Supersingular Mumford–Tate conjecture

Prove that, for a smooth connected variety X mapping to the mod-p Siegel moduli space and containing a point x mapping to a supersingular abelian variety, the connected \mathbb{Q}_\ell-monodromy group H_\ell equals H_{\mathbb{Q}_\ell} for every prime \ell\ne p and the crystalline monodromy group H_p equals H_p as defined by the common \mathbb{Q}-form.

Background

The paper formulates a positive-characteristic analogue of the Mumford–Tate conjecture for subvarieties meeting the supersingular locus. The group H is defined as the smallest subgroup of the rational endomorphism group containing the connected \ell-adic monodromy groups for all \ell\ne p and the crystalline monodromy group at p. The conjecture asserts that these realizations already coincide with the corresponding monodromy groups. The paper proves the conjecture under a generically ordinary GSpin Shimura-subvariety hypothesis, so the general formulation remains the explicitly stated conjectural problem.

References

We make the following conjecture \begin{conj}[Supersingular Mumford-Tate]\label{conj: ss MT} We have $H_{Q_\ell} = H_\ell$, $H_{} = H_p$. \end{conj}

— S-integrality for families of ordinary K3 surfaces and algebraicity theorems  (2609.25703 - Jiang et al., 22 Sep 2026) in Section 1, subsection “Basic Mumford–Tate,” Conjecture 1 (Conjecture \ref{conj: ss MT})

\begin{conj}\label{conj: Basic MT} We have $H_{Q_\ell} = H_\ell$, and $H_{} = H_p$. \end{conj}

— S-integrality for families of ordinary K3 surfaces and algebraicity theorems  (2609.25703 - Jiang et al., 22 Sep 2026) in Section 5, subsection “Basic points,” Conjecture \ref{conj: Basic MT}