On the Shimura variety having Mumford's fake projective plane as a connected component
A fake projective plane (FPP) is a compact complex surface with the numerical invariants \(c_1^2=3c_2=9, p_g=q=0\) and with an ample canonical class \(K_X\). FPP has the same Betti numbers as \(\mathbb{C}\mathbb{P}^2\) but not homeomorphic to it. The first example of FPP was constructed by \textit{D. Mumford} [Am. J. Math. 101, 233--244 (1979; Zbl 0433.14021)]. Any FPP can be realized via \(p\)-adic uniformization as an arithmetic quotient of the complex unit ball. The main purpose of this paper is to give an explicit description of the Shimura variety that has Mumford's FPP as a connected component. A Shimura variety is constructed as a disjoint union of finitely many arithmetic quotients of a complex unit ball. The reflex field associated to it is \(K\) is the imaginary quadratic field \(\mathbb{Q}(\sqrt{-7})\). Theorem. The Shimura variety consists of three connected components, and has the canonical model defined over the reflex field \(K\). Furthermore, these connected components are all isomorphic to each other as complex surfaces, and one of which is isomorphic to Mumford's FPP. These connected components have models defined over the \(7\)-th cyclotomic field \(\mathbb{Q}(\zeta_7)\) over \(\mathbb{Q}\).
scientific article; zbMATH DE number 5279025
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