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On the reduction of the Hilbert-Blumenthal-moduli scheme with \(\Gamma_0(p)\)-level structure. (English) Zbl 0916.14022

Fix a totally real number field \(F/\mathbb{Q}\) of degree \(g\) and a prime number \(p\), remaining prime in \(F\). In this paper we study the reduction mod \(p\) of Hilbert-Blumenthal varieties of level \(\Gamma_0(p)\) where \(p\) denotes a fixed prime number. This is a special case of a Shimura variety \(\text{Sh}_C (G,X)\) for which the \(p\)-primary part \(C_p\subset G(\mathbb{Q}_p)\) of the subgroup \(C\subset G(\mathbb{A}_f)\) is of parahoric type. For these Shimura varieties, one may address the two problems:
(1) Determine the local structure of the natural model \(M_C/\mathbb{Z}_{(p)}\) of this Shimura variety.
(2) Determine the global structure of the reduction mod \(p\) of this model.
Treating the second of these questions, the so called “supersingular locus” on \(M_C\otimes\mathbb{F}\) (where \(\mathbb{F}\) is an algebraic closure of \(\mathbb{F}_p)\) is of particular interest. It is always a closed subset of \(M_C\). The most well-known example of these Shimura varieties, given by the data \(G=\text{GL}(2)\), \(C_p=\left\{ \left(\begin{smallmatrix} a & b\\ c & d \end{smallmatrix} \right)\in\text{GL}(2,\mathbb{Z}_p) | c\equiv 0\pmod p\right\}\), is studied by P. Deligne and M. Rapoport [in: Modular Functions one variable. II, Proc. Int. Summer School, Antwerp 1972, Lect Notes Math. 349, 143-316 (1973; Zbl 0281.14010)] and serves as a prototype in this paper. The Shimura variety associated to these data – the elliptic moduli curve of level \(\Gamma_0(p)\) – possesses a model \(M_C\) over \(\mathbb{Z}_{(p)}\). Putting \(C_p'=\text{GL}(2,\mathbb{Z}_p)\) and \(C'=C^pC_p'\) and denoting by \(M_{C'}\) the model of \(\text{Sh}_{C'}(G,X)\) over \(\mathbb{Z}_{(p)}\) (classifying elliptic curves with level-\(C^p\) structure), the main results of the Deligne-Rapoport paper cited above with respect to the two questions above are: (1) The scheme \(M_C\) is regular, has the relative dimension one over \(\mathbb{Z}_{(p)}\), and possesses semi-stable reduction. (2) Over \(M_{C'} \otimes \mathbb{F}_p\) there is a section \(Fr\) to the canonical projections \(p_1:M_C \otimes\mathbb{F}_p\to M_{C'}\otimes\mathbb{F}_p\) (resp. a section \(Ver\) to \(p_2:M_C\otimes \mathbb{F}_p\to M_{C'}\otimes\mathbb{F}_p)\) being given by the Frobenius morphism (resp. the Verschiebung). The special fiber \(M_C\otimes\mathbb{F}_p\) is the union of \(Fr(M_{C'} \otimes\mathbb{F}_p)\) and \(Ver(M_{C'}\otimes\mathbb{F}_p)\), these two closed one-dimensional subschemes intersecting transversally in exactly the supersingular points.
The aim of this paper is to generalize these results to Hilbert-Blumenthal varieties. We succeed to give answers to the questions posed above in the two-dimensional case. An \(F\)-cyclic isogeny \(f: (A,i)\to (A',i')\) between abelian schemes with real multiplication in \(F\) is an isogeny \(f:(A,i)\to(A',i')\) of degree \(p^g\) satisfying \(\ker(f)\leq{}_pA\). It is shown that the moduli scheme \(M_{\Gamma_0(p)}/\mathbb{Z}_{(p)}\) classifying \(F\)-cyclic isogenies (+ additional data) has a model with semi-stable reduction.
For \(g=2\), the paper gives a global description of the supersingular locus on \(M_{\Gamma_0(p)}\) and on \(M_{\text{abs}}\) where \(M_{\text{abs}}/\mathbb{Z}_{(p)}\) denotes the moduli scheme classifying abelian schemes with real multiplication: All irreducible components \(C_i\) of \((M_{\text{abs}}\otimes\mathbb{F}_{p^2})^{ss}\) are isomorphic to \(\mathbb{P}^1\), two components meeting in at most one point and each point in \(C_i(\mathbb{F}_{p^2})\) being the intersection of \(C_i\) with another component. The irreducible components \(R_i\) of \((M_{\Gamma_0(p)} \otimes \mathbb{F}_{p^2})^{ss}\) are all smooth surfaces; the set of these components is in bijection with the components \(C_i\). Each one of the canonical projections \(p_j\), \(j=1,2\), gives \(R_i\) the structure of a rationally ruled surface over some component \(C_i\).

MSC:

14K10 Algebraic moduli of abelian varieties, classification
14G35 Modular and Shimura varieties

Citations:

Zbl 0281.14010
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