The Siegel moduli variety is the coarse moduli variety of principally polarized abelian varieties of dimension . An abelian variety polarization comes from an ample line bundle and gives a homomorphism from the abelian variety to its dual abelian variety, which parametrizes algebraically trivial line bundles. The polarization is principal when this homomorphism is an isomorphism. Over , the analytic space is the quotient of the Siegel upper half-space by . The upper half-space consists of complex symmetric matrices whose imaginary parts are positive definite. Its complex dimension is .

In particular, parametrizes elliptic curves, while is a threefold parametrizing principally polarized abelian surfaces. An algebraic curve in is Hodge generic when it is contained in no proper special subvariety. The Zilber-Pink prediction for such a curve concerns finiteness of its intersections with the union of all special curves and special points.

See also

Abelian Variety, Abelian Variety Polarization, Complex Multiplication, Dual Abelian Variety, Elliptic Curve, Moduli Space, Special Subvariety, Zilber-Pink Conjecture

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References

Daw, C. and Orr, M. "Unlikely Intersections with Curves in ." Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22, 1705-1745, 2021. https://doi.org/10.2422/2036-2145.202006_014.Milne, J. S. "Shimura Varieties and Moduli." In Handbook of Moduli, Vol. II (Ed. G. Farkas and I. Morrison). Somerville, MA: International Press, pp. 467-548, 2013. https://www.jmilne.org/math/xnotes/svhIP.pdf.

Cite this as:

Weisstein, Eric W. "Siegel Moduli Variety." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SiegelModuliVariety.html

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