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12 votes
0 answers
811 views

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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8 votes
0 answers
603 views

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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7 votes
1 answer
1k views

"Weight-monodromy" for open varieties

Suppose that $X/\mathbb{Q}_p$ is a smooth, projective variety, and choose a prime $\ell\neq p$. Then the weight-monodromy conjecture says that the graded pieces $\mathrm{Gr}_k^M$ of the monodromy ...
ChrisLazda's user avatar
  • 1,838
7 votes
1 answer
689 views

Two motivic complexes, compared

Bloch defines the motivic complexes $\mathbf{Z}(n)$ in his paper "Algebraic Cycles and Higher K-Theory" (1986). Some references (that I currently am unable to track down) use $$\check{\mathbf{Z}}(n) :...
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6 votes
1 answer
525 views

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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5 votes
1 answer
348 views

Spectral sequence in Betti cohomology

Let $X$ be a smooth projective algebraic variety over the complex numbers, and let us name $$f : X_{\rm an}\to X_{\rm Zar}$$ the morphism of sites induced by sending a Zariski open $U\subset X$ to $...
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4 votes
0 answers
265 views

Explicit linear object underlying $l$-adic cohomology for almost all $l$

If you are working with closed manifolds you can consider cohomology with any coefficients you like but ultimately everything is determined by the singular cohomology with $\mathbb{Z}$-coefficients. ...
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3 votes
0 answers
307 views

Semisimplicity conjecture

In this short note Ben Moonen proves that over fields of characteristic zero that are of finite type over their prime field, the Tate conjecture about surjectivity of cycle maps implies the semi-...
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2 votes
0 answers
245 views

What unramified Galois representations come from geometry?

I think we don't know what crystalline representations come from geometry. What about the unramified ones? Specifically let $\phi:\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)\to GL_n(\mathbb{Q}...
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1 vote
0 answers
290 views

Coniveau in étale motivic cohomology

Let $X$ be a smooth variety over a field. Is there a spectral sequence: $$E_1^{p,q} := \bigoplus_{x\in X^{(p)}}H^{q-p}(\kappa(x)_{\rm ét},\mathbf{Z}(n))\Rightarrow H^{q-p}(X_{\rm ét},\mathbf{Z}(n))$$...
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