9 problems

Soulé's finiteness conjecture. The group $H^2_p$ is always finite.

Kato's weak Tamagawa number conjecture. There is a subspace $H^{\operatorname{constr}}_{{\cal M}}$ such that $r_{{\cal D}}$ and $r_p$ restricted to it are isomorphisms and $H^2_p$…

The local Tamagawa number conjecture. For every positive integer $j$ there exists a unique $\mathbb{Z}_p$-basis

Let $L/K$ be a finite Galois extension of number fields with Galois group $G$, let $r>1$ be an integer, and let $p$ be an odd prime. Let $\phi_r$, $A_{\phi_r}^S$, $\psi_r$,…

Let $V_l^i$ be the $l$-adic representation attached to the $i$th cohomology of the generic fibre, and let $R\Gamma_f(\mathbb Q_p,V_l^i)$ be the corresponding local finite-cohomolog…

Let $U$ be the open arithmetic base over which the smooth $l$-adic sheaf $V_l^i$ is defined, let $T_l^i\subseteq V_l^i$ be a Galois-stable $\mathbb Z_l$-lattice, and let…

The $p$-part of the Tamagawa number conjecture. The element $T\Omega(\mathbb{Q}(m)_{F'/F})_p$ vanishes:

Let $F'/F$ be the finite extension and let $G_{F'/F}$ be its Galois group. For the Tate motive $\mathbb{Q}(m)_{F'/F}$, let $L^*(\mathbb{Q}(m)_{F'/F})$ be its equivariant Artin $L$-…

Let $M$ be one of the motives $M(F)$ or $M(\eta)$. Let $\Xi(M)$ be the fundamental line … and let $\vartheta_\infty:E_\infty\simeq\Xi(M)\otimes_\mathbb{Q}\mathbb{R}$ be the isomorp…