Read e-book online Compactification of Siegel Moduli Schemes PDF

Algebraic Geometry

By Ching-Li Chai

The Siegel moduli scheme classifies largely polarised abelian forms and its compactification is a crucial bring about mathematics algebraic geometry. the most results of this monograph is to end up the lifestyles of the toroidal compactification over Z (1/2). This consequence must have extra purposes and is gifted right here with enough history fabric to make the ebook appropriate for seminar classes in algebraic geometry, algebraic quantity idea or automorphic types.

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In the above definition we can replace SP29(Z) by any subgroup r of finite index. 2). acts on H; continuously. t. exist neighborhoods U of x and U' r, then there of x' such that r·u n u' = ~. t. 9 H* r\ 9 yU = U if y E rx and yU n U = ~ y ~ rx . (i) For any subgroup is a compact Hausdorff space. 22 r c SP29(Z) of finite index, * (ii) (r\Hg )-(r\Hg) has a natural finite stratification. stratum is a quotient of some Hr , r < g, (iii) Each stratum of r~g* Each by a congruence subgroup. has a natural structure of normal analytic space.

N),2in. 7 : G(L) ~ G(8) Over Spec C, automorphisms of G(8) the center Gm(C) = C*. is a symmetric isomorphism. 28)(C) is canonically isomorphic to 37 r(8,28)\ Hg , and the [WJ. theta null morphism N= g II d. - 1 i=l 1 is just the one defined by This furnishes the link between the algebraic and classical theory of theta constants. 38 §5. Toroidal compactification In [SC], Mumford and his coworkers constructed, for each bounded symmetric domain X and an arithmetic subgroup r of Aut(X), class of explicit toroidal compactifications of r\X.

13 ~-operators E: Hg as a Siegel domain of third kind [B-B] for general discussion; see also §4 Then we restrict the canonical extension ~F' * The projectivity of r\Hg and justice to the Rk (r) , f extends in a canonical series. Poincar~-Eisenstein Poincar~-Eisenstein [F], [B-B] for more details. is proved by using Siegel We won't be able to do any series, and instead refer to [Ca], These series produce a lot of modular forms, and R(r) is integral over the subalgebra generated by them. Since the ~-operator Poincar~-Eisenstein maps Poincare-Eisenstein series on Hg Hr , r series on < to g.

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