## New answers tagged motives

3
votes

### What is motivic sheaf intuitively?

Q1. At least partially yes.:) There should exist a nice system of connecting functors. This text https://arxiv.org/abs/1801.10129 should give much of this theory; I don't know why it is not published ...

4
votes

### Motive of CM elliptic curve and modular forms

Since this question has come alive again, let me point out that the Hecke operators cannot give a splitting of $h^1(E)$ into two pieces over $F$, since the Hecke correspondences on a modular curve are ...

1
vote

### Motive of CM elliptic curve and modular forms

Question 1: The field generated by the Fourier coefficients of an elliptic curve associated to a modular form is $\mathbb Q$. (For example, since the Fourier coefficients can be calculated by counting ...

4
votes

### Which varieties are sums of tensor powers of the Lefschetz motive?

One class of examples is already indicated in the comments, and the question itself. I thought it would be good to include this in an official answer.
Proposition. Let $X$ be smooth projective ...

25
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,...

5
votes

### Functoriality conjectures on the slice filtration

This is also investigated in Shane Kelly's thesis, see Section 4.2.

8
votes

Accepted

### Homotopy invariance of $\ell$-adic cohomology

Proof of homotopy invariance: This follows from a base change/Kunneth type statement and the calculation of the cohomology of $\mathbb A^1$.
Specifically, Lemma 7.6.7 of Lei Fu's etale cohomology ...

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