The motives tag has no wiki summary.

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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 ...

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664 views

### Motivic cohomology vs. K-theory for singular varieties

As far as I understand, for a smooth variety $X$ its motivic cohomology could be described as the corresponding piece of the $\gamma$-filtration of (Quillen's) $K^*(X)$; this is completely true for ...

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468 views

### Weak Lefschetz for torsion motivic cohomology (or torsion Chow groups)?

I am trying to prove the following Weak-Lefschetz-type statement for motivic cohomology: if $Z$ is a smooth hyperplane section of a smooth projective $X$ of dimension $d$ (say, over the field of ...

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779 views

### 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 ...

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### Voevodsky's counterexample to the existence of a motivic t-structure

I have been trying to unravel some of the known relationships between various ideas on mixed motives. I find the litterature quite hard to follow -"from experts, for experts".
Voevodsky in ...

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496 views

### The Shafarevich Conjecture and motivic Langlands stacks.

Hi, I recently learned about an amazing conjecture of Shafarevich
(proved by Faltings) about the finiteness of the number
of curves of a fixed genus with good reduction outside a
finite number of ...

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547 views

### 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 ...

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339 views

### General cohomology groups and motives

Let $X$ be a variety over $\mathbb{Q}$. Let $\mathcal{F}$ be a sheaf on $X$. Then we have an action of $Gal(\mathbb{Q})$ on $H_{et}^i(X,\mathcal{F})$. In certain cases we can say a lot about this ...

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654 views

### 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 ...

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701 views

### Motivic Galois group and Shimura varieties

Hi,
Suppose that one has a Shimura variety $Sh(G,X)$ where $(G,X)$ is the corresponding Shimura datum and suppose that it can be interpreted as a moduli space of motives (e.g. PEL type Shimura ...

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421 views

### 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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349 views

### 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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461 views

### Decomposition of Motives of cellular varieties

Dear community,
in his 2005 Inventiones Paper "On motivic decompositions arising from the method
of Białynicki-Birula" P. Brosnan deduced from the classical (?) theorem of Bialynicki-Birula on ...

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148 views

### The vanishing of homotopy invariant $K$-theory of dg-categories

In my previous question The vanishing of non-connective K-theory in negative degrees
I asked when one can be sure that the negative non-connective $K$-groups of a differential graded category vanish. ...

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559 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 ...

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750 views

### 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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729 views

### How does the conjectural Langlands group fit into the Tannakian point of view?

I've read that one way to formulate the Langlands program is the following:
Let $\mathcal{L}_ {\mathbb{Q}}$ be the conjectural Langlands group. Then the category of semi-simple (continuous) ...

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1k views

### How would a motivic proof of the Riemann hypothesis over finite fields go?

It is well known that Grothendieck had a different idea than Deligne about how one should go about proving the Riemann hypothesis for finite fields. However, since Grothendieck's desired proof never ...

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602 views

### Is the “L-function of the complex cohomology” of a motive equal to the L-function of its l-adic realization?

Let's say I have a motive in $\mathcal{M}_{num}(K)$ ($K$ a number field). For each prime $l$ there is a realization of this motive in terms of etale cohomology with coefficients in $\mathbb{Q}_l$. ...

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231 views

### (why) Are the following two constructions of zeta functions equal?

Let $X$ be a variety defined over $\mathbb{Q}$. One has the usual Hasse-Weil zeta function.
Now, let's do a different construction. Base change $X$ to $\mathbb{C}$: $X_{\mathbb{C}}$. Now look at its ...

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320 views

### Motivic interpretation of genus 2 Siegel forms induced by lifts of Maass and Skoruppa

Background: There are several known lifts from integral weight modular forms to Siegel forms of genus 2, among them the Saito-Kurokawa lift. Another lift construction that is important for ...

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850 views

### Why would the category of Motives be Tannakian?

After reading the answer to my previous question: What are the different theories that the motivic fundamental group attempts to unify?
I decided to read up on Tannakian formalism.
Given the ...

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1k views

### What are the different theories that the motivic fundamental group attempts to unify?

I must preface by confessing complete ignorance in the subject. I've read introductory texts about the theory of motives, but I am certainly no expert.
In ...

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205 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). ...

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748 views

### Motives from the fundamental group made nilpotent

I am reading the fascinating paper of Deligne on "le groupe fondamental de la droite projective moins trois points", and other stuffs related to anabelian geometry. This suggested the following ...

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1k views

### “Modular forms from Feynman integrals ”?

I would like to learn more about the background of this talk, but found no text on that theme. Do you know more? Edit: An interesting talk by Miranda Cheng (slides).
Edit: A talk today on the theme, ...

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629 views

### Crystalline realizations of Artin motives

What are the crystalline realizations of Artin motives?
In a paper by Kisin and Wortmann, "A note on Artin motives" (google it and you'll find it immediately), they define a suitable category of ...

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### How is the Ising model an example of a lattice model as per Kontsevich?

In section 3.2 of Kontsevich's very interesting paper "Notes on motives in finite characteristic,", he gives an axiomatic definition of a "lattice model" attached to a Boltzmann datum (V_1,V_2,R), ...

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### commutativity constraint in Grothendieck's motives

This is a basic question about Grothendieck's conjectural category $M_k$ of pure motives (over a field $k$). This construction first produces a category (the "false category of motives") which need ...

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### Why is the zeta function of a variety over a finite field not a polynomial? (question about motives)

I've been doing some light(?) reading on motives and the standard conjectures in an attempt to put various things that I tangentially know in perspective.
The question is this: the Weil conjectures ...

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695 views

### Is there a connection between the theory of motives and homotopy theory?

I have read that motives were designed to be the common part of the many homology theories, a way of unifying them. But as I understand it: homotopy is closely related to homology, there is only 1 ...

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479 views

### Is there a 'classical' definition for the support of a perverse sheaves.

I would like to define the support of a mixed motivic sheaf. This should be something similar to the support of a perverse sheaf.:) Is there any 'classical' definition for the latter?
I suspect that ...

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### Are two conjectural descriptions of the motivic t-structure (via cohomology and via affine varieties) known to be equivalent?

I know of two conjectural descriptions of the motivic $t$-structure for Voevodsky's motives.
The first one is based on the conjecture that Weil cohomology theories should yield exact and ...

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285 views

### Could the Kunneth decomposition of a motif depend on the choice of $l$?

Suppose that over some (algebraically closed) field $K$ of characteristic $p>0$ we have: numerical equivalence of cycles coincides with homological one with respect to ${\mathbb{Q}}_{l}$ and ...

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894 views

### If the numerical equivalence of cycles coincides with the homological one, does the Hodge standard conjecture follow?

Suppose that over an algebraically closed field $K$ of finite characteristic the numerical equivalence of cycles relation (for algebraic cycles of smooth projective varieties) coincides with the ...

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145 views

### $G_m$-cohomology of a motif (that corresponds to a stack?)

As in the question For a G-variety, what could one say about the motif of the corresponding simplicial variety
I am in the following situation: $G$ is an algerbraic group, and X is a smooth ...

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249 views

### Do $Q_l$-etale Euler characteristics of Chow motives coincide for all $l$?

I am interested in (Chow) motives over (algebraically closed) characteristic $p>0$ fields. For $H$ being $\mathbb{Q}_l$-adic cohomology, one can consider $Ch_l(M)=\sum (-1)^i\dim_{\mathbb{Q}_l} ...

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### A motivic complex

By definition, Voevodsky's motivic complex (an object of his $DM^{eff}_-$) is a complex of sheaves with transfers whose cohomology sheaves are homotopy invariant. Now, I consider the complex (of ...

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266 views

### For a G-variety, what could one say about the motif of the corresponding simplicial variety

Let G be an algerbraic group, and X be a G-variety (that I will assume to be smooth). Then one can consider a simlicial variety whose terms are $G^i\times X$. This simplicial variety yields a 'complex ...

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497 views

### Questions on standard (motivic) conjectures

Over an (algebraically closed) characteristic $p$ field, is it known that the cohomological equivalence of cycles relation (with respect to $\mathbb{Q}_l$-adic \'etale cohomology) does not depend on ...

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405 views

### Are finite correspondances flat?

In Voevodsky&Mazza&Weibel's book on motivic cohomology they define "an elementary correspondance from X (Smooth connected scheme over $k$) to Y (separated scheme over $k$) as an irreducible ...

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292 views

### Explain the relation between $K_0$ and morphisms of Chow motives

The Chern class yields an isomorphism $K_0(X)\otimes \mathbb Q\cong \bigoplus_{i\ge 0} Chow^i(X)\otimes \mathbb Q$ (for a smooth variety $X$ over a field?), whereas the latter group is isomorphic to ...

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509 views

### 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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274 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 ...

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### The conjectural relation between mixed motivic sheaves and the perverse t-structure.

As far as I remember, there 'should exist' an exact etale realization functor from the category of mixed motivic sheaves (over a base scheme $S$) to the category of perverse $l$-adic sheaves over $S$. ...

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585 views

### 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 $a_p=1$ ...

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305 views

### A composition of a finite morphisms with the transpose correspondence: is it the multiplication by the degree?

Let $X,Y$ be smooth varieties over a field $k$ (which in my case is perfect of finite characteristic $p$; we may also assume that $X,Y$ are connected); $s:Y\to X$ is a finite morphism of degree $d$. ...

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170 views

### A Hodge substructure with ''nice' weight factors that does not correspond to a mixed submotif?

Suppose that we have a mixed motif $M$ with only two (subsequent) weights, and have a subobject for each weight factor. How we can control whether there exists a submotif of $M$ with these two ...

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131 views

### On 'graded polarizable' triangulated categories; are there any mixed Galois module analogues known? Also on mixed realizations

There is a construction by Beilinson (in section 3 of "Notes on absolute Hodge cohomology") of the derived category of graded polarizable mixed Hodge complexes; he also proved that this category is ...

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### Which pure categories related with Weil cohomology theories are semi-simple?

As far as I understand, the category of pure polarizable Hodge modules is semi-simple, whereas the cohomology of the corresponding schemes is graded polarizable. Is it true that one doesn't have any ...