Hunt, Bruce The geometry of some special arithmetic quotients. (English) Zbl 0904.14025 Lecture Notes in Mathematics. 1637. Berlin: Springer. xiii, 332 p. (1996). Let \(X\) be a Riemannian symmetric space such that the associated semisimple Lie group \(G= \text{Aut} (X)\) of automorphisms of \(X\) is non-compact. Let \(\Gamma\) be a discrete subgroup of \(G\) with finite covolume which acts on \(X\) properly discontinuously. Then \(\Gamma \backslash X\) is a locally symmetric space of non-positive curvature and finite volume. If \(\Gamma'\) is a normal subgroup of \(\Gamma\) of finite index, then the finite group \(\Gamma / \Gamma'\) acts on the locally symmetric spaces \(\Gamma'\). The monograph under review concerns the geometry of such locally symmetric space with automorphism group under the condition that the symmetric space \(X\) is Hermitian and that the discrete subgroups \(\Gamma\) and \(\Gamma'\) are arithmetic. It studies a number of highly interesting examples of such spaces by applying the general theory of interpreting arithmetic quotients of Hermitian symmetric spaces as parameter spaces of families of abelian varieties. Chapter 1 gives a review of Shimura’s theory which interpretes arithmetic quotients as moduli spaces of abelian varieties with given polarization, endomorphism ring and level structure. Chapter 2 studies the split \(\mathbb R\) case in which a maximal \(\mathbb Q\)-split torus is also a maximal \(\mathbb R\)-split one. In this case, the well-known geometry of the Hermitian symmetric domain is reflected in the geometry of its arithmetic quotient. Chapter 3 discusses the geometry of particular algebraic varieties which turn out to be Baily-Borel embeddings of arithmetic quotients by employing the method of uniformization rather than the more traditional method of automorphic forms. Chapter 4 is devoted to some geometry which canonically arises from the configuration of the 27 lines on a general cubic surfaces.The heart of this monograph consists of chapters 5 and 6 which discuss two very interesting varieties. One is the Burkhardt quartic threefold \(\mathcal B_4 \subset \mathbb P^4\) and the other is the invariant quintic fourfold \(\mathcal I_5 \subset \mathbb P^4\). The automorphism group of \(\mathcal I_5\) and \(\mathcal B_4\) are the Weyl group \(W(E_6)\) of \(E_6\) and the normal subgroup \(G_{25,920}\) of \(W(E_6)\) of index two, respectively. Finally, reviews of rational groups of Hermitian type and some classical algebraic geometry are given as appendices. Reviewer: Min Ho Lee (Cedar Falls) Cited in 4 ReviewsCited in 62 Documents MSC: 14K10 Algebraic moduli of abelian varieties, classification 32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) 14M17 Homogeneous spaces and generalizations 14J30 \(3\)-folds 22E40 Discrete subgroups of Lie groups 11F55 Other groups and their modular and automorphic forms (several variables) Keywords:arithmetic quotients of Hermitian symmetric spaces; arithmetic groups; moduli varieties; Burkardt quartic threefold; cubic surfaces; parameter spaces of families of abelian varieties; quintic fourfold; automorphism group × Cite Format Result Cite Review PDF Full Text: DOI