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18
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0answers
846 views

Grothendieck's letter to Faltings: On the yoga of motives and the degeneration of Leray spectral sequence

In Grothendieck's letter to Faltings, he writes There exists at this time a kind of “yoga des motifs”, which is familiar to a handful of initiates, and in some situations provides a firm support ...
15
votes
0answers
309 views

$\zeta(n)$ as a mixed Tate motive

I am trying to understand why there exists, for each $n \geq 2$, a mixed Tate motive $M$ over $\mathbb{Q}$ such that $M \in Ext^1_{MT(\mathbb{Q})}(\mathbb{Q}(0), \mathbb{Q}(n))$ and $\zeta(n)$, ...
13
votes
0answers
531 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
votes
0answers
616 views

Motivic derived algebraic geometry

In algebraic geometry one studies spaces locally modelled on commutative algebras, i.e. commutative algebra objects in the symmetric monoidal category Ab of abelian groups. Now supposedly in the ...
11
votes
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 ...
10
votes
0answers
312 views

The status of automorphic induction

Background: Let $K/F$ be a degree $r$ extension of number fields. It is conjectured that an automorphic representation of GL$_n$ associated to $K$ induces an automorphic representation of GL$_{rn}$ ...
10
votes
0answers
887 views

Are there analogues of Beilinson's conjectures for motives with coefficients?

There's a body of wisdom (following Beilinson, Bloch, Deligne, ...) relating mixed Tate motives, motivic cohomology, algebraic K-theory, special values of L-functions, and polylogarithms. My ...
9
votes
0answers
576 views

Has there been any progress on the standard conjectures on algebraic cycles?

What's the current state of these conjectures? Who is working on them? In "Standard conjectures on algebraic cycles" Grothendieck says: "They would form the basis of the so-called "theory of ...
8
votes
0answers
148 views

Stack of Tannakian categories? Galois descent?

I'm having trouble finding a reference for something that I'm guessing the experts worked out long ago. Let's take a local or global field $F$ for this post, and fix a separable algebraic closure ...
8
votes
0answers
176 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 ...
8
votes
0answers
340 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 ...
8
votes
0answers
610 views

Kato's log motives

What are they and what are their intended uses? Does anyone have notes/slides of this talk? I am curious about "log motives" because there seems to exist a "log motivic yoga" among experts in ...
7
votes
0answers
406 views

Integral decomposition of the diagonal (Chow motives)

Let $k$ be a field of characteristic zero and let $X$ be a smooth proper varity over $k$ of dimension $d$. The Künneth standard conjecture conjectures that there exist projectors $e_0, e_1, \ldots, ...
7
votes
0answers
1k views

What are “fractional motives”?

Kirti Joshi's musings mention "fractional motives". Do you know what are they good for and what the current state of constructions is for them? Edit: Further cases of "fractional motives" as ...
6
votes
0answers
154 views

Nice references for injective model structures and Quillen functors between motivic homotopy categories

It seems that for the injective model structures for the categories simplicial presheaves and simplicial Nisnevich sheaves (on the category of smooth varieties over a perfect field $k$) there exist ...
6
votes
0answers
268 views

Does one need l to be invertible in S in order to consider the l-adic cohomology of S-schemes and motives?

When Ivorra defines the $l$-adic realization of $S$-motives (i.e. of Voevodsky's motives over a scheme $S$) he demands $l$ to be invertible in $S$. Is this condition really necessary? What happens ...
6
votes
0answers
333 views

Is singular cohomology representable by a (Voevodsky's) motivic complex?

For any $c>0$ does there exist an object $C$ of Voevodsky's $DM^{eff}_-$ (over the field of complex numbers) such that for any $i\le c$ and any smooth variety $X$ the $i$-th singular cohomology of ...
5
votes
0answers
195 views

Any counterexamples known for the Generalized Tate conjecture?

One can state the generalized Tate conjecture over arbitrary finitely generated fields; to this end one should just define Galois representation to be effective if the eignevalues of the actions of ...
5
votes
0answers
486 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 ...
5
votes
0answers
681 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 ...
5
votes
0answers
199 views

Are Tate twists of t-positive motives positive with respect to the Voevodsky's homotopy t-structure?

Let $X$ be a Voevodsky's motif (over a perfect field) that belongs to the positive part of the homotopy $t$-structure (i.e. its cohomology as an object of $D^-(ShSmCor)$ is zero in negative degrees). ...
5
votes
0answers
196 views

Which pure categories related with Weil cohomology theories are semi-simple?

As far as I understand, the category of pure polarizable Hodge modules is semi-simple, whereas the cohomology of the corresponding schemes is graded polarizable. Is it true that one doesn't have any ...
5
votes
0answers
295 views

Generalized Hodge conjecture for cohomology of smooth non-proper varieties?

Is there such a statement known i.e. does there exist a conjectural description of the coniveau filtration for singular cohomology of a smooth non-proper variety over the field of complex numbers (in ...
4
votes
0answers
97 views

The 'most general' papers on rational Borel-Moore motivic homology and K'-theory?

There are two ways to define Borel-Moore motivic homology (of schemes) with rational coefficients: one should either consider certain complexes of algebraic cycles, or the $\gamma$-filtrations of ...
4
votes
0answers
149 views

Finer motivic decomposition in a bigger motivic category

In his Ph.D. thesis, Semenov shows that the motivic decomposition of a variety in general is not unique. He works in the category of Chow-Motives and not in a bigger category of motives. Is there an ...
4
votes
0answers
151 views

Is there a motive attached to half weight modular forms?

The question is kind of contained in the title but let me add a few words. If $f$ is a cusp form of weight $k$ for $SL(2, \mathbb{Z})$ then Scholl constructed a Grothendieck motive $M(f)$. In this ...
3
votes
0answers
259 views

Use of derivators to the theory of motives?

This is a rather imprecise question but i think this could become a interesting pool of ideas and comments. The theory of motives has evolved to a complex field of research the moment Voevodsky (and ...
3
votes
0answers
327 views

Two questions on motivic homotopy theory

I am a beginner in motivic homotopy theory and motivic cohomology and have just begun reading books by Dundas et al. and Mazza-Voevodsky-Weibel. I have two basic questions: Why is the question of ...
3
votes
0answers
155 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 ...
3
votes
0answers
309 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 ...
3
votes
0answers
450 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 ...
3
votes
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. ...
3
votes
0answers
314 views

Motivic interpretation of genus 2 Siegel forms induced by lifts of Maass and Skoruppa

Background: There are several known lifts from integral weight modular forms to Siegel forms of genus 2, among them the Saito-Kurokawa lift. Another lift construction that is important for ...
3
votes
0answers
571 views

Gauss--Manin connection for de Rham realisation

Let $X$ and $S$ be smooth schemes of finite type over a field $k$ and let $\pi:X\to S$ be a smooth morphism of finite type. The relative de Rham cohomology $H^i_{dR}(X/S)$ of $X$ over $S$ is a ...
2
votes
0answers
76 views

Galois group for 0-dimensional motives

$\newcommand{\M}{\mathcal{M}_0}$$\newcommand{\Q}{\mathbb{Q}}$ It is my understanding that in dimension 0, the theory of motives should just be Galois theory for fields. I am hoping to find a reference ...
2
votes
0answers
146 views

What is the relation between Beilinson's conjectures and Standard conjectures of algebraic cycles?

Do Standard conjectures on the K-theory of varieties over finite field have implications in the motivic cohomology of Z where exist the correct formalism of Beilinson's conjectures? What is the ...
2
votes
0answers
186 views

tensor of motives

Let $X$ and $Y$ be two smooth projective varieties over $k$. Consider the Chow motives $M(X\times\mathbb{P}^n)\simeq M(X)\otimes M(\mathbb{P}^n)$ and $M(Y\times\mathbb{P}^n)\simeq M(Y)\otimes ...
2
votes
0answers
274 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 ...
2
votes
0answers
170 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 ...
2
votes
0answers
324 views

A motivic complex

By definition, Voevodsky's motivic complex (an object of his $DM^{eff}_-$) is a complex of sheaves with transfers whose cohomology sheaves are homotopy invariant. Now, I consider the complex (of ...
2
votes
0answers
164 views

A Hodge substructure with ''nice' weight factors that does not correspond to a mixed submotif?

Suppose that we have a mixed motif $M$ with only two (subsequent) weights, and have a subobject for each weight factor. How we can control whether there exists a submotif of $M$ with these two ...
2
votes
0answers
130 views

On 'graded polarizable' triangulated categories; are there any mixed Galois module analogues known? Also on mixed realizations

There is a construction by Beilinson (in section 3 of "Notes on absolute Hodge cohomology") of the derived category of graded polarizable mixed Hodge complexes; he also proved that this category is ...
2
votes
0answers
154 views

Transformation rule for motivic integration: divisors or divisors classes?

In the Kontsevich's 'change of variables for motivic integral' rule does one consider any possible choices for the corresponding canonical divisors, or is it necessary to fix certain representatives ...
1
vote
0answers
66 views

relations between nori motives and pure motives

The category of Nori motives $NMM$ is defined by tools of graph category. I think one can similarly define a category $EM$ as the graph category of $P(k)$ (the graph of projective smooth k-varieties) ...
1
vote
0answers
131 views

Which “concrete” morphisms of varieties and motives induce bijections of their lower Chow groups?

This question is a continuation of Varieties with Chow groups supported in positive codimension: examples and properties? What examples are known of morphisms of varieties and Chow motives (say, over ...
1
vote
0answers
112 views

The right (not the left 'Suslin complex' one) adjoint to the embedding of $ DM^{eff} $ into $D(ShvTr) $?

Inside the derived category of Nisnevich sheaves with transfers there is the category $DM^{eff} $ of Voevodsky's effective motivic complexes (actually, Voevodsky only considered bounded above ...
1
vote
0answers
176 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 ...
1
vote
0answers
132 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 ...
1
vote
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 ...
1
vote
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 ...