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29 votes

Progress on the standard conjectures on algebraic cycles

For future references. Feel free to edit to include new cases, or any improvements. As for 2015, the standard conjectures on algebraic cycles is unconditionally (at lest) known for $X$: Lefschetz ...
Myshkin's user avatar
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28 votes
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How are motives related to anabelian geometry and Galois-Teichmuller theory?

Two clarifications: For anabelian geometry, you should ask how much information about a variety is contained in the Galois action on its etale fundamental group. While it's true that the motivic ...
Will Sawin's user avatar
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26 votes
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Hilbert's 3rd problem,number theory, motives, cyclic homology,...

This circle of topics is certainly one of my favourite surprising connections in mathematics. I will try to outline what little I understand of the big picture. Apologies for the length. Hilbert's ...
Matthias Wendt's user avatar
26 votes

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

(1) We have $$ N_n(X) = \sum_{k = 0}^{2d} (-1)^k \mathrm{tr}\left(\mathrm{Frob}^n \colon H^k(X) \to H^k(X) \right) = \sum_{k = 0}^{2d} (-1)^k \sum_{i=1}^{ h^k(X)} \lambda_{k,i}^n$$ where $\lambda_{k,...
Will Sawin's user avatar
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24 votes

How are motives related to anabelian geometry and Galois-Teichmuller theory?

There's a very deep connection between motives and Grothendieck-Teichmüller theory but it isn't well-understood yet. I can't even frame it precisely in higher genus, but at least I can frame a ...
Leila Schneps's user avatar
22 votes
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The modularity theorem as a special case of the Bloch-Kato conjecture

That is not what the link says. To quote (emphasis mine): ... in which this conjecture was reduced to a special instance of the Bloch-Kato conjecture for the symmetric square motive of an elliptic ...
Jack Sprat's user avatar
20 votes

Reference - motives of curves

This phenomenon is specific to the fields $\mathbb F_q$ (in this case you need the Artin motives as well) and $\bar{\mathbb F}_q$. Roughly, the outline is as follows. If the Tate conjecture is true ...
R. van Dobben de Bruyn's user avatar
19 votes

Why presheaves with transfer?

I hope someone can provide a better answer with more of an eye towards the motivic world, but for now let me outline that the exact same phenomenon exists in classical stable homotopy theory. Here the ...
Denis Nardin's user avatar
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17 votes
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Voevodsky's Triangulated Categories of Motives and their Relationships

I'm not sure that it is possible to compress the big picture into one answer; yet I will try to give a hint. Firstly, one can hardly hope to have a "reasonable" motivic $t$-structure for motives with ...
Mikhail Bondarko's user avatar
17 votes
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What exactly do the standard conjectures in characteristic zero refer to?

To prove that the standard conjectures are true for any Weil cohomology over a given field $k$, it suffices to prove them for an arbitrary chosen Weil cohomology over each subfield of finite type of $...
D.-C. Cisinski's user avatar
16 votes
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Conjecture of relation between residues of Feynman integrals and mixed Tate motives

1) Counterexamples were found in the paper Brown, Francis; Schnetz, Oliver: "A $K3$ in $\phi^4$". Duke Math. J. 161 (2012), no. 10, 1817–1862. It is now the general feeling that most $\phi^4$-Feynman ...
Friedrich Knop's user avatar
16 votes
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Tate twists and cohomology of $\mathbf{P}^1$

The Tate twist is what we need to express Poincaré duality without making any choice. Such a choice appears in the choice of an orientation of the affine line minus the origin, and have shadows in the ...
D.-C. Cisinski's user avatar
15 votes
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Is the Hodge Conjecture an $\mathbb{A}^1$-homotopy invariant?

Isn't this very easy? If the varieties are $\mathbb{A}^1$-homotopy equivalent, then their Voevodsky motives are isomorphic also (since there is a connecting functor making the obvious diagram ...
Mikhail Bondarko's user avatar
15 votes
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What's motivic about $\mathbb{A}^1$-homotopy theory? What's motivic about correspondences?

There is no need a priori to define these categories of motives starting from correspondences. The stable homotopy theory of schemes $SH$ may be characterized by a universal property saying that it is ...
D.-C. Cisinski's user avatar
14 votes
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Hodge standard conjecture for étale cohomology

I'm very interested myself on a better answer to this question, but let me point out the obvious: the main problem is that there is no Hodge theory on positive characteristic. The proof in ...
Myshkin's user avatar
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14 votes
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How to think about infinite generatedness of motivic cohomology

While waiting for someone more competent than me to answer, let me turn the question right back to you. Why should motivic cohomology be finitely generated? The answer is, of course, that there's no ...
Denis Nardin's user avatar
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13 votes

Why would one "attempt" to define points of a motive as $\operatorname{Ext}^1(\mathbb{Q}(0),M)$?

In the spirit of You Could Have Invented Spectral Sequences by T.Chow, I claim you could have invented $\operatorname{Ext}^{1}(\mathbb Q(0),M)$ as group of "rational points" of a motive. Here is how. ...
Olivier's user avatar
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13 votes
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Realization Functor From $SH$ to Derived Category of $Gal$-Modules

The most general functor of this form was constructed by Ayoub in La réalisation étale et les opérations de Grothendieck. Ayoub considers the ∞-category $DA^{et}(S,\Lambda)$ which is defined exactly ...
Marc Hoyois's user avatar
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13 votes
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Intuition behind the definition of finite correspondences

Traditionally correspondences were defined simply as cycles on the product, but then you need a moving lemma just to define composition. This limits you to working on smooth varieties. The beauty of ...
Donu Arapura's user avatar
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13 votes
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What are Motivic homotopy types?

Not sure about later developments, but the idea is mentioned in a famous passage of Grothendieck's Récoltes et Semailles. I quote from Roy Lisker's translation: Thus, the motive presents itself as ...
Myshkin's user avatar
  • 17.3k
13 votes

Voevodsky's Triangulated Categories of Motives and their Relationships

I won't embark on the difficult question of what one wants out of a category of motives, but I can make some comments on what might motivate the various choices of topologies. Nisnevich (aka ...
Shane Kelly's user avatar
13 votes
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Why linearization leads to arithmetization?

I think: The category of varieties over $\mathbb Q$ is already very arithmetic. One reason that the linearization is considered arithmetic is that so much of the tractable arithmetic information is ...
Will Sawin's user avatar
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13 votes
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Motives associated to a Number Field

Write $X=\mathrm{Spec}\, k$, which is a $0$-dimensional variety. Motives of $0$-dimensional varieties are called Artin motives, and they are pure. The Betti realization is the Betti cohomology of $X$, ...
Julian Rosen's user avatar
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13 votes
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Motivic vs Deligne cohomology

The existence of a cycle class map from motivic cohomology is a general fact to every cohomology satisfying certain axioms. For example, to every mixed Weil theory in the terminology of Cisinski-...
Tintin's user avatar
  • 2,711
12 votes

$\zeta(n)$ as a mixed Tate motive

Partially inspired by an unpublished work of Wojtkoviak for $\zeta (3)$, all of this follows from this result by Deligne: Theorem. $\pi_1(\mathbb{P}^1-\{0,1,\infty\})$ is a smooth mixed Tate motive ...
Myshkin's user avatar
  • 17.3k
12 votes
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Does the Grothendieck ring of varieties contain torsion?

As per Theo Johnson-Freyd's request, I'm converting my comment to an answer. Larsen-Lunts show that if $k$ is algebraically closed of characteristic zero, then there is a natural isomorphisms $$K_0(\...
Daniel Litt's user avatar
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12 votes
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A question on Voevodsky´s categories

One could say that the story begins with Beilinson's conjectures on the existence of a theory of motivic cohomology. In accordance with the insights of the Grothendieck school that cohomology ...
AAK's user avatar
  • 5,779
12 votes

Chow Groups of varieties over number fields

The statement you want follows fairly straightforwardly from Bass' conjecture -- sufficiently straightforwardly that it may well not have a separate name of its own. If $\Sigma$ is a sufficiently ...
David Loeffler's user avatar
11 votes
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Reference for Nori motives

Probably the best introduction has already been mentioned by Donu Arapura, and it is available online: Marc Levine, Mixed Motives (2005) Section 1 ("Essentials of Nori Motives") of this paper might ...
Myshkin's user avatar
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11 votes
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Derived version of equivalence between motives and representations of Motivic galois groups?

Let $k$ be a field and $\operatorname{DM}_{gm}(k)_{\mathbb Q}$ the ∞-category of rational geometric motives over $k$. A mixed Weil cohomology theory induces a symmetric monoidal exact functor $$ R: \...
Marc Hoyois's user avatar
  • 8,462

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