Canonical subgroups and p-adic vanishing cycles for abelian varieties
Let \(k\) be an algebraically closed field of characteristic \(p>0\), \(W=W(k)\) its ring of Witt vectors and \(\sigma\) the endormorphism of Frobenius on \(k\) and \(W\). Let \(A\) be an ordinary \(k\)-abelian variety of dimension \(g\) and \(\mathfrak M\) the moduli space of formal deformations of \(A\) over \(W\)-artinian local algebras with residue field \(k\). It follows from Serre-Tate theory [\textit{N. Katz}, in Surfaces algébriques, Lect. Notes Math. 868, 138--202 (1981; Zbl 0477.14007)] that there exists a canonical isomorphism of \(W\)-formal schemes \(\mathfrak M\overset{\cong}\longrightarrow\text{Hom}_{\mathbb Z_p}(T_pA(k)\otimes T_p\hat{A}(k),\hat{\mathbb G}_m)\), where \(\hat{A}\) is the dual abelian variety of \(A\) and \(T_p\) denotes the \(p\)-adic Tate module. \textit{B. Dwork} showed in [appendix of \textit{P. Deligne}, \textit{L. Illusie}, in: Surfaces algébriques, Lect. Notes Math. 868, 80--137 (1981; Zbl 0537.14012)] that the structure of the formal toric group over \(\mathfrak M\) is imposed by a given \(W\)-morphism \(\Phi:\mathfrak M\to\mathfrak M^{(\sigma)}\) that lifts the Frobenius morphism. In particular, the structure of the formal group of Serre-Tate is completely determined by the canonical lifting of Frobenius \(\Phi_{\text{can}}:\mathfrak M\to\mathfrak M^{(\sigma)}\). In this case Dwork conjectured that the canonical lifting is overconvergent (the problem of the excellent lifting). More precisely, let \(\mathcal X\) be a formal \(W\)-scheme topologically of finite type and \(\mathcal A\to\mathcal X\) a formal abelian scheme such that \(\mathcal A_k\to\mathcal X_k\) is a family of ordinary abelian varieties. Let \({}_p\mathcal A\) be the kernel of multiplication by \(p\), \({}_p\mathcal A^{\text{ét}}\) the greatest étale quotient and \({}_p\mathcal A^0\) the kernel of the natural morphism \({}_p\mathcal A\to{}_p\mathcal A^{\text{ét}}\). We suppose given a cartesian diagram \[ \begin{tikzcd} \mathcal A/{}_p\mathcal A^0\ar[r,"\Theta"]\ar[d] & \mathcal A\ar[d]\\ \mathcal X\ar[r,"\Phi" '] & \mathcal X \end{tikzcd} \] such that \(\Phi\) lifts \(\Phi_k\) and \(\Theta\) lifts the natural morphism \((\mathcal A/{}_p\mathcal A^0)_k=\mathcal A_k^{(\Phi_k)}\to\mathcal A_k\). This diagram exists for certain moduli spaces of abelian varieties. In this case, Dwork conjectured that \(\Phi\) is overconvergent. \textit{P. Deligne} and \textit{B. Dwork} [Publ. Math., Inst. Hautes Étud. Sci 37, 27--115 (1969; Zbl 0284.14008)] and \textit{N. Katz} [in: Modular Forms in One Variable III, Lect. Notes Math 350, 69--190 (1973; Zbl 0271.10033)] proved this conjecture for families of elliptic curves. Later \textit{B. Dwork} [Invent. Math. 12, 249--256 (1971; Zbl 0219.14014)] used this result to prove that the unit \(L\)-function of a universal family of ordinary Legendre elliptic curves admits meromorphic continuation to \(\mathbb C_p\). In this paper the authors show the overconvergence for arbitrary dimension under the hypothesis that \(p\geq3\) and deduce from it an application to the study of certain unity \(L\)-functions. In order to state the main result of the paper, let us assume that \(K\) is a complete valued field of characteristic 0 with perfect residue field \(k\) of characteristic \(p>0\), let \(\mathcal O_K\) be its ring of integers and \(S=\text{Spec}(\mathcal O_K)\). Let \(A\) be an \(S\)-abelian scheme of relative dimension \(g\), \(S_1=\text{Spec}(\mathcal O_K/p\mathcal O_K)\), \(A_1=A\times_SS_1\) and \({}_pA\) the kernel of the multiplication by \(p\). Given a finite and flat \(S\)-group scheme \(G\), we define a canonical decreasing exhaustive filtration \((G_a,a\in\mathbb Q_{\geq0})\) by finite flat closed subgroup schemes. For each real number \(a\geq0\), let \(G^{a+}=\bigcup_{b>a}G^b\), where \(b\in\mathbb Q\). The main result says the following. Suppose \(p\geq3\), denote by \(e\) the absolute ramification index of \(K\) and \(j=e/(p-1)\). Let \(A\) be an abelian \(S\)-scheme of relative dimension \(g\) such that the Hodge height of \(H^1(A_1,\mathcal O_{A_1})\) is strictly smaller than \(\inf\left( \frac1{p(p-1)},\frac{p-2}{(p-1)(2g(p-1)-p)}\right)\). Then the hole \({}_pA^{j+}\) of the canonical filtration of \({}_pA\) is finite and flat over \(S\) with rank \(p^g\). Furthermore, if \(A_k\) is ordinary, then \({}_pA^{j+}\) is the connected neutral component of \({}_pA\). This theorem implies the conjecture of the excellent lifting of Dwork for the moduli spaces of abelian varieties. In the paper under review the authors treat the case of \(\mathfrak A_{g,N}\) which is the moduli space parametrizing abelian schemes of relative dimension \(g\) which are principally polarized and have a level \(N\)-structure. By \textit{D. Mumford} [Geometric Invariant Theory (Springer-Verlag) (1965; Zbl 0147.39304)] this moduli problem is representable by a quasi-projective connected and smooth \(W\)-scheme. Its formal completion along the special fiber contains a formal open set \(Q\) which represents ordinary abelian varieties. By the universal property, \(Q\) comes equipped with a canonical Frobenius lifting \(Q_k\). The authors prove that this lifting is overconvergent.
scientific article; zbMATH DE number 2103609
scientific article