Integral crystalline Tate classes of codimension two at supersingular points of Mumford's fourfolds
For a supersingular abelian variety $X$ over $\bar{\mathbb{F}}_p$, let $T^4(X)$ be the lattice of integral crystalline Tate classes of codimension $2$ and $S(X)$ the span of products of divisor classes. Let $p\ge5$ and $\mathscr{G}_p$ be the $p$-divisible group of the supersingular points of Mumford's families of abelian fourfolds at primes inert in their cubic field, for the hyperspecial levels given by tensor-induced lattices. For every abelian fourfold $X$ with $X[p^\infty]\cong\mathscr{G}_p$ we show that $T^4(X)/S(X)\cong(\mathbb{Z}/p)^4$, and that the integral crystalline Tate…
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