We first provide a detailed proof of Kato's classification theorem of log $p$-divisible groups over a noetherian henselian local ring. Exploring Kato's idea further, we then define the notion of a standard extension of a classical finite étale group scheme (resp. classical étale $p$-divisible group) by a classical finite flat group scheme (resp. classical $p$-divisible group) in the category of finite Kummer flat group log schemes (resp. log $p$-divisible groups), with respect to a given chart on the base. These results are then used to prove that log $p$-divisible groups are formally log smooth. We then study the finite Kummer flat group log schemes $T_n(\mathbf{M}):=H^{-1}(\mathbf{M}\otimes_{\mathbb{Z}}^L\mathbb{Z}/n\mathbb{Z})$ (resp. the log $p$-divisible group $\mathbf{M}[p^{\infty}]$) of a log 1-motive $\mathbf{M}$ over an fs log scheme and show that they are étale locally standard extensions. Lastly, we give a proof of the Serre-Tate theorem for log abelian varieties with constant degeneration.
We use the $p$-divisible group attached to a 1-motive to generalize the conjugate $p$-adic uniformization of Iovita--Morrow--Zaharescu to arbitrary $p$-adic formal semi-abelian schemes or $p$-divisible groups over the ring of integers in a $p$-adic field. This mirrors a mixed Hodge theory construction of the inverse uniformization map for complex semi-abelian varieties. We also highlight the geometric structure of the target of the conjugate uniformization map, which is an étale cover of a negative Banach--Colmez space in the sense of Fargues--Scholze.
Rigid analytic 1-motives and conjugate uniformization of abeloid varieties
Morningside Center of Mathematics, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；Institute for Advanced Study in Mathematics, Harbin Institute of Technology（哈尔滨工业大学数学研究所）
As a generalization of Drinfeld modules, Greg Anderson introduced abelian t-modules and t-motives over a perfect field. In this article we study relative versions of these over rings. We investigate isogenies among them. Our main results state that every isogeny possesses a dual isogeny in the opposite direction, and that a morphism between abelian t-modules is an isogeny if and only if the corresponding morphism between their associated t-motives is an isogeny. We also study torsion submodules of abelian t-modules which in general are non-reduced group schemes. They can be obtained from the a…
We develop the analog of crystalline Dieudonné theory for p-divisible groups in the arithmetic of function fields. In our theory p-divisible groups are replaced by divisible local Anderson modules, and Dieudonné modules are replaced by local shtukas. We show that the categories of divisible local Anderson modules and of effective local shtukas are anti-equivalent over arbitrary base schemes. We also clarify their relation with formal Lie groups and with global objects like Drinfeld modules, Anderson's abelian t-modules and t-motives, and Drinfeld shtukas. Moreover, we discuss the existence of …
Pure t-motives were introduced by G. Anderson as higher dimensional generalizations of Drinfeld modules, and as the appropriate analogs of abelian varieties in the arithmetic of function fields. In this article we develop their theory regarding morphisms, isogenies, Tate modules, and local shtukas. The later are the analog of p-divisible groups. We investigate which pure t-motives are semisimple, that is, isogenous to direct sums of simple ones. We give examples for pure t-motives which are not semisimple. Over finite fields the semisimplicity is equivalent to the semisimplicity of the endomor…
We develop a theory of Tannakian Galois groups for t-motives and relate this to the theory of Frobenius semilinear difference equations. We show that the transcendence degree of the period matrix associated to a given t-motive is equal to the dimension of its Galois group. Using this result we prove that Carlitz logarithms of algebraic functions that are linearly independent over the rational function field are algebraically independent.
This is a revised and slightly expanded version. We point out that in the previous summary, "without cohomology" should really read "almost without cohomology" because of the proof of Lemma 2, that the idea to consider effective motives divisible by the Lefschetz motive was anticipated by Serre in letters to Gilles Lachaud and Marc Perret, and finally that the birational invariance of the number of points modulo q in fact follows from a 1983 Comptes Rendus note of Torsten Ekedahl (I am grateful to Antoine Chambert-Loir for making this observation and indicating this reference).
We define ket abelian schemes, ket 1-motives, and ket log 1-motives, and formulate duality theory for these objects. Then we show that tamely ramified strict 1-motives over a complete discrete valuation field can be extended to ket log 1-motives over the corresponding discrete valuation ring. As an application, we present a proof to a result of Kato stated in one of his preprint without proof.
We define the categories of log motives and log mixed motives. The latter gives a new formulation for the category of mixed motives. We prove that the former is a semisimple abelian category if and only if the numerical equivalence and homological equivalence coincide, and that it is also equivalent to that the latter is a Tannakian category. We discuss various realizations, formulate Tate and Hodge conjectures, and verify them in curve case.
We connect two developments aiming at extending Voevodsky's theory of motives over a field in such a way to encompass non-$\mathbf{A}^1$-invariant phenomina. One is theory of reciprocity sheaves introduced by Kahn-Saito-Yamazaki. Another is theory of the triangulated category $\operatorname{\mathbf{logDM}}^{\operatorname{eff}}$ of logarithmic motives launched by Binda, Park and Østvær. We prove that the Nisnevich cohomology of reciprocity sheaves is representable in $\operatorname{\mathbf{logDM}}^{\operatorname{eff}}$.
Let $R$ be a complete discrete valuation ring with perfect residue field $k$ of positive characteristic $p$ and field of fractions $K$ of characteristic 0. In this paper we consider a $K$-1-motive $M_K$ as in [Ra] and its associated Barsotti-Tate group. This last does not in general extend to a Barsotti-Tate group over $R$. However, with some assumptions, it extends to a logarithmic Barsotti-Tate group over $R$. This follows from [Ra] and Kato's results on finite logarithmic group schemes. Once chosen a uniformizing parameter $π$ of $R$, any logarithmic Barsotti-Tate group over $R$ is describe…
We introduce the notion of log motivic triangulated categories, which is the theoretical framework for understanding the motivic aspect of cohomology theories for fs log schemes. Then we study the Grothendieck six operations formalism for log motivic triangulated categories.
In this work we develop a theory of motives for logarithmic schemes over fields in the sense of Fontaine, Illusie, and Kato. Our construction is based on the notion of finite log correspondences, the dividing Nisnevich topology on log schemes, and the basic idea of parameterizing homotopies by $\overline{\square}$, i.e. the projective line with respect to its compactifying logarithmic structure at infinity. We show that Hodge cohomology of log schemes is a $\overline{\square}$-invariant theory that is representable in the category of logarithmic motives. Our category is closely related to Voev…
We prove that a variation of mixed Hodge structure is embedded in a logarithmic variation of pure Hodge structure, and a generalized version of this result. These results suggest some simple construction of the category of mixed motives by using log pure motives.
The goal of this paper is to extend the work of Voevodsky and Morel on the homotopy $t$-structure on the category of motivic complexes to the context of motives for logarithmic schemes. To do so, we prove an analogue of Morel's connectivity theorem and show a purity statement for $(\mathbf{P}^1, \infty)$-local complexes of sheaves with log transfers. The homotopy $t$-structure on $\mathbf{logDM}^{\textrm{eff}}(k)$ is proved to be compatible with Voevodsky's $t$-structure i.e. we show that the comparison functor $R^{\overline{\square}}ω^*\colon \mathbf{DM}^{\textrm{eff}}(k)\to \mathbf{logDM}^{\…
The zeta function of a motive over a finite field is multiplicative with respect to the direct sum of motives. It has beautiful analytic properties, as were predicted by the Weil conjectures. There is also a multiplicative zeta function, which instead respects the tensor product of motives. There is no analogue of the Weil conjectures, and we give a sufficient criterion for an analytic continuation to exist. This happens, for example, for cellular varieties, abelian varieties, or genus g > 1 curves with a supersingular Jacobian.
In this article, we will give the Deligne 1-motives up to isogeny corresponding to the $\mathbb{Q}$-limiting mixed Hodge structures of semi-stable degenerations of curves, by using logarithmic structures and Steenbrink's cohomological mixed Hodge complexes associated to semi-stable degenerations of curves.
We define higher pro-Albanese functors for every effective log motive over a field $k$ of characteristic zero, and we compute them for every smooth log smooth scheme $X=(\underline{X}, \partial X)$. The result involves an inverse system of the coherent cohomology of the underlying scheme as well as a pro-group scheme $\mathrm{Alb}^{\log}(X)$ that extends Serre's semi-abelian Albanese variety of $\underline{X}-|\partial X|$. This generalizes the higher Albanese sheaves of Ayoub, Barbieri-Viale and Kahn.
