On the Hilbert-Blumenthal moduli problem.

*(English)*Zbl 1087.14022Let \(F\) be a totally real number field of dimension \(g\) over \({\mathbb Q}\), with ring of integers \({O}_F\); we consider schemes \(S\) over \({\mathbb Z}_{(p)}\). Roughly speaking, a Hilbert-Blumenthal moduli space parametrizes abelian schemes \(A/S\) equipped with a suitable action \(\iota: {O}_F \rightarrow \text{End}_S(A)\). Several (possibly) different notions of “suitable” appear in the literature, including:

(R) The Lie algebra \(\text{ Lie}(A)\) is a locally free \({O}_F \otimes {O}_S\)-module of rank 1;

(DP) Étale-locally on \(S\) there exists an \({O}_F\)-linear isogeny \(A \rightarrow A^\vee\) of degree prime to \(p\);

(K) For each \(x\in F\), the characteristic polynomial of \(\iota(x)\), acting on \(\text{ Lie}(A)\), is equal to a certain standard polynomial.

These notions are due, respectively, to M. Rapoport [Compos. Math. 36, 255–335 (1978; Zbl 0386.14006)], P. Deligne and G. Pappas [Compos. Math. 90, No. 1, 59–79 (1994; Zbl 0826.14027)], and R. E. Kottwitz [J. Am. Math. Soc. 5, No. 2, 373–444 (1992; Zbl 0796.14014)]. If \(S\) is actually a \({\mathbb Q}\)-scheme, these notions trivially coincide; if \(F\) is unramified at \(p\), these conditions remain equivalent. In characteristic dividing the discriminant, however, the relation between these three conditions is more subtle, and untangling this relation is important for constructing integral models of Hilbert-Blumenthal moduli spaces.

C.-F. Yu has proved [Ann. Inst. Fourier 53, No. 7, 2105–2154 (2003; Zbl 1117.14027)] that if \(S\) is the spectrum of a field, then (DP) and (K) are equivalent. The present paper proves (Theorem 1) that (DP) and (K) are equivalent conditions on an abelian scheme over an arbitrary base.

After introducing appropriate notions of level structure, one can define moduli problems \({M}^{\text{R}}\), \({M}^{\text{DP}}\), and \({M}^{\text{K}}\) corresponding to the three conditions listed above. Yu has shown that \({M}^{\text{R}}\) coincides with the smooth locus of (a space representing the functor) \({M}^{\text{DP}}\). The second main result of this paper (Theorem 2) is that \({M}^{\text{DP}}\) and \({M}^K\) are isomorphic.

One step in the proof of Theorem 2 is the study of local models for \({M}^{\text{DP}}\) and \({M}^{\text{K}}\). Étale-locally, each of these spaces is a closed subscheme of a certain Grassmann variety. The author proves (Theorem 3) that in fact these closed subschemes coincide.

(R) The Lie algebra \(\text{ Lie}(A)\) is a locally free \({O}_F \otimes {O}_S\)-module of rank 1;

(DP) Étale-locally on \(S\) there exists an \({O}_F\)-linear isogeny \(A \rightarrow A^\vee\) of degree prime to \(p\);

(K) For each \(x\in F\), the characteristic polynomial of \(\iota(x)\), acting on \(\text{ Lie}(A)\), is equal to a certain standard polynomial.

These notions are due, respectively, to M. Rapoport [Compos. Math. 36, 255–335 (1978; Zbl 0386.14006)], P. Deligne and G. Pappas [Compos. Math. 90, No. 1, 59–79 (1994; Zbl 0826.14027)], and R. E. Kottwitz [J. Am. Math. Soc. 5, No. 2, 373–444 (1992; Zbl 0796.14014)]. If \(S\) is actually a \({\mathbb Q}\)-scheme, these notions trivially coincide; if \(F\) is unramified at \(p\), these conditions remain equivalent. In characteristic dividing the discriminant, however, the relation between these three conditions is more subtle, and untangling this relation is important for constructing integral models of Hilbert-Blumenthal moduli spaces.

C.-F. Yu has proved [Ann. Inst. Fourier 53, No. 7, 2105–2154 (2003; Zbl 1117.14027)] that if \(S\) is the spectrum of a field, then (DP) and (K) are equivalent. The present paper proves (Theorem 1) that (DP) and (K) are equivalent conditions on an abelian scheme over an arbitrary base.

After introducing appropriate notions of level structure, one can define moduli problems \({M}^{\text{R}}\), \({M}^{\text{DP}}\), and \({M}^{\text{K}}\) corresponding to the three conditions listed above. Yu has shown that \({M}^{\text{R}}\) coincides with the smooth locus of (a space representing the functor) \({M}^{\text{DP}}\). The second main result of this paper (Theorem 2) is that \({M}^{\text{DP}}\) and \({M}^K\) are isomorphic.

One step in the proof of Theorem 2 is the study of local models for \({M}^{\text{DP}}\) and \({M}^{\text{K}}\). Étale-locally, each of these spaces is a closed subscheme of a certain Grassmann variety. The author proves (Theorem 3) that in fact these closed subschemes coincide.

Reviewer: Jeff Achter (Fort Collins)

##### MSC:

14G35 | Modular and Shimura varieties |

11F41 | Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces |

11G18 | Arithmetic aspects of modular and Shimura varieties |

14K10 | Algebraic moduli of abelian varieties, classification |