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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Grothendieck-Teichmüller conjecture

(1) In "Esquisse d'un programme", Grothendieck conjectures Grothendieck-Teichmüller conjecture: the morphism $$ G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T}) $$ is an isomorphism. Here $...
AFK's user avatar
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Are we better in computing integrals than mathematicians of 19th century?

When I started to learn mathematics, I was fascinating by legendary «Демидович»: problems in mathematical analysis. Fifteen years later, when I open chapters about integrals, I see a long list of ...
13 votes
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Stack of Tannakian categories? Galois descent?

I'm having trouble finding a reference for something that I'm guessing the experts worked out long ago. Let's take a local or global field $F$ for this post, and fix a separable algebraic closure $\...
Marty's user avatar
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Kato's log motives

What are they and what are their intended uses? Does anyone have notes/slides of this talk? I am curious about "log motives" because there seems to exist a "log motivic yoga" among experts in ...
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12 votes
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Number field analog of Artin-Tate $\Rightarrow$ BSD?

What is the difference between the alternating product of the Hasse-Weil $L$-functions of the generic fiber of an arithmetic scheme $X\to\text{Spec}(\mathbf{Z})$ and the zeta function of $X$? (each ...
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What is missing in the current constructions of pure and mixed motives?

Yo! Maybe this question is too broad, so maybe it should be community wiki? In summary, I want to known all the known comparisons between all the constructions of pure and mixed motives and what make ...
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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 ...
nxir's user avatar
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Are there analogues of Beilinson's conjectures for motives with coefficients?

There's a body of wisdom (following Beilinson, Bloch, Deligne, ...) relating mixed Tate motives, motivic cohomology, algebraic K-theory, special values of L-functions, and polylogarithms. My ...
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Linear algebra of elliptic curves over p-adic fields

Unfortunately, the following question is somewhat ill-posed. However, I hope to make what I'm looking for sufficiently clear. Given two elliptic curves (or Abelian varieties) over $\mathbb{C}$, one ...
Moosbrugger's user avatar
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What are "fractional motives"?

Kirti Joshi's musings mention "fractional motives". Do you know what are they good for and what the current state of constructions is for them? Edit: Further cases of "fractional motives" as ...
Thomas Riepe's user avatar
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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" ...
Andreas Holmstrom's user avatar
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What is the precise definition of "Hypergeometric motives over $\mathbb{Q}$"?

The question is as in the title, but here is some background: Section 4 of this paper by Beukers, Cohen and Mellit is called "Hypergeometric motive over $\mathbb{Q}$" but no actual (pure) ...
naf's user avatar
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Uncountably many non-isomorphic Tate modules

Do there exist uncountably many abelian surfaces with good reduction over $\mathbb{Q}_p$ with pairwise non-isomorphic rational $p$-adic Tate modules? If we took $l$-adic Tate modules there would be ...
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9 votes
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Would full resolution of singularities have cohomological implications beyond the alteration theory?

De Jong's result on alterations allows one to show the potential semistability of certain Galois representations arising from cohomology of varieties (among other things). If we knew the existence of ...
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Which field extensions do not affect Chow groups?

Let $X$ be a (say, smooth projective) variety over a field $k$. For which $K$ it is known that the ("ordinary", that is, not higher) Chow groups of $X$ map onto that of $X_K$ bijectively? ...
Mikhail Bondarko's user avatar
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Searching for hypergeometric motives that split

Motivation: It seems that the splitting of a hypergeometric motive is closely related to some highly non-trivial hypergeometric identities discovered by Ramanujan, Guillera et al. The splitting of ...
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Beilinson regulators and Bloch's mythological algebraic intermediate Jacobians

In the paper introducing his motivic cycle complexes, Bloch outlines a project he says he was going to return to in the future: Towards the end of page 270, he says, given a smooth projective variety ...
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Motivic Galois theory and Betti realizations?

Why Motivic Galois groups are defined with Betti realizations? (In fact Absolute Galois groups can be defined in this way (with Betti realizations), why they are so related?).
peter's user avatar
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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}}(\...
Anandam Banerjee's user avatar
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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 ...
Will Sawin's user avatar
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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 ...
Neil Strickland's user avatar
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Values of cohomology theory on a point

$\DeclareMathOperator\Sm{Sm}$It is a well-known fact that in algebraic topology, generalized cohomology theories are determined by their values on the point. I was wondering whether anything similar ...
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Reference request: Motivic Cohomology and Cycle class maps

For a smooth projective variety $X$ over any field $K$, Voevodsky showed in his paper ``Motivic Cohomology Groups Are Isomorphic to Higher Chow Groups in Any Characteristic" that the motivic ...
Jesse Silliman's user avatar
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Computing motivic Galois group

Suppose I have a motive $M$ over $\mathbb{Q}$, and can compute the Euler factor of the associated $L$-function for any good prime $p$. How can I compute the Zariski closure of the image of the Galois ...
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A Generalization of the Tate-Shafarevich/Tate/Fontaine-Mazur Conjectures

Let $A$ be an abelian variety over a number field $k$. The Tate-Shafarevich conjecture says that the Tate-Shafarevich group of $A$ is finite. A weakening of this conjecture states that the $\ell$-...
David Corwin's user avatar
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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-...
Frédéric Paugam's user avatar
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Corresponding notion of unramified for motives (or de Rham cohomology)

The etale cohomolgoy of a variety $X$ over a number field $K$ is a Galois representation of $\mathrm{Gal}(\overline K/K)$ with some properties coming from $X$, e.g., it is unramified outside $S$ if $X$...
Masse's user avatar
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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-...
David Corwin's user avatar
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In what sense do the real and complex places correspond to setting q equal to 1 or -1?

It often happens that if we have a scheme $X/\mathbb Z$ (or an open subset thereof) and we denote by $p(q) = X(\mathbb F_q)$, then $p(1)$ and $p(-1)$ compute the euler characteristic of $X(\mathbb C)$ ...
Asvin's user avatar
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Status of the conjectured vanishing of Bloch-Kato H^2

There is a folklore conjecture that $\operatorname{Ext}^2$ vanishes in the category of geometric $p$-adic Galois representations (i.e. representations that are unramified almost everywhere and de Rham ...
David Corwin's user avatar
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7 votes
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249 views

Uniqueness of Mixed Tate Motive

I am reading the book Periods and Nori Motives by Huber and Muller-Stach et al. A question comes up to me. Suppose $\text{DM}_{gm}(k,\mathbb{Q})$ is the triangulated category of geometric mixed ...
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Roadmap to study (Deligne) Algebraic geometry over Tannakian categories

I would like to know the way to proceed in the first lecture of Deligne's Le groupe fondamental de la droite projective moins trois points. General advices for reading Deligne's paper. What should I ...
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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 ...
Cristian D. Gonzalez-Aviles's user avatar
7 votes
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306 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 ...
Mikhail Bondarko's user avatar
7 votes
0 answers
706 views

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, ...
Xandi Tuni's user avatar
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6 votes
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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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6 votes
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204 views

Motives in tropical geometry

Is there a notion of motives in tropical geometry? Similar like the notion introduced by Grothendieck in algebraic geometry.
Raoul's user avatar
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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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392 views

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 ...
Wenzhe's user avatar
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6 votes
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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
6 votes
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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
6 votes
0 answers
241 views

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
6 votes
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
0 answers
120 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 ...
Tintin's user avatar
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5 votes
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175 views

Can a Chow motif be isomorphic to its own direct summand?

Let $M$ be a $R$-linear Chow motif over a field $k$ that is perfect but not necessarily algebraically closed. Can one prove that $M$ is not a direct summand of itself (that is, $M\not\cong M\bigoplus ...
Mikhail Bondarko's user avatar
5 votes
0 answers
370 views

What is the motive of $\operatorname{Bun}_G(X)$?

$\DeclareMathOperator\Bun{Bun}$Let $X$ be a scheme over algebraically closed field $k$, $G$ a reductive group and $\Bun_G(X)$ the stack of $G$ bundles on $X$. Write $[\Bun_G(X)]\in K_\text{st}$ for ...
Pulcinella's user avatar
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5 votes
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264 views

Reference request: Tate's conjecture for L functions of motives

What's a good reference for the most general form of Tate's conjecture for the order of poles of the L function of a motive? Thanks!
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