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.
459 questions
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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 ...
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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?
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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 ...
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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$_{rn}$ ...
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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 ...
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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 ...
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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?
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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 "...
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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*$_1$...
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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 ...
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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 ...
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"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 ...
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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 ...
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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(\...
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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 ...
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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 $H^i(X, \overline{\mathbb Q}_\ell (j))$ ...
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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, ...
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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 ...
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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 $\...
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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 ...
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Rost Correspondence and minimal Pfister-Neighbors
In http://www.math.uiuc.edu/K-theory/0357/ Karpenko utters the following:
Conjecture 1.6. If an anisotropic quadric $X = Q$ possesses a Rost correspondence,
then the quadratic form (defining $X$) ...
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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 $DM(k)$?
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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?
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Equivariant motivic sheaves
Thanks to the work of Cisinski-Deglise: http://arxiv.org/abs/0912.2110, we now have a triangulated category of `motivic sheaves' available that admits the standard yoga of the six functors.
Is there ...
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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 ...
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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" ...
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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]=[...
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Analogue of Tate or Hodge conjecture for varieties over $\mathbb 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 background;...
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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)$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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About embedding pure motives into the triangulated category of mixed motives and some further questions about motivic cohomology
I have tried reading some texts about motives, mainly motivic cohomology (Bloch's " Lectures on Algebraic Cycles", Voevodsky's "Motivic Cohomology" etc). However some things confused me. I don't know ...
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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 ...
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What can one say about zero-cycle groups for products of Chow motives
What can one say about the Chow group of zero-cycles (up to rational equivalence) for a product of smooth projective varieties and Chow motives (so, I am interested in the kernel $Chow_0(P)\otimes ...
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Are Anderson $T$-motives motives for the function field analogy?
this question is related to this one Geometry for Anderson's motives?, though the previous one doesn't answer exactly my question.
Let $\mathbb{C}_{\infty}$ be the function field analog of $\...
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Stable motivic cohomology with finite coefficients?
In this question, which attracted no responces so far, I've asked whether it is possible to extend the Beilinson-Lichtenbaum etale descent rule for motivic cohomology to singular varieties, in ...
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Standard conjectures on positive characteristic
In this MO answer of M. Bondarko, he says:
"the Hodge conjecture implies all the Grothendieck's standard conjectures over base fields of characteristic 0..."
and in Remarks on Grothendieck's ...
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t-structures and higher categories?
I'm curious to find out where the viewpoint of higher categories may be useful so here is a somewhat vague question (which may or may not have a reasonable answer).
Given a triangulated category, one ...
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Dimension of binary motives of a quadric
Let $Q$ be a anisotropic quadric of dimension $d$ over $k$.
We work in the category of effective Chow-Motives over $k$.
Let $T$ be the Tate-Motive.
For a motive $M$ we write $M(l)$ for its $l$-th Tate-...
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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 ...
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Connections between Standard, Hodge and Tate conjectures on algebraic cycles?
What implications would a solution of the Standard Conjectures have on the Hodge and Tate Conjectures and reverse?
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Differences and relationships between Motivic cohomology and Universal cohomology theory? [duplicate]
Differences and relationships between Motivic cohomology (Beilinson, Lichtenbaum and Voevodsky) and Universal cohomology theory (Grothendieck)?
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Motivic L-function vs motivic zeta function
Let $M$ be a pure motive over a field $k$. Roughly speaking, the L-function of $M$ is the product over all primes $p$ of
$$L_p(M,s)=\det(I-Fr_p|_{M_\ell^I} N(p)^{-s})^{-1}$$
where $Fr_p$ is a ...
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Is the Tate conjecture known for etale covers of products of curves
Let $X$ be a (smooth projective geometrically connected) surface over a finitely generated field $k$. The Tate conjecture predicts that, for $l$ a prime number invertible in $k$, the Chern class map ...
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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 ...
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Articles about Weil cohomology theory by Grothendieck and Artin
In "The Standard Conjectures" Kleiman says that the following properties of Weil cohomology theory were proved in 1963 for étale cohomology by Artin and Grothendieck, except for the last one that it ...
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