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0answers
174 views

Virtual Lefschetz motive

Hi there, I have a question which popped up while reading papers on motives. Let $V_k$ be the category of (projective) k-varieties, and let $K_0(V_k)$ be the Grothendieck ring of $V_k$; then ...
3
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1answer
447 views

“Motivic structure on higher homotopy of non-nilpotent spaces” ?

Has anyone an idea where one can read more about Deepam Patel's talk "Motivic structure on higher homotopy of non-nilpotent spaces" http://www.ihes.fr/~abbes/SGA/patel.html ? Edit/Answer: The video ...
21
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1answer
693 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 ...
9
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2answers
877 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 ...
1
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0answers
127 views

Chow-Künneth decomposition for hypersurfaces

Short version: is the Chow-Künneth motivic decomposition known for $X \hookrightarrow \mathbb{P}^n_k$ a hypersurface over a field $k$? Long version: let $M(X)$ be the Chow motive of $X$ with ...
3
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2answers
221 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 ...
5
votes
1answer
439 views

Analogue of Tate or Hodge conjecture for varieties over $\Q_p$

I've been learning about p-adic Hodge theory recently (I'm a beginner), and I've been wondering about the following question the past couple of weeks. Sorry for the long setup, it's mainly ...
3
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0answers
153 views

Correspondences of schemes induced by a finite Galois extension

I am trying to do the first exercises of the Mazza-Voevodsky-Weibel book Lecture Notes on Motivic Cohomology, which are some elementary results about correspondences between schemes. Recall that if ...
0
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1answer
166 views

Pull-back of algebraic cycles under holomorphic maps

Let $f:X \to Y$ be a holomorphic map between two smooth complex projective manifolds. Is there a good notion of pull-back of algebraic cycles by $f$ which preserves degree in the following sense: ...
0
votes
1answer
250 views

Realisation functor

Let k be a field. Is there a realization functor $DM_{gm}(k,\mathbb{Z}/n)^{op} \to D^b_c(k, \mathbb{Z}/n)$ from category of motives to category of complexes of étale sheaves of $\mathbb{Z}/n$ ...
1
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1answer
175 views

Definition of Hurewicz map relating $SH(k)$ with $DM_\_^{eff}(k)$

In "Motivic Homotopy Theory" on page 153 it is stated that there exists a canonical Hurewicz map relating the motivic stable homotopy category with the category $\operatorname{DM}_\_^{eff}(k)$. ...
4
votes
1answer
504 views

Should the etale cohomology of a smooth projective variety (over rationals) be semi-simple; why?

$\DeclareMathOperator{\char}{char}\DeclareMathOperator{\gal}{Gal}$ Let $P$ be a smooth projective variety over a field $K$ (one may certainly assume that $K$ is perfect; the case $K=\mathbb{Q}$ ...
7
votes
1answer
314 views

Ring structure for the motivic spectrum/complex that represents singular cohomology?

As the discussion here Is singular cohomology representable by a (Voevodsky's) motivic complex? shows, the singular cohomology of (smooth) complex varieties is represented by a motivic complex ...
8
votes
0answers
174 views

Corresponding notion of unramified for motives (or de Rham cohomology)

The etale cohomolgoy of a variety $X$ over a number field $K$ is a Galois representation of $\mathrm{Gal}(\overline K/K)$ with some properties coming from $X$, e.g., it is unramified outside $S$ if ...
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0answers
162 views

chow kunneth motivic decomposition for dummies

Hi everybody, I have just realized that, if I had to explain Chow-Kunneth decomposition for Chow motives to someone who is a newbie of the subject, I would hardly find a way which is far from the ...
2
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0answers
273 views

Voevodsky's 'split standard triple' argument: an explanation; does it work with $Z/nZ$-coefficients?

For varieties over a perfect field (of some characteristic $p$ that could be 0) Voevodsky defines the notion of a 'standard triple' (see ...
3
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0answers
303 views

Good morphisms of distinguished triangles: can Neeman's method be applied to the motivic stable homotopy category?

It is well known that non-uniqueness of a cone for a morphism in a triangulated category $C$ makes constructing exact functors (of triangulated categories) a difficult task. In section 3 of his "Some ...
7
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1answer
303 views

The vanishing of $MGL^{2n+i,n}(X)$; do spectra of smooth projective varieties generate $SH_{l}$?

I have two questions related to the stable motivic homotopy categories of Morel-Voevodsky. The first is probably simple; I wonder what is known on the second one. For the algebraic cobordism theory ...
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1answer
403 views

Can there exist Chow motives/motivic cohomology for compact Kähler manifolds?

Can there exist a 'reasonable' extension of the (higher) Chow groups of complex smooth projective algebraic varieties to functors on the category of compact Kähler manifolds? Are there any ...
4
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1answer
701 views

(Mixed) Tate motives

Hi there, in recent times I was reading texts about motives, and I want to ask something about Tate motives which is not clear to me (as I came across different definitions in different texts). Let ...
2
votes
1answer
281 views

Chow group of a (particular) motive [+ reference request]

I have two (not unrelated) questions. Let me first give a short introduction. Introduction For a general overview of the setup I refer to the introduction (§1) of [Zhang]. Let $k$ be a number field ...
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2answers
622 views

Motivic generalisation of Neron-Ogg-Shaferevich criterion

Given a variety $X$ over $\mathbb{Q}$ with good reduction at $p$, proper smooth base change tells us that its $l$-adic cohomology groups are unramified at $p$ (and I'd guess some $p$-adic Hodge theory ...
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1answer
616 views

Grothendieck ring of “varieties carrying a function”

Fix a base ring $R$, and consider pairs $(X,f)$ where $X$ is a scheme of finite type over $R$ and $f:X\to R$ is an $R$-valued algebraic (not constructible!) function on $X$. I want to consider a ...
6
votes
1answer
366 views

motivic t-structure and realisations

Let $k$ be a field and $DM_k$ denote the triangulated category of geometric motives with $ \mathbb{Q}$ coeffients over $k$. Recall that there exists a motive functor $M: Var_k\rightarrow DM_k$, which ...
39
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1answer
2k views

What is the relationship between motivic cohomology and the theory of motives?

I will begin by giving a rough sketch of my understanding of motives. In many expositions about motives (for example, www.jmilne.org/math/xnotes/MOT102.pdf), the category of motives is defined to be ...
6
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1answer
577 views

Motivic proof of Weil-conjectures?

Assuming the standard conjectures (and whatever is needed in addition), is there a nice proof of the Weil-conjectures written completely in the language of motives?
3
votes
1answer
405 views

Realization of Voevodsky Motives over a perfect field in mixed categories.

Let $k$ be a finite field and $l$ be different from characteristic of $k$. Is there a realization functor from the Voevodsky's category with $\mathbb Q$ coefficients to the constructible mixed étale ...
3
votes
1answer
358 views

A question about the Tannakian etale fundamental group of a curve

Let $X$ be a smooth connected quasi-projective curve over $\mathbf{Q}$. Let $U$ be the pro-unipotent etale fundamental group of $X$ over $\mathbf{Q}_p$. $U^1 = U$ and let $U^n =[U,U^{n-1}]$. Let ...
16
votes
1answer
862 views

Connes-Kreimer Hopf algebra and cosmic Galois group

Hi, I'm interested in the relation between the two following constructions motivated by renormalization: Connes-Kreimer gives an interpretation of the renormalization procedure in the framework of ...
2
votes
0answers
168 views

CM abelian variety from an algebraic Hecke character?

Hi, Given an algebraic Hecke character $\chi$ of a number field $k$ there should be a "rank 1 CM-motive" $M$ with $\overline Q$-coefficients such that $L(s,M) = L(s,\chi)$. This follows from the ...
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0answers
511 views

Status of conjectures in Serre's 1969 expose on Galois representations on l-adic cohomology

In [S]: Serre, Jean-Pierre. Facteurs locaux des fonctions zeta des varietes algebriques (definitions et conjectures), Seminaire Delange-Pisot-Poitou, 1969-70 Serre presents nine conjectures ...
11
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1answer
631 views

Motivic cohomology vs. K-theory for singular varieties

As far as I understand, for a smooth variety $X$ its motivic cohomology could be described as the corresponding piece of the $\gamma$-filtration of (Quillen's) $K^*(X)$; this is completely true for ...
3
votes
0answers
448 views

Weak Lefschetz for torsion motivic cohomology (or torsion Chow groups)?

I am trying to prove the following Weak-Lefschetz-type statement for motivic cohomology: if $Z$ is a smooth hyperplane section of a smooth projective $X$ of dimension $d$ (say, over the field of ...
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3answers
754 views

The historical development of automorphic geometry

Background: Today the notion of automorphic geometry is often framed in the context of the Langlands program, in particular what is sometimes called the Langlands reciprocity conjecture. This is ...
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1answer
1k views

Voevodsky's counterexample to the existence of a motivic t-structure

I have been trying to unravel some of the known relationships between various ideas on mixed motives. I find the litterature quite hard to follow -"from experts, for experts". Voevodsky in ...
5
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0answers
482 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
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1answer
531 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 ...
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0answers
335 views

General cohomology groups and motives

Let $X$ be a variety over $\mathbb{Q}$. Let $\mathcal{F}$ be a sheaf on $X$. Then we have an action of $Gal(\mathbb{Q})$ on $H_{et}^i(X,\mathcal{F})$. In certain cases we can say a lot about this ...
11
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0answers
642 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 ...
5
votes
0answers
674 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 ...
2
votes
1answer
411 views

Hilbert's 3rd problem,number theory, motives, cyclic homology,…

This talk by Jinhyun Park connects a lot of interesting themes, making me curious to read more about that. Do you know where?
8
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0answers
336 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 ...
6
votes
2answers
432 views

Decomposition of Motives of cellular varieties

Dear community, in his 2005 Inventiones Paper "On motivic decompositions arising from the method of Białynicki-Birula" P. Brosnan deduced from the classical (?) theorem of Bialynicki-Birula on ...
3
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0answers
145 views

The vanishing of homotopy invariant $K$-theory of dg-categories

In my previous question The vanishing of non-connective K-theory in negative degrees I asked when one can be sure that the negative non-connective $K$-groups of a differential graded category vanish. ...
5
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2answers
545 views

Is there for every variety X an abelian variety A such that their 1st l-adic cohomologies are isomorphic?

This question is somewhat inspired by Kevin Buzzard's answer to What is the interpretation of complex multiplication in terms of Langlands? and somewhat from my own curiosity about such topics. Let ...
3
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1answer
692 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 ...
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1answer
639 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) ...
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2answers
994 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 ...
1
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1answer
579 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 $\mathbb{Q}_l$. ...
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0answers
225 views

(why) Are the following two constructions of zeta functions equal?

Let $X$ be a variety defined over $\mathbb{Q}$. One has the usual Hasse-Weil zeta function. Now, let's do a different construction. Base change $X$ to $\mathbb{C}$: $X_{\mathbb{C}}$. Now look at its ...