# Questions tagged [motives]

for questions about motives in algebraic geometry, including constructions of categories of motives and motivic sheaves, and aspects of the standard conjectures.

454
questions

2
votes

0
answers

247
views

### Why is the weight monodromy hard in mixed characteristics?

I know very little about the conjecture, beyond Grothendieck's monodromy theorem perhaps (a dense open subgroup of inertia acting unipotently on pure motives). But I heard that it was completely ...

24
votes

2
answers

2k
views

### Foundations and contradictions of Scholze's work: the category of presentable infinity categories contains itself

Preface: I am not an expert in the work of Scholze, or anything for that matter.
Question
Has Scholze stated what axioms he is using to develop his theory of motives and analytic geometry. In the ...

2
votes

0
answers

161
views

### Reference for facts used in Bloch, "Algebraic cycles and L-functions II"

The proof of lemma 1.1 in [1] does not give references for a few statements it uses.
In the setting of the proof, $X$ is a smooth projective variety over a number field $k$ with a fixed embedding to $\...

8
votes

0
answers

321
views

### Triple comparison of cohomology in algebraic geometry

Let $X$ be a smooth proper variety over $\mathbb{Q}$ and $p$ a prime number. For an integer $k$, we have:
a finitely-generated abelian group $H^k(X^{\mathrm{an}}(\mathbb{C});\mathbb{Z})$
a finitely-...

0
votes

0
answers

118
views

### Roots of weight of a characteristic polynomial of Frobenius

We are expected to solve a conjecture of the title. Reference is Jean-Pierre Serre — Propriétés conjecturales des groupes de Galois motiviques et des représentations l-adiques.
Precisely;
Conjecture A:...

3
votes

1
answer

341
views

### Elementary questions on motives

Motives are objects that appear in algebraic geometry, which is closely related to algebraic cycles. It was considered by Grothendieck, and developed notably by Deligne, Voevodsky etc.
I have the ...

3
votes

0
answers

96
views

### Chow-Künneth conjecture and Galois base change

Consider $K'|K$ a finite galois extension of degre $m$ and galois group $G$. Recall the Chow-Künneth conjecture :
Conjecture
For any (smooth projective) variety over a field $k$ and $H$ a Weil ...

7
votes

0
answers

222
views

### How are connected the hypergeometric motives of WZ-Pairs?

If $\small{(F,G)}$ is a WZ-pair and general asymptotic conditions $\lim_{k\rightarrow\infty}\small{G(n,k)=0}$ and $\lim_{n\rightarrow\infty}\small{F(n,k)=0}$ hold, then we have the certified ...

3
votes

1
answer

334
views

### Bloch–Beilinson conjecture for varieties over function fields of positive characteristic

Is there a version of the Bloch–Beilinson conjecture for smooth projective varieties over global fields of positive characteristic? The conjecture I’m referring to is the “recurring fantasy” on page 1 ...

5
votes

0
answers

192
views

### Motivic $L$-functions came from automorphic representations

Langlands in his 1978 ICM talk made a conjecture that all motivic $L$-functions should arise as automorphic $L$-functions. A part of this conjecture, namely for some Hasse-Weil $\zeta$ functions is a ...

5
votes

1
answer

452
views

### Any "motive"(-like) theory which can catch that cusp $y^2=x^3$ (and similar) are non-trivial?

Consider cusp $y^2=x^3$ which can also be described as $k[z]~without~z$ , taking $x=z^3,y=z^2$.
Algebraically its $Spec$ is quite different from $k$. For example:
it has plenty non-trivial "line-...

12
votes

1
answer

1k
views

### Relation between motives and geometric Langlands

When working over a number field (or a function field over a finite field), one predicts that the Langlands program is related to the theory of motives over this field. There are several ways I have ...

12
votes

1
answer

2k
views

### Who proved the motivic 6-functor formalism?

In the recent beautiful talk "Motives and ring stacks" Peter Scholze states the theorem saying that there exists an initial 6-functor formalism on $\mathit{Sch}_\mathbb{Z}$ such that
when $...

1
vote

0
answers

361
views

### Amitsur's theorem for generalized Severi–Brauer varieties

Let $k$ be a field of characteristic zero and assume that $A$ is a central simple algebra of index $2^n > 2$. We denote by $\operatorname{SB}_i(A)$ the $i$-th (generalized) Severi–Brauer variety of ...

29
votes

4
answers

3k
views

### What is the status of the theory of motives?

It has been almost 60 years since Grothendieck conceived the conjectural theory of motives in order to grasp the common behavior of the most important (Weil) cohomology theories.
But what is the ...

2
votes

0
answers

141
views

### Compatibility of system of $\ell$-adic representations associated to Voevodsky motives

Let $M$ be an object of Voevodsky's category $DM_{gm}(K,\mathbb{Q})$ for a number field $K$. For each prime number $\ell$, there is an $\ell$-adic realization $M_{\ell}$ in the bounded derived ...

3
votes

0
answers

166
views

### Simplicial resolution for commutative group scheme

Let $X$ be a quasi-projective $k$-variety. In this case the symmetric power $S^d(X)$ is well-defined. If $S^\bullet(X)=\bigsqcup_{n>0}S^d(X)$, where we suppose $S^0(X)=\operatorname{spec}(k)$, then ...

2
votes

1
answer

213
views

### Removing quasi-projective assumption in the formalism of four operations

In Ayoub's thesis, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (I), Ayoub proved that given a stable homotopical $2$-functor (Definition 1.4.1) $\...

3
votes

0
answers

177
views

### Direct images commute with homotopy colimits

In Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (II), Ayoub defined the notion of a stable homotopical algebraic derivators; roughly, for a ...

6
votes

0
answers

251
views

### Correspondence between motives and automorphic representations

What I know:
I understand motives via its realization; in Coates' and Perrin-Riou's paper On $p$-adic L-functions Attached to Motives over $\mathbb{Q}$ (see http://doi.org/10.2969/aspm/01710023), the ...

2
votes

1
answer

241
views

### Motives of resolutions of singularities

Suppose $X'$ is a resolution of singularities of a projective variety $X$ over a field $k$ of characteristic 0 that is functorial for smooth morphisms.
How are the (mixed) motives of $X$ related to (...

3
votes

0
answers

142
views

### Grothendieck ring of varieties in positive characteristic, away from the characteristic

In "The universal Euler characteristic for varieties of characteristic zero", Bittner shows that over a field $k$ of characteristic zero, the Grothendieck ring $K_{0}(Var_{k})$ of varieties ...

3
votes

0
answers

156
views

### Applications of the theory of derivators to constructing cone functors

One of main reasons that the theory of derivators was introduced is to fix the non-functoriality of the cone construction of triangulated categories. I know that today derivator theory is broad and ...

2
votes

0
answers

155
views

### Map between Mordell-Weil group and Ext of (Mixed) Motives

We know that the motivic cohomology of an abelian variety $A$ over a number field $k$ computes the Mordell-Weil group up to torsion, and so if we were to grant the existence and nice behaviour of ...

3
votes

0
answers

136
views

### A Galois equivariant Weil cohomology theory with coefficients in the rational numbers and a variation of the Tate/Hodge conjecture

A well-known example of Serre shows that there can be no Weil cohomology theory with $\mathbb Q$ coefficients for schemes over $\mathbb F_{p^2}$. However, this example is no obstruction to a Weil ...

8
votes

0
answers

522
views

### Has Grothendieck's motivic vision been realised?

Apparently (https://twitter.com/stewartbrand/status/1635057392814821376) Bing's AI search thinks that "the full theory of motives remains elusive". My impression was that the current ...

6
votes

0
answers

246
views

### Rank $2$ motivic local systems on a curve

This question is about the article "Motivic local systems on curves and Maeda's conjecture" by Yeuk Hay Joshua Lam.
In the proof of Theorem 1.1 it is claimed (on lines 4-5 of p. 7) that any ...

4
votes

0
answers

201
views

### What are the modularity conjectures for Artin motives?

Classically, singular cohomology is an important tool for studying topological spaces, in particular, complex varieties. In the mid-twentieth century it was realized that there are many analogues of ...

6
votes

0
answers

215
views

### Motives in tropical geometry

Is there a notion of motives in tropical geometry? Similar like the notion introduced by Grothendieck in algebraic geometry.

2
votes

0
answers

1k
views

### When is a subspace of the cohomology of a smooth projective scheme on $k$ a motive?

Let $X$ be a smooth projective scheme over a number field $k$, and $V_{p}$ (resp. $V_{\text{dR}}, V_{\text{B}}$) a sub-space of $H_{et,p}^{\ast}(X)$ (resp. $H^{\ast}_{\text{dR}}(X), H^{\ast}_{\text{B} ...

2
votes

0
answers

92
views

### Manifestation of Hecke operator on the category of abelian varieties (or motives)

If we are given some postivie integer $N$ and a prime $p$, then we have the Hecke operator $T_p$ on modular forms, which is a cohomological manifestation of the Hecke correspondence
$$X_0(N)\leftarrow ...

3
votes

2
answers

423
views

### Functor between categories of motives

$\DeclareMathOperator\Var{Var}\DeclareMathOperator\Motives{Motives}$Let us assume for the moment that we have a "nice" category of motives, that is for fields $k$ we have a contravariant ...

4
votes

0
answers

418
views

### In which "sense" unramified Milnor-Witt K-groups are unramified

Let $X$ be an integral locally noetherian smooth
scheme over base field $k$. Then for every $x \in X^{(1)}$ point of codimension $1$, the stalk $\mathcal{O}_{X,x}$ is a discrete valuation
ring. ...

12
votes

2
answers

1k
views

### What exactly do the standard conjectures in characteristic zero refer to?

As the title suggest it seems standard conjectures mean different things depending on the context. I had the impression that in characteristic 0 they are a list of conjectures about varieties over an ...

7
votes

0
answers

174
views

### Has anyone written about filtered Tannakian categories?

tl;dr Is there any source that discusses the concept of a filtered Tannakian category? I'm writing a paper with this notion and want to know if it's ever been discussed.
The original book by Saavedra-...

3
votes

0
answers

172
views

### Boundedness indices in Voevodsky's smash nilpotence conjecture in family

Let $X$ be a smooth projective variety over an algebraically closed field $k$. Voevodsky introduced the following notion : an algebraic cycle $Z$ in $X$ is smash nilpotent if there exist $N>0$ such ...

7
votes

1
answer

497
views

### Is there any theory of "étale cohomology" with algebraic coefficients?

For simplicity, I will restrict attention to untwisted coefficients.
Let $k$ be a finite field of characteristic $p$, and $\ell\ne p$ prime. Can one define a cohomology theory with $\mathbb{Q}_\ell^{\...

4
votes

0
answers

333
views

### Voevodsky's motives and Deligne's systems of realizations

$\newcommand{\gm}{\mathrm{gm}}$Let $\mathbf{DM}_{\gm}(\mathbb{Q},\mathbb{Z})$ be Voevodsky's category of geometric motives over $\mathbb{Q}$ with coefficients in $\mathbb{Z}$ (e.g. as on p.124 of ...

7
votes

2
answers

531
views

### Finite generation of motivic cohomology of number fields

Let $F$ be a number field ($F=\mathbb Q$ is fine for my purposes) and let $n\geq2$ be an integer. Is it known whether the first motivic cohomology groups
$$\mathrm H^1(\mathrm{Spec}(F),\mathbb Z(n))$$
...

5
votes

1
answer

230
views

### Motive associated to a cuspidal representation of $GSp_{4}$

In the paper by L.Clozel in this book (a French text), there is this conjecture (conjecture 4.5 p139)
Conjecture: Given $\pi$ an algebraic cuspidal representation of $Gl(n)$ of weight $w$ and denote ...

5
votes

0
answers

129
views

### Examples of comonoids (coalgebras) in the stable homotopy category $\mathbf{SH}$

My question is both for the topological and for the algebraic/motivic version of the stable homotopy category $\mathbf{SH}$.
It is well known that most cohomologies are represented in $\mathbf{SH}$ by ...

4
votes

1
answer

505
views

### What is motivic sheaf intuitively?

I am not very familiar with motif theory, but I do know a little about Hodge theory.
I view (mixed) motif theory as an enhancement of (mixed) Hodge structures.
Q1. Is (mixed) motivic sheaf theory an ...

5
votes

1
answer

380
views

### Which varieties are sums of tensor powers of the Lefschetz motive?

Any smooth projective variety $X$ gives an object $h(X)$ in the category of pure Chow motives. If $X$ is a generalized flag variety, i.e. a quotient $G/P$ where $G$ is semisimple linear algebraic ...

23
votes

1
answer

1k
views

### Can we state the Riemann Hypothesis part of the Weil conjectures directly in terms of the count of points?

For algebraic curves we can state the Riemann hypothesis part of the Weil conjectures directly as a formula for the number of points on the curves, sidestepping the zeta function. Namely, given a ...

4
votes

1
answer

230
views

### Functoriality conjectures on the slice filtration

Voevodsky wrote on his paper "Open Problems in the Motivic Stable Homotopy Theory, I" that
Three other groups of conjectures in motivic homotopy theory, not included in to this paper, seem ...

3
votes

1
answer

515
views

### Homotopy invariance of $\ell$-adic cohomology

In the end of the Voevodsky’s lectures on cross functors, P. Deligne considers a couple of axioms which define (using the vocabulary of Ayoub's thesis) a stable homotopical 2-functor. Among them, we ...

1
vote

0
answers

78
views

### Localization with or without transfers

Let $Sh_{Nis}^{tr}$ be the category of Nisnevich sheaves with transfers of abelian groups over a perfect field. Let $u\colon Sh_{Nis}^{tr}\to Sh_{Nis}$ be the functor “forget transfers” and let $h_0^{\...

9
votes

1
answer

455
views

### Why is the category of motives generated by varieties?

I'm reading Ayoub's paper Motifs des varietes analytiques rigides, but I'm not quite familiar with motives. In this paper, he defines the category of motives to be $\mathbf{RigDM}^{\rm eff}_{\rm Nis}(...

3
votes

0
answers

395
views

### Stable $\infty$-category of motives

In nLab motive, it defines the derived category of motives as the full sub-$\infty$-category of the $\infty$-category of functors $\mathop{\mathrm{Fun}}(\mathrm{Cor}_k^{\mathrm{op}}, \mathcal S)$ ...

6
votes

1
answer

365
views

### Nisnevich topology inspired by Adeles

I'm quite a newbe in the field of motives & A1 homotopy theory,
so please forgive me if the question is too elementary:
In the intro from wikipedia on Nisnevish topology
is remarked that it's ...