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.

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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 ...
Mikhail Bondarko's user avatar
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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, ...
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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.
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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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Solving "a, b, a+b have given divisors" problem

I've read an interesting article, math.NT/0409456 where you're just trying to solve a simple problem: For a given (finite) set of primes S find all solutions to an equation ...
Ilya Nikokoshev's user avatar
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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 ...
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Difference of Beilinson conjecture and equivariant Tamagawa number conjecture

As stated in the title, I am wondering the main difference between Beilinson conjecture and eTNC. If I read correctly, I can see that there are many literature treating both conjectures in the same ...
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How does $\zeta^{\mathfrak{m}}(2)$ and relate to $\zeta(2)$?

EDIT There appears to be a numerical zeta function $\zeta(2)$ as well as at least two different "motivic" zeta function realizations (Betti and de Rham) $\zeta^{\mathfrak{m}}(2)$. The "period map" of ...
john mangual's user avatar
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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 ...
user267839's user avatar
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$l$-adic periods?

For an algebraic variety $X$ over $\mathbb{Q}$ the comparison isomorphism between Betti and de Rham cohomologies provides the theory of periods with a motivic context whose reformulation as motivic ...
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Virtual Motives Infinitely Divisible by Lefschetz Motive

Let $K_0(Var_k)$ be the grothendieck group of the category of $k$-varieties, and call its elements virtual motives. $\mathbb{L}:=[\mathbb{A}^1_k]$ is called the Lefschetz motive. I think that if a ...
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Virtual mixed Tate motives

Let $\mathbf{Sch}_k$ be the category of $k$-schemes of finite type, and let $K_0(\mathbf{Sch}_k)$ be the Grothendieck ring of $k$-schemes. Let $\mathbb{Z}[\mathbb{L}]$ the subring generated by the ...
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Interesting implications on the theory of motives if the Hodge conjecture holds

For example, Under the Hodge conjecture the Motivic galois group coincides with Mumford-Tate group. The Hodge conjecture implies the Lefschetz and Kunneth standard conjectures, as well as ...
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Relationship between motivic Galois groups and Langlands program [duplicate]

I would like to know if there is any relationship between the motivic Galois groups and the Langlands program. Many thanks.
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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 ...
Rex's user avatar
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When is the K-theory presheaf a sheaf?

Let $F$ be a Deligne-Mumford stack that is of finite type, smooth and proper over $\mathrm{Spec~}k$ for a perfect field $k$. Consider $K_m$, the presheaf of $m$-th $K$-groups on $F_{et}$, the etale ...
expz's user avatar
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Néron theory for motives of arbitrary weight

SGA 7, tome 1, exp. IX, contains in its introduction and in section 13.4 remarks about ideas and conjectures of Deligne on a “théorie de Néron pour motifs de poids quelconque”. Would someone please ...
Thanos D. Papaïoannou's user avatar
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actions of the absolute Galois group and the motivic Galois group on étale cohomology

Let $K$ be a field of characteristic $0$; let $\ell$ be any prime; and let $\mathrm{Mot}(K, \mathbb{Q}_{\ell})$ be a Tannakian category of motives over $K$ with coefficients in $\mathbb{Q}_{\ell}$. So,...
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Intuition for polarized Hodge structures

A Hodge structure can be defined as a real, algebraic representation of the Deligne torus ${Res}^\mathbb{C}_{\mathbb{R}}\mathbb{G}_m$. Coming from Kahler manifolds the intuition for this is clear. The ...
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Motives and homotopy theories of algebraic varieties

The theory of motives is an attempt to cope with the fact that there are many reasonable cohomology theories of algebraic varieties. Now, sometimes your cohomology theory does not just give you a ...
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Are exotic affine spaces motivic/whatever equivalent to affine space?

This question is inspired by this MO question; in turn by this MO; in turn by these MO, MO. An exotic affine space is an affine variety $V$ whose $\mathbb{C}$-points are diffeomorphic to $\mathbb{R}^{...
Alexander Chervov's user avatar
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Homotopy equivalence between two basepoints of the etale homotopy type of the one-torus

Let $T = \mathbb{G}_m$ be the torus, and let $\tilde{T}$ be its étale universal cover (a pro-object in schemes of finite type). Then both $T$ and $\tilde{T}$ have a well-defined étale homotopy type. ...
Dmitry Vaintrob's user avatar
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Functoriality for $\ell$-adic cohomology - a question

This should a be basic enough question, but I’m a little confused. In proving that $H^*(X,\mathbf{Q}_{\ell})$ is functorial (in the sense of Weil cohomology theories: see axiom D2 here) as $X$ ranges ...
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Followup questions about the relationship between modular forms and motives

It occurs more and more that I ask a question on math stackexchange and then realize that it is more appropriate to mathoverflow. Hopefully this reflects well on myself... In any case, I copy here ...
Nicole's user avatar
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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 ...
naf's user avatar
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Motives in tropical geometry

Is there a notion of motives in tropical geometry? Similar like the notion introduced by Grothendieck in algebraic geometry.
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Cohomology theories for algebraic varieties over number fields

There is a standard line which is repeated by anyone writing/talking about motives and cohomology of algebraic varieties over number fields: namely, there are many such cohomologies and then the ...
Dr. Evil's user avatar
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Current state of Serre's Motives conjectures in Seattle

It would be worth if we have a current state of the conjectures of Propriétés conjecturales des groupes de Galois motiviques et des représentations l-adiques. J P Serre. In Motives, Seattle And ...
Jean's user avatar
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Hodge Realisation of Mixed Tate Motives

For a field $k$ which satisfies Beilinson-Soule vanishing conjecture, the from Levine's paper, https://www.uni-due.de/~bm0032/publ/TateMotives.pdf There exists an abelian category of mixed Tate ...
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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 ...
Mikhail Bondarko's user avatar
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0 answers
341 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 $...
Yohann Ségalat's user avatar
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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). ...
Mikhail Bondarko's user avatar
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0 answers
303 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 ...
Mikhail Bondarko's user avatar
6 votes
0 answers
461 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 $...
Mikhail Bondarko's user avatar
6 votes
0 answers
371 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 ...
Mikhail Bondarko's user avatar
5 votes
2 answers
821 views

Is there for every variety X an abelian variety A such that their 1st l-adic cohomologies are isomorphic?

This question is somewhat inspired by Kevin Buzzard's answer to What is the interpretation of complex multiplication in terms of Langlands? and somewhat from my own curiosity about such topics. Let $...
Makhalan Duff's user avatar
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2 answers
1k views

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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Motivation for Karoubi envelope/ idempotent completion

This is the second part of my venture to become more comfortable with the concept of idempotent elements and idempotent splittings from category theoretical viewpoint. In the first part we considered ...
user267839's user avatar
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The historical development of automorphic geometry

Background: Today the notion of automorphic geometry is often framed in the context of the Langlands program, in particular what is sometimes called the Langlands reciprocity conjecture. This is ...
Laie's user avatar
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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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2 answers
701 views

On triangulated categories of pro-objects

Which term is used for model categories whose homotopy categories are triangulated? Stable proper model categories? I want $Ho(Pro-M)$ to be triangulated ($Pro-M$ is the category of pro-objects of M) ...
Mikhail Bondarko's user avatar
5 votes
1 answer
283 views

A question on motivic zeta-function

It's well-known that over $\mathbb F_q$ every smooth projective conic $C$ is isomorphic to a projective line. So the formula for the motivic zeta-function $Z_{mot}(C)$ is evident since $S^n\mathbb P^1 ...
John S.'s user avatar
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1 answer
527 views

Is this Mayer-Vietoris sequence motivic?

Suppose $Y$ is a variety defined over $\mathbb{Q}$ and $pt$ is a rational point of $Y$. Let $\pi:X \rightarrow Y$ be the blow up of $Y$ at $pt$ and $D$ be the exceptional divisor. For simplicity let's ...
Wenzhe's user avatar
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2 answers
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Automorphic form encoding the orders of $N$ modulo $p$.

Let $N$ be a nonzero rational number. For every prime number $p$ with $v_p(N)=0$, let $a_p$ denote the index in $\mathbb Z/p\mathbb Z$ of the subgroup generated by $N$ modulo $p$. So we have $a_p=1$ ...
Xandi Tuni's user avatar
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Why is the triangulated category of motives easier than the abelian one?

There are several expository articles with the title "You could have invented [insert something mysterious here]" (a notable one being about spectral sequences, possibly it even started this genre). ...
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5 votes
1 answer
465 views

Around algebraic equivalence of cycles

Let $k$ be a finitely generated field, $X$ a smooth projective $k$-variety, $\ell$ a prime number, $\ell\in k^{\times}$, $r\ge 0$ an integer. The Tate conjecture asserts surjectivity of the cycle ...
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5 votes
1 answer
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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-...
nxir's user avatar
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motives of elliptic curves, modular forms, Hecke characters

Let $E$ be an elliptic curve over $\mathbb{Q}$. By the modularity theorem, $L(E, s)$ is the $L$-function of some modular form $f$. Now one has the following motives: (a) The Chow motive $h^1(E)$ ...
paL's user avatar
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1 answer
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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 ...
John Baez's user avatar
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1 answer
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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 ...
Marsault Chabat's user avatar

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