**5**

votes

**1**answer

207 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

**2**answers

500 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.

**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 ...

**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 $[S] \in \mathbb{Z}[\mathbb{L}]$,...

**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 ...

**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 ...

**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 ...

**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 ...

**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

**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 $\...

**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 ...

**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) $...

**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 ...

**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, ...

**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 "...

**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 ...

**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)?

**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 $H^i(X, \overline{\mathbb Q}_\ell (j))$ ...

**5**

votes

**0**answers

222 views

### Hyperplane sections of principal homogeneous spaces

Let $P_i$ denote the $i$-th vertex in the Dynkin diagramm of an algebraic group. It symbolizes a parabolic subgroup of $G$ corresponding to the other vertices, meaning $G/P_i$ is a smooth, projective, ...

**1**

vote

**1**answer

160 views

### Chow groups of locally trivial affine fibrations

Let $X$ be an algebraic variety over an algebraically closed field $k$ of characteristic $0$.
A locally trivial $\mathbf{A}^n$-fibration is a morphism $\pi \colon Y \to X$ such that $\pi^{-1}(U)\cong ...

**2**

votes

**0**answers

312 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 $\...

**5**

votes

**1**answer

269 views

### Motives of a variety of type D4

Over the last decade Nikita Semenov, Skip Garibaldi and others have made some progress in the theory of cohomological invariants, (Rost)-motives and motivic decompositions of algebraic groups. For ...

**1**

vote

**1**answer

196 views

### motive of the general linear group

Let $k$ be a perfect field. Let $GL_n$ be the general linear group over $k$. Does anybody know a reference for the computation of the motive
$$
M(GL_n)
$$ in Voevodsky's category $DM(k)$?

**5**

votes

**2**answers

432 views

### Should the Grothendieck ring of varieties be K_0 of numerical motives?

Assuming the Standard Conjectures, should the Grothendieck ring of varieties be the $K_0$ of the abelian category of numerical motives?

**6**

votes

**1**answer

262 views

### Is Scholl construction of modular motives related to Deligne's construction of $\ell$-adic representations?

first of all, I need to declare my extreme ignorance on the topic of modular forms, so , please, does not assume that I know Deligne's construction in details.
In https://www.dpmms.cam.ac.uk/~ajs1005/...

**8**

votes

**0**answers

258 views

### Geometric vs combinatorial motives over Spec Z

Consider the category of reduced schemes of finite type over $\mathbb{Z}$. Take the Grothendieck group of this category, i.e. the free abelian group on isomorphism classes, modulo the usual "syzygy" ...

**24**

votes

**2**answers

1k views

### On Grothendieck's idea on his Standard Conjecture B

Let me recall the Standard Conjecture B (see [1,2] below):
The $\Lambda$-operation of Hodge theory is algebraic.
It more or less says that the partial inverse to “cupping with the class of a ...

**9**

votes

**1**answer

435 views

### Why do we need localization by Leftschetz motive?

Definition of the Grothendieck group and Leftschetz motive. The Grothendieck group of varieties is a free abelian group generated by classes of algebraic varieties with the following relation:
$$
[X]=[...

**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 ...

**1**

vote

**1**answer

275 views

### Splitting varieties of two Galois cohomology symbols

One characteristic property of the so called norm varieties defined by Rost, is that they split pure symbols of Galois cohomology groups $H^n(k,\mu_p)$, meaning:
For some $\alpha \in H^n(k,\mu_p)$ ...

**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 ...

**2**

votes

**1**answer

425 views

### Algebraic equivalence vs linear equivalence

Maybe the question is too general, but nevertheless:
under what conditions on algebraic variety $X$, algebraic equivalence of divisors coincide with linear equivalence?
What are typical classes of ...

**3**

votes

**1**answer

284 views

### Etale Realization and Gysin Sequence

Ivorra defined a tensor triangulated functor from Voevodsky's triangulated category of motives to the derived category of complexes of etale sheaves of $\mathbb{Z}/n$ modules with bounded cohomology ...

**12**

votes

**2**answers

656 views

### Are there known cases of the Mumford–Tate conjecture that do not use Abelian varieties?

(For a formulation of the Mumford–Tate conjecture, see below.)
The question
As far as I know, all non-trivial known cases of the Mumford–Tate conjecture more or less depend on the Mumford–Tate ...

**4**

votes

**0**answers

144 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 ...

**7**

votes

**0**answers

419 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 ...

**12**

votes

**2**answers

2k views

### Mysterious quotes (at least for me)

I heard two quotes, one from Alain Connes and an other one from Orlov.
Alain Connes was talking about noncommutative geometry and he said the following:
" a noncommutative algebra creates its own ...

**13**

votes

**3**answers

915 views

### why are motives more serious than “naive” motives?

I know my question is a bit vague, sorry for this.
Let $k$ be a field of characteristic zero. Consider the Grothendieck ring of varieties over $k$, usually denoted by $K_0(Var_k)$. This is generated ...

**5**

votes

**0**answers

230 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 ...