math.AG — Algebraic Geometry

Integral crystalline Tate classes of codimension two at supersingular points of Mumford's fourfolds

Contributed by Antti Hippeläinen

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 conjecture holds in codimension $2$ on $X$ if statements in a preprint of Markman hold. At these primes, Kisin's crystalline tensors make the Dieudonné module a tensor product of three rank-$2$ modules permuted cyclically by Frobenius, and at supersingular points Frobenius maps a tensor basis to multiples of basis vectors. For such monomial modules all lattices involved are spanned by multiples of basis vectors, with exponents given by min-plus formulas independent of $p$. Under the endomorphisms, $T^4/S$ is a non-split extension of $(\mathbb{Z}/p)^2$ by $(\mathbb{Z}/p)^2$, generated by one Weil class of an inert imaginary quadratic action. Assuming these statements, an algebraic such class comes from specializing a Weil cycle on a fourfold with complex multiplication. A projector that is a polynomial in an endomorphism, and an intersection matrix with unit determinant, fix its $p$-adic normalization without a comparison theorem. At unramified non-inert primes with such levels, $T^4=S$ at the supersingular points. Of the fourteen isomorphism classes of monomial supersingular Dieudonné modules of height $8$, the conjecture holds unconditionally for nine, under the same assumption for that of $\mathscr{G}_p$, and is open for four. The indices are computed exactly with integers.