Let $F$ be a totally real field in which a prime number $p>2$ is inert. We continue the study of the (generalized) Goren–Oort strata on quaternionic Shimura varieties over finite extensions of ${\mathbb{F}}_p$. We prove that, when the dimension of the quaternionic Shimura variety is even, the Tate conjecture for the special fiber of the quaternionic Shimura variety holds for the cuspidal $\pi$-isotypical component, as long as the two unramified Satake parameters at $p$ are not differed by a root of unity.

We establish interior $C^2$ estimates for convex solutions of the scalar curvature equation and the ${\sigma }_2$-Hessian equation. We also prove interior curvature estimates for isometrically immersed hypersurfaces $(M^n,g)\subset {\mathbb{R}}^{n+1}$ with positive scalar curvature. These estimates are consequences of interior estimates for these equations obtained under a weakened condition.

Let $\pi$ traverse a sequence of cuspidal automorphic representations of ${GL}_2$ with large prime level, unramified central character, and bounded infinity type. For $G\in \{{GL}_1,{PGL}_2\}$, let $H(G)$ denote the assertion that subconvexity holds for $G$-twists of the adjoint $L$-function of $\pi$, with polynomial dependence upon the conductor of the twist. We show that $H({GL}_1)$ implies $H({PGL}_2)$.

In geometric terms, $H({PGL}_2)$ corresponds roughly to an instance of arithmetic quantum unique ergodicity with a power savings in the error term, $H({GL}_1)$, to the special case in which the relevant sequence of measures is tested against an Eisenstein series.

Let $k$ be an algebraically closed field of characteristic $p>0$. We give a birational characterization of ordinary abelian varieties over $k$: a smooth projective variety $X$ is birational to an ordinary abelian variety if and only if ${\kappa }_S(X)=0$ and $b_1(X)=2dimX$. We also give a similar characterization of abelian varieties as well: a smooth projective variety $X$ is birational to an abelian variety if and only if $\kappa (X)=0$, and the Albanese morphism $a:X\to A$ is generically finite. Along the way, we also show that if ${\kappa }_S(X)=0$ (or if $\kappa (X)=0$ and $a$ is generically finite), then the Albanese morphism $a:X\to A$ is surjective and in particular $dimA\le dimX$.