Bad reduction of the Siegel moduli scheme of genus two with \(\Gamma_ 0(p)\)-level structure.

*(English)*Zbl 0734.14010As the authors state the theory of bad reduction of elliptic moduli schemes can be regarded as being matured by now in contrast to the theory of bad reduction of Siegel moduli schemes with level structures. The main reason for this discrepancy lies in the fact that the scheme of n-torsion points is a Cartier divisor in the elliptic case whereas for \(g>1\) one gets just a 0-cycle.

Following an idea of Katz and Mazur there is a formulation of the notion of Drinfel’d level structure with the aid of the so-called “full set of sections” which is a substitute for Cartier divisors and which works in a certain sense in higher dimensions as well. - More precisely for any natural number N let \({\mathcal A}_{g,\Gamma_ 0(N)}\) denote the algebraic stack such that for any scheme S, \({\mathcal A}_{g,\Gamma_ 0(N)}(S)\) will be the category of isogenies \(A_ 1\to_{\phi}A_ 2\to S\) of principally polarized abelian schemes \(\lambda\) : A/S\(\overset \sim \rightarrow A^ t/S\) such that \(\phi^*(\lambda_ 2)=N\cdot \lambda_ 1\). Then there are two natural proper projections \({\mathcal A}_{g,\Gamma_ 0(N)}\to {\mathcal A}_ g\) and in prime case a direct tangent space calculation shows that the ordinary locus \({\mathcal A}_{g,\Gamma_ 0(p)}\) is smooth over \(spec({\mathbb{Z}}_{(p)})\). In case \(g=2\) Dieudonné theory forces the stack \({\mathcal A}_{2,\Gamma_ 0(p)}\) to be regular except a finite number of points which is the core of this long article. In addition these finitely many isolated singularities are actually Cohen-Macaulay and occur when the three irreducible components of the special fibre \({\mathcal A}_{2,\Gamma_ 0(p)}\otimes {\bar {\mathbb{F}}}_ p\) meet and as the authors remark “are the points one would think of as being the worst”.

Following an idea of Katz and Mazur there is a formulation of the notion of Drinfel’d level structure with the aid of the so-called “full set of sections” which is a substitute for Cartier divisors and which works in a certain sense in higher dimensions as well. - More precisely for any natural number N let \({\mathcal A}_{g,\Gamma_ 0(N)}\) denote the algebraic stack such that for any scheme S, \({\mathcal A}_{g,\Gamma_ 0(N)}(S)\) will be the category of isogenies \(A_ 1\to_{\phi}A_ 2\to S\) of principally polarized abelian schemes \(\lambda\) : A/S\(\overset \sim \rightarrow A^ t/S\) such that \(\phi^*(\lambda_ 2)=N\cdot \lambda_ 1\). Then there are two natural proper projections \({\mathcal A}_{g,\Gamma_ 0(N)}\to {\mathcal A}_ g\) and in prime case a direct tangent space calculation shows that the ordinary locus \({\mathcal A}_{g,\Gamma_ 0(p)}\) is smooth over \(spec({\mathbb{Z}}_{(p)})\). In case \(g=2\) Dieudonné theory forces the stack \({\mathcal A}_{2,\Gamma_ 0(p)}\) to be regular except a finite number of points which is the core of this long article. In addition these finitely many isolated singularities are actually Cohen-Macaulay and occur when the three irreducible components of the special fibre \({\mathcal A}_{2,\Gamma_ 0(p)}\otimes {\bar {\mathbb{F}}}_ p\) meet and as the authors remark “are the points one would think of as being the worst”.

Reviewer: F.W.Knoeller (Marburg)

##### MSC:

14G35 | Modular and Shimura varieties |

14K10 | Algebraic moduli of abelian varieties, classification |

14K02 | Isogeny |

11G18 | Arithmetic aspects of modular and Shimura varieties |