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Canonical subgroups and p-adic vanishing cycles for abelian varieties. (Sous-groupes canoniques et cycles évanescents $$p$$-adiques pour les variétés abéliennes.) (French) Zbl 1062.14057
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 [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. B. Dwork showed in [appendix of P. Deligne, 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. P. Deligne and B. Dwork [Publ. Math., Inst. Hautes Étud. Sci 37, 27–115 (1969; Zbl 0284.14008)] and 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 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 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.

MSC:
 14K10 Algebraic moduli of abelian varieties, classification 14K15 Arithmetic ground fields for abelian varieties 14F30 $$p$$-adic cohomology, crystalline cohomology
Keywords:
vanishing cycles; abelian varieties
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References:
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