
8k
views
What would a "moral" proof of the Weil Conjectures require?
At the very end of this 2006 interview (rm), Kontsevich says
"...many great theorems are originally proven but I think the proofs are not, kind of, "morally right." There should be better proofs......
- 1,764
17
votes
2
answers
2k
views
Why does the Gamma function satisfy a functional equation?
In question #7656, Peter Arndt asked why the Gamma function completes the Riemann zeta function in the sense that it makes the functional equation easy to write down. Several of the answers were from ...
- 118k
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-...
- 41.4k
18
votes
2
answers
3k
views
References for Artin motives
I find the following description of Artin motives in Wikipedia. Since these seem to be quite related to number theory, I am interested to learn more in that context. I request the experts available in ...
- 7,442
6
votes
3
answers
601
views
Solving "a, b, a+b have given divisors" problem
I've read an interesting article, math.NT/0409456 where you're just trying to solve a simple problem:
For a given (finite) set of primes S find all solutions to an equation ...
- 15.1k