Comparison between analytic and algebraic constructions of toroidal compactifications of PEL-type Shimura varieties (Q2879872)

A construction of degenerating abelian varieties was given by D. Mumford. It generalises Tate's elliptic curve. Over a normal ring \(R\) complete in a \(I\)-adic topology he divides an algebraic torus \(T\) by a group of periods. These are given by a symmetric bilinear form \(b\) on the character group \(X\) of \(T\) satisfying \(b(x, x)\in I\) for \(x\neq 0\). For this one embeds \(T\) into a relatively complete model \(\widetilde P\), divides the associated formal scheme by the periods (which operate properly in the Zariski-topology), and then shows that the quotient is algebraic. Everything is parallel to the complex theory except that one does not divide a complex vector-space by \(2g\) periods but first divides by \(g\) of them and then the resulting algebraic torus by the remaining \(g\). For example sections of the ample line bundle defining the polarisation are given by theta-functions which have the same Fourier-expansion. This can be used to construct a converse to Mumford, by recovering the periods from the Fourier-coefficients. Finally this converse allows to construct the toroidal compactification of the Siegel moduli space, as a parameter space for definite bilinear forms \(b(\, ,\,)\).NEWLINENEWLINE The purpose of the paper under review is to show that the associated complex space coincides with the complex toroidal compactification defined by analytic means. For this the key fact is that the formal completions along the strata coincide. For this the author shows that one gets the same theta series. Another possibility (which the author dislikes) would be to perform the Mumford construction in the analytic context, that is to show that the periods operate properly discontinuously over a suitable open subset (``\(|b(x, x)|<1\)'') of \(\widetilde P\).NEWLINENEWLINE The presentation of the subject is obscured by some unavoidable difficulties: Firstly, one has to consider semi-abelian degenerations, not just tori. Secondly, the author has discovered in his thesis that for subgroups of the symplectic group level structures are more complicated than one might think at first glance, and of course, he tries to incorporate them here. Both of these complications enforce a heavy board of notations.