Siegel modular variety

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, the 20th-century German number theorist who introduced the varieties in 1943. Siegel modular varieties are the most basic examples of Shimura varieties. Siegel modular varieties generalize moduli spaces of elliptic curves to higher dimensions and play a central role in the theory of Siegel modular forms, which generalize classical modular forms to higher dimensions. They also have applications to black hole entropy and conformal field theory. 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. 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. A Siegel modular variety may also be constructed as a Shimura variety defined by the Shimura datum associated to a symplectic vector space. The Siegel modular variety Ag has dimension g(g + 1)/2. Furthermore, it was shown by Yung-Sheng Tai, Eberhard Freitag, and David Mumford that Ag is of general type when g ≥ 7. Siegel modular varieties can be compactified to obtain projective varieties. In particular, a compactification of A2(2) is birationally equivalent to the Segre cubic which is in fact rational. Similarly, a compactification of A2(3) is birationally equivalent to the Burkhardt quartic which is also rational.