All Questions

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 ...

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 $\ell$-...

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. ...

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-...

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}...

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