**5**

votes

**1**answer

208 views

### Constructing groups of Type E7 with certain Tits Index

In a new survey on $E_8$, namely
Skip Garibaldi - E8 the most exceptional group
, the author gives an example (Example 8.4., page 15) on how to construct a group of type E8 with a prescribed Tits-...

**5**

votes

**3**answers

502 views

### References - Voevodsky motives are the derived category of Nori motives?

First I would like to know if this has been worked out, and if the answer is affirmative I would like to know some references.

**10**

votes

**2**answers

1k 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 ...

**12**

votes

**1**answer

498 views

### Is there a yoga of effectivity for motives and their realizations?

Inside the 'usual' categories of motives (Chow, Voevodsky ones) there are the categories of effective motives. Similarly, there are effective Hodge structures; they seem to be closely related with ...

**23**

votes

**1**answer

760 views

### Big list - Equivalent descriptions of Hodge conjecture?

I would like to know equivalent descriptions of the Hodge conjecture (with references).
Dan Freed's Version:
Consider a topological cycle (boundary less chains that are free to deform) on a ...

**3**

votes

**0**answers

100 views

### Non-multiplicative Euler-Poincaré Characteristics

Are there known examples of a non-multiplicative Euler-Poincaré characteristic on varieties?
Let $\mathbf{Var}/k$ be the category of varieties over a filed $k$, i.e. the category of reduced separated ...

**14**

votes

**1**answer

757 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 ...

**27**

votes

**3**answers

1k 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 $\zeta(n)$, ...

**4**

votes

**0**answers

96 views

### Polynomially countable varieties and virtual mixed Tate motives

Let $K_0(Var_k)$ be the Grothendieck ring of $k$-varieties for a field $k$. Let $\mathbb{L}$ denote the class of the affine line over $k$. Let $S$ be a $k$-variety and $[S] \in \mathbb{Z}[\mathbb{L}]$,...

**15**

votes

**4**answers

983 views

### Applications of homotopy purity theorem of Morel-Voevodsky

One of the most important theorems in motivic homotopy theory is the homotopy purity theorem of Morel-Voevodsky which says that the motivic Thom space of the normal bundle $\mathcal N_{Z/X}$ of a ...

**3**

votes

**0**answers

173 views

### Passing motivic decompositions from rational to algebraic equivalence

It is well known that there are several adequate equivalence relations for algebraic cycles (see https://en.wikipedia.org/wiki/Adequate_equivalence_relation for a list including definitions).
The ...

**19**

votes

**1**answer

520 views

### Why is the CM closure of $\mathbb{Q}$ the “ultimate” coefficient field for motives?

In a rough way, a category of motives over a field $k$ with coefficients in a field $K$ gives a universal cohomology theory with coefficients in $K$ for algebraic varieties defined over $k$. I had the ...

**11**

votes

**2**answers

436 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 $...

**5**

votes

**1**answer

217 views

### Intuition behind the definition of finite correspondences

Finite correspondences were introduced by Suslin-Voevodsky (if I am not wrong) to define motivic complexes that compute motivic cohomology. Let $X$ and be smooth separated schemes of finite type over ...

**5**

votes

**0**answers

227 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 $...

**2**

votes

**0**answers

142 views

### Does the numerical equivalence relation coincide with the homological one for 1-cycles (in positive characteristic)?

Is the Grothendieck's Standard Conjecture D (stating that the numerical equivalence relation for algebraic cycles with rational coefficients coincides with the homological one) known to be true for ...

**8**

votes

**0**answers

187 views

### Motivic cohomology of a point

I was wondering how much is known about the integral motivic cohomology groups of $\mathrm{Spec}\, k$, $H^{n,p}_{\mathrm{mot}}(\mathrm{Spec}\, k,\mathbb{Z})$. One knows that $H^{n,n}_{\mathrm{mot}}(\...

**1**

vote

**0**answers

92 views

### On “splitting off small weights” from Chow motives

I am interested in certain properties Chow motives that seem to be (more or less) "classical" (so, I mostly need references; the base field may be assumed to be algebraically closed).
So, consider ...

**7**

votes

**0**answers

111 views

### Can the failure of the multiplicativity of archimedean L-factors be corrected?

My question is parallel to J. Borger' question:
Can the failure of the multiplicativity of Euler factors at bad primes be corrected?
As emphasized by Scholbach in his paper on special values of L-...

**8**

votes

**2**answers

377 views

### Reference for Nori motives

I would like to study Nori motives and I am a complete outsider of the subject. I do, however, have background on Chow motives, Voevodsky motives $\mathrm{DM}$ and his stable homotopy category $\...

**5**

votes

**2**answers

943 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 ...

**7**

votes

**0**answers

219 views

### Quadratic twists of 1-motives

Quadratic twists of elliptic curves (or, more generally, abelian varieties) are familiar objects in arithmetic geometry. I would like to extend that definition to the category of 1-motives over global ...

**26**

votes

**4**answers

3k views

### difference between equivalence relations on algebraic cycles

For the definitions of the equivalence relations on algebraic cycles see http://en.wikipedia.org/wiki/Adequate_equivalence_relation.
I want to know how far away from each other the equivalence ...

**17**

votes

**0**answers

317 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 ...

**4**

votes

**0**answers

166 views

### Derived equivalent varieties with differing integral Mukai-Hodge structures?

For a smooth projective complex variety $X$ of dimension $n$, let $H^i(X)$ denote its integral Hodge structure of weight $i$. Define $\tilde{ H^0}(X) = \bigoplus H^{2i}(X)\otimes \Bbb Z(i)$ and $\...

**5**

votes

**1**answer

235 views

### Generalized Euler characteristics of non-motivic origin

By a generalized Euler characteristic $\chi$, I mean an isomorphism invariant $\chi(V)$ inside some abelian group $A$, defined for every varietiy $V$ over a field $k$, with the property that, for all ...

**10**

votes

**1**answer

432 views

### Does the Grothendieck ring of varieties contain torsion?

Let $K_0(Var_k)$ be the abelian group generated by the isomorphism classes of varieties over the field $k$ with the relations
$$[X]=[U]+[X\setminus U]$$
for every variety $X$ and open subvariety $U$.
...

**8**

votes

**1**answer

181 views

### Realization Functor From $SH$ to Derived Category of $Gal$-Modules

Let $k$ be a field. I would like a reference for realization functors from Morel-Voevodksy's stable category $SH(k)$ to the derived categories of $Gal(\bar{k}/k)$-modules. Has something like this been ...

**3**

votes

**1**answer

189 views

### About the decomposition of a Chow group of a variety

I would like that someone helps me to find an article on the net treating the following decomposition : $ \mathrm{CH}^k (X)_{ \mathbb{Q} } = \displaystyle\bigoplus_{ i + j = k } \mathrm{CH}^{i,j} (X) $...

**22**

votes

**1**answer

2k views

### 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 ...

**11**

votes

**1**answer

285 views

### Non-algebraic Hecke characters

Algebraic Hecke characters are ubiquitous in modern number theory. They are in 1-1 correspondence with one dimensional complex Galois representations, and in some precise sense they are the building ...

**7**

votes

**2**answers

574 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?

**15**

votes

**1**answer

542 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 ...

**13**

votes

**2**answers

618 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$_{rn}$ ...

**2**

votes

**0**answers

127 views

### Residual Representation of a Motive

Suppose we have $M$ a hypergeometric motive, and $\rho$ its associated Galois rep over $\mathbb{Q}_{l}$. Is there any easy/concrete way to find $\bar{\rho}$, the residual representation at a prime (in ...

**4**

votes

**0**answers

91 views

### Norm variety for n=5, p=2 not isomorphic to a quadric

In the paper "Motivic construction of cohomological invariants", the author displays a list of known norm varieties for several $n,p$ on page $11$. For $p=2, n=5$ it says that a norm variety is given ...

**20**

votes

**1**answer

524 views

### Why would one “attempt” to define points of a motive as $\operatorname{Ext}^1(\mathbb{Q}(0),M)$?

I'm a novice when it comes to motives. (I've read multiple introductory texts.)
I'm attempting to read Galois Theory and Diophantine geometry by Minhyong Kim. In it, he says that "One might attempt, ...

**4**

votes

**1**answer

667 views

### Grothendieck's letter to Serre on the Standard Conjectures

Is it the letter dated in 27/08/1965 of Grothendieck where he presents to Serre the Standard conjectures on algebraic cycles?

**6**

votes

**1**answer

751 views

### Hodge Standard Conjecture in Positive Characteristic

In the Wikipedia article on the Hodge Standard Conjecture it is written (note: it has since been fixed):
In characteristic zero the Hodge standard conjecture holds, being a consequence of Hodge ...

**8**

votes

**1**answer

287 views

### Motivic cohomology and pushforward maps

I have a question about pushforward maps for the motivic cohomology groups $H^p(X, \mathbf{Q}(q))$, for $X$ a smooth variety over a characteristic 0 field.
According to Mazza--Voevodsky--Weibel "...

**19**

votes

**1**answer

717 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 *C*$_1$...

**1**

vote

**1**answer

278 views

### Rost-Motive for n > 2

Have a look on the paper
F. Morel, Voevodsky's proof of Milnor's conjecture, Bull. Amer. Math. Soc. 35 (1998), 123-143, doi:10.1090/S0273-0979-98-00745-9.
and go to example 6.5 please.
In this ...

**6**

votes

**0**answers

287 views

### Motivic fundamental group of the moduli space of curves?

Suppose I have a smooth projective family of varieties of varieties over $\mathcal M_g$ - i.e. a universal functor, commuting with deformations, from curves to smooth projective varieties. Can I ...

**5**

votes

**1**answer

337 views

### “Weight-monodromy” for open varieties

Suppose that $X/\mathbb{Q}_p$ is a smooth, projective variety, and choose a prime $\ell\neq p$. Then the weight-monodromy conjecture says that the graded pieces $\mathrm{Gr}_k^M$ of the monodromy ...

**6**

votes

**1**answer

463 views

### Pure motives and compatible systems of $\ell$-adic representations

I am trying to understand the statement of the conjectures of Deligne on special values of certain $L$-functions, from his article titled, "Valuers de Fonctions L et periodes d'integrales" which ...

**1**

vote

**0**answers

408 views

### Are there connections between Homotopy type theory and Grothendieck's theory of motives? [closed]

Are there any "visions" (maybe "dreams"), future plans or connections between Homotopy type theory and Grothendieck's theory of motives (or at least "connections" with universal cohomology theory)?

**4**

votes

**1**answer

312 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 M(\...

**14**

votes

**4**answers

840 views

### Number of $\mathbb F_p$ points constant mod $p$?

I have some affine varieties $X$ defined over $\mathbb Z$, and associated integers $c(X)$, with the property that $\# X_{\mathbb Z/p} \equiv c(X) \bmod p$ for all $p$. (In particular $c(X)$ is usually ...

**10**

votes

**1**answer

481 views

### Which degree does a motivic Galois representation show up in?

Consider a representation $\rho: \operatorname{Gal} (\overline{\mathbb Q} | \mathbb Q ) \to GL_n ( \overline{\mathbb Q}_\ell)$ that is a subrepresentation of $H^i(X, \overline{\mathbb Q}_\ell (j))$ ...