On the supersingular locus of the \(\mathrm{GU}(2,2)\) Shimura variety (Q466557)

This paper approaches a problem of explicitly describing the supersingular locus of Shimura varieties. This problem is important intrinsically for the sake of understanding Shimura varieties but also because of its applications to Kudla's program. The authors solve the problem for \(\mathrm{GU}(2,2)\) Shimura varieties by using previous methods of \textit{I. Vollaard} and \textit{T. Wedhorn} [Invent. Math. 184, No. 3, 591--627 (2011; Zbl 1227.14027)] together with a new input and exploiting the exceptional isomorphism \(\mathrm{SU}(2,2) \simeq \mathrm{Spin}(4,2)\).NEWLINENEWLINELet \(E\) be a quadratic imaginary field and let \(p>2\) be inert in \(E\). Let \(\mathcal{O} \subset E\) be the integral closure of \(\mathbb{Z}_{(p)}\) and let \(V\) be a free \(\mathcal{O}\)-module of rank \(4\) endowed with a perfect \(\mathcal{O}\)-valued Hermitian form of signature \((2,2)\). Let \(G = \mathrm{GU}(V)\) be the group of unitary similitudes of \(V\). This is a reductive group over \(\mathbb{Z} _{(p)}\). Fix a compact open subgroup \(\mathrm{U}^p \subset G(\mathbb{A}_f ^p)\) which is sufficiently small and define \(\mathrm{U}_p = G(\mathbb{Z}_p)\) and \(\mathrm{U} = \mathrm{U}_p \mathrm{U}^p \subset G(\mathbb{A}_f)\).NEWLINENEWLINEUsing this data one defines the Shimura variety \(M_{\mathrm{U}}\) which is smooth and of relative dimension \(4\) over \(\mathbb{Z}_{(p)}\). It is a moduli space of abelian fourfolds, up to prime-to-\(p\) isogeny, with an additional structure. We denote by \(M_{\mathrm{U}} ^{\mathrm{ss}}\) the reduced supersingular locus of the geometric special fiber. One of the main results of the article states that \(M_{\mathrm{U}} ^{\mathrm{ss}}\) has pure dimension \(2\) and for \(\mathrm{U}^p\) sufficiently small all irreducible components of \(M_{\mathrm{U}} ^{\mathrm{ss}}\) are isomorphic to the Fermat hypersurface NEWLINE\[NEWLINEx_0 ^{p+1} + x_1 ^{p+1} +x_2 ^{p+1} +x_3 ^{p+1} = 0.NEWLINE\]NEWLINE Moreover if two irreducible components intersect non-trivially, the reduced scheme underlying their scheme-theoretic intersection is either a point or a projective line.NEWLINENEWLINEThis result is deduced from the local analogue of it. Recall that by Rapoport-Zink uniformization theorem we can express \(M_{\mathrm{U}} ^{\mathrm{ss}}\) as a disjoint union of quotients of the Rapoport-Zink space \(\mathcal{M}_{\mathrm{red}}\). Hence it suffices to prove the analogous result for \(\mathcal{M} _{\mathrm{red}}\). This takes most of the paper. The authors analyze \(\mathcal{M} _{\mathrm{red}}\) by decomposing it further using ``vertex lattices'' in the group of special quasi-endomorphisms and exploiting the linear algebra behind it.