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Let be a totally real field in which a prime number is inert. We continue the study of the (generalized) Goren–Oort strata on quaternionic Shimura varieties over finite extensions of . 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 -isotypical component, as long as the two unramified Satake parameters at are not differed by a root of unity.
We establish interior estimates for convex solutions of the scalar curvature equation and the -Hessian equation. We also prove interior curvature estimates for isometrically immersed hypersurfaces with positive scalar curvature. These estimates are consequences of interior estimates for these equations obtained under a weakened condition.
Let traverse a sequence of cuspidal automorphic representations of with large prime level, unramified central character, and bounded infinity type. For , let denote the assertion that subconvexity holds for -twists of the adjoint -function of , with polynomial dependence upon the conductor of the twist. We show that implies .
In geometric terms, corresponds roughly to an instance of arithmetic quantum unique ergodicity with a power savings in the error term, , to the special case in which the relevant sequence of measures is tested against an Eisenstein series.
Let be an algebraically closed field of characteristic . We give a birational characterization of ordinary abelian varieties over : a smooth projective variety is birational to an ordinary abelian variety if and only if and . We also give a similar characterization of abelian varieties as well: a smooth projective variety is birational to an abelian variety if and only if , and the Albanese morphism is generically finite. Along the way, we also show that if (or if and is generically finite), then the Albanese morphism is surjective and in particular .