Siegel modular variety

A 2D slice of a Calabi–Yau quintic. One such quintic is birationally equivalent to the compactification of the Siegel modular variety A1,3(2).[1]

In mathematics, a Siegel modular variety or Siegel moduli space is an algebraic variety that parametrizes certain types of abelian varieties of a fixed dimension. More precisely, Siegel modular varieties are the moduli spaces of principally polarized abelian varieties of a fixed dimension. They are named after Carl Ludwig Siegel,[2] a 20th century German mathematician who specialized in number theory. He introduced[2] Siegel modular varieties in a 1943 paper.[3]

Siegel modular varieties are the most basic examples of Shimura varieties.[4] Siegel modular varieties generalize moduli spaces of algebraic curves to higher dimensions and play a central role in the theory of Siegel modular forms, which generalize classical modular forms to higher dimensions.[1] They also have applications to black hole entropy and conformal field theory.[5]

Construction

The Siegel modular variety Ag, which parametrize principally polarized abelian varieties of dimension g, can be constructed as the complex analytic spaces constructed as the quotient of the Siegel upper half-space of degree g by the action of a symplectic group. Complex analytic spaces have naturally associated algebraic varieties by Serre's GAGA.[1]

The Siegel modular variety Ag(n), which parametrize principally polarized abelian varieties of dimension g with a level n-structure, arises as the quotient of the Siegel upper half-space by the action of the principal congruence subgroup of level n of a symplectic group.[1]

A Siegel modular variety may also be constructed as a Shimura variety defined by the Shimura datum associated to a symplectic vector space.[4]

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