Three things I learned from colleagues this semester

1 (from Patrick Allen) Let $F$ be a number field, and let $\pi$ be a cohomological cuspidal automorphic representation of some $\mathrm{GL}_n(\mathbf{A}_F)$.  Suppose that $\rho_\pi : G_{F,S} \to \mathrm{GL}_n(\overline{\mathbf{Q}_p})$ exists and satisfies local-global compatibility at all places, and that $H^1_f(F, \mathrm{ad}\,\rho_\pi) = 0$ as predicted by Bloch-Kato.  Then the following are equivalent:

a) $H^2(G_{F,S}, \mathrm{ad}\,\rho_\pi) = 0$, as predicted by Jannsen’s conjecture;

b) $H^1(G_{F,S}, \mathrm{ad}\,\rho_\pi)$ has the right dimension;

c) The product of restriction maps $H^1_f(F,\mathrm{ad}\,\rho_\pi(1))\to \prod_{v|p} H^1_f(F_v,\mathrm{ad}\,\rho_\pi(1))$ is injective.

The equivalence of a) and b) follows from Tate’s global Euler characteristic formula, but their equivalence with c) was news to me.  The question of whether or not c) holds came up incidentally in my work with Jack on Venkatesh’s conjecture, so it was very pleasing to learn that it follows from Bloch-Kato + Jannsen.

2 (from Keerthi Madapusi Pera) If $\mathbf{G} / \mathbf{Q}_p$ is semisimple and simply connected, and isotropic (i.e. contains some $\mathbf{Q}_p$-split torus), then $\mathbf{G}(\mathbf{Q}_p)$ has no proper finite-index subgroups.

3 (from Stefan Patrikis) Let $\pi$ be as in 1) again. There are two number fields naturally associated with $\pi$ (besides $F$): the field $\mathbf{Q}(\pi)$ generated by its Hecke eigenvalues, and the “reflex field” $E\subseteq F$ of its cohomological weight.  Is there any chance that $E$ is always a subfield of $\mathbf{Q}(\pi)$?, I asked SP.  Yes, said he.