Moduli of abelian varieties

Moduli of abelian varieties
Name	Moduli of abelian varieties
Field	Algebraic geometry

Moduli of abelian varieties — The moduli of abelian varieties parametrize isomorphism classes of principally polarized abelian varieties and their variants, forming rich geometric objects that connect arithmetic, topology, and representation theory. These moduli spaces appear as algebraic stacks and complex analytic quotients, relate to automorphic representations and Hodge theory, and play central roles in questions originating with Bernhard Riemann, David Mumford, and Igor Dolgachev.

Introduction

The classical construction of moduli of abelian varieties synthesizes ideas from Bernhard Riemann's theta functions, André Weil's work on complex tori, and the scheme-theoretic foundations introduced by Alexander Grothendieck. Over C one obtains quotients of Siegel upper half-spaces by symplectic groups closely tied to Hodge theory and the Torelli theorem for curves studied by Rita Gunning and Farkas & Kra. In arithmetic contexts, contributions of Jean-Pierre Serre, John Tate, and Gerd Faltings connect these moduli to the Langlands program and Diophantine finiteness results such as Faltings's theorem.

Moduli problem and moduli functors

Formally, the moduli problem is phrased in terms of a functor from schemes to groupoids assigning to a scheme S the category of abelian schemes over S with polarizations and level structures, following the stack-theoretic framework pioneered by Alexander Grothendieck and developed further by Pierre Deligne and Jean-Marc Fontaine. The representability and coarse moduli space properties were established in work of David Mumford and Michael Artin using geometric invariant theory associated with actions of groups like GSp(2g) and Sp(2g). For principal polarizations one obtains the moduli stack A_g (an algebraic stack) whose coarse moduli variety is often denoted A_g as well in classical literature associated with Igusa and Igusa's modular forms.

Siegel modular varieties and moduli spaces

Over the complex numbers the analytic incarnation of these moduli spaces is the Siegel modular variety, realized as the quotient of the Siegel upper half-space by arithmetic subgroups of Sp(2g, Z), a perspective developed by Carl Ludwig Siegel and analyzed through modular forms by Hecke and Eichler. The connection to automorphic forms and representation theory was systematized in work of Harish-Chandra, Robert Langlands, and James Arthur, making Siegel modular varieties central to the study of automorphic representations and Eisenstein series investigated by Stephen Gelbart and Henniart. Geometric compactifications and cohomology computations involve methods due to Ash, Mumford, and Pink.

Level structures and compactifications

Level structures—introduced in the arithmetic moduli literature by Igusa and formalized by Grothendieck and Mumford—rigidify the moduli to produce fine moduli schemes for congruence subgroups of Sp(2g, Z). Toroidal and minimal (Baily–Borel) compactifications, constructed by Mumford, Faltings, Chai, and Baily & Borel, provide projective models that accommodate degenerations to semiabelian varieties studied by Alexeev and Künnemann. These compactifications interface with mixed Hodge structures developed by Wilfred Schmid and Deligne, and with the boundary components classified by Ash and Mumford.

Arithmetic and reduction modulo p

Reduction modulo primes and integral models were explored by Tate and Grothendieck and later refined by Kottwitz, Rapoport, and Zink in the theory of models with parahoric level structure. The study of Newton polygons, Ekedahl–Oort strata, and p-divisible groups uses the contributions of Manin, Oort, and Faltings, and is essential for understanding the special fibers of Shimura varieties of type A_g as in work of Kisin and Vasiu. p-adic Hodge theory techniques due to Fontaine and Faltings relate crystalline cohomology of abelian varieties to Galois representations significant for Wiles's modularity methods and ongoing Langlands-related reciprocity.

Torelli theorem and relations to other moduli spaces

The Torelli theorem identifies the locus of Jacobians of algebraic curves inside the moduli of principally polarized abelian varieties, a statement classically due to Rita Gunning's modern expositions and attributed to ideas of Riemann and proven in full by Andreotti and Frankel in many accounts. This embedding relates the moduli of curves M_g, constructed by Deligne and Mumford, to A_g and connects to Prym varieties studied by Mumford and Beauville. Relations with Hilbert modular varieties of David Hilbert and unitary Shimura varieties studied by Shimura and Taniyama arise from additional endomorphism structures, linking to the theory of complex multiplication developed by Shimura and Tate.

Applications and examples

Examples include the classical case g=1 where the moduli is the modular curve studied by Modular group actions, Hecke operators introduced by Hecke, and the arithmetic of elliptic curves central to Andrew Wiles and Freeman Dyson's broader narratives. Higher-dimensional examples appear in the work of Igusa on theta constants, Mumford's examples of degenerations, and explicit families used by Faltings in Diophantine applications. Applications span the proof strategies for cases of the Langlands program, investigations of rational points influenced by Faltings and Raynaud, and computational aspects in explicit class field theory going back to Kronecker and Weber.

Category:Algebraic geometry