Let $A$ be an abelian variety defined over a field $k$. In this paper we define a filtration $F^{r}$ of the group $CH_{0}(A)$ and prove an isomorphism $\frac{K(k;A,...,A)}{\Sym}\otimes\mathbb{Z}[\frac{1}{r!}]\simeq F^{r}/F^{r+1}\otimes\mathbb{Z}[\frac{1}{r!}]$, where $K(k;A,...,A)$ is the Somekawa K-group attached to $r$-copies of the abelian variety $A$.\\ In the special case when $k$ is a finite extension of $\mathbb{Q}_{p}$ and $A$ has split multiplicative reduction, we compute the kernel of the map $CH_{0}(A)\otimes\Z[\frac{1}{2}]\rightarrow \rm{Hom}(Br(A),\Q/\Z)\otimes\Z[\frac{1}{2}]$, induced by the pairing $CH_{0}(A)\times Br(A)\rightarrow\mathbb{Q}/\Z$.
We prove the equality of two non-logarithmic ramification filtrations defined by Matsuda and Abbes-Saito for the abelianized absolute Galois group of a complete discrete valuation field in positive characteristic. We also compute the refined Swan conductor and the characteristic form of a character of the fundamental group of a smooth separated scheme over a perfect field of positive characteristic by using sheaves of Witt vectors.
Assuming the Mumford-Tate conjecture, we show that the center of the endomorphism ring of an abelian variety defined over a number field can be recovered from an appropriate intersection of the fields obtained from its Frobenius endomorphisms. We then apply this result to exhibit a practical algorithm to compute this center.
A refined non-vanishing of the $p$-adic logarithm of a rational point on an abelian variety
University of Texas at Austin（德克萨斯大学奥斯汀分校）；Princeton University（普林斯顿大学）；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）
Given an integral p-adic variety, we observe that if the integral Hodge--de Rham spectral sequence behaves nicely, then the special fiber knows the Hodge numbers of the generic fiber. Applying recent advancements of integral p-adic Hodge theory, we show that such a nice behavior is guaranteed if the p-adic variety can be lifted to an analogue of second Witt vectors and satisfies some bound on dimension and ramification index. This is a (ramified) mixed characteristic analogue of results due to Deligne--Illusie and Fontaine--Messing. Lastly, we discuss an example illustrating the necessity of t…
In this short note we extend some results obtained in \cite{Gazaki2015}. First, we prove that for an abelian variety $A$ with good ordinary reduction over a finite extension of $\mathbb{Q}_p$ with $p$ an odd prime, the Albanese kernel of $A$ is the direct sum of its maximal divisible subgroup and a torsion group. Second, for a semi-abelian variety $G$ over a perfect field $k$, we construct a decreasing integral filtration $\{F^r\}_{r\geq 0}$ of Suslin's singular homology group, $H_0^{sing}(G)$, such that the successive quotients are isomorphic to a certain Somekawa K-group.
We study duality theorems for the relative logarithmic de Rham-Witt sheaves on semi-stable schemes $X$ over a local ring $\mathbb{F}_q[[t]]$, where $\mathbb{F}_q$ is a finite field. As an application, we obtain a new filtration on the maximal abelian quotient $π^{\text{ab}}_1(U)$ of the étale fundamental groups $π_1(U)$ of an open subscheme $U \subseteq X$, which gives a measure of ramification along a divisor $D$ with normal crossing and $\text{Supp}(D) \subseteq X-U$. This filtration coincides with the Brylinski-Kato-Matsuda filtration in the relative dimension zero case.
We prove that the ring of Siegel modular forms of weight divisible by g+n+1 is isomorphic to the ring of (log) pluricanonical forms on the n-fold Kuga family of abelian varieties and its certain compactifications, for every arithmetic group for a symplectic form of rank 2g>2. We also give applications to the Kodaira dimension of the Kuga variety. In most cases, the Kuga variety has canonical singularities.
Subset-Sum Density Realization in Locally Finite Abelian Groups
Eötvös University（厄特沃什·罗兰大学）；Alfréd Rényi Institute（阿尔弗雷德·雷尼研究所）；Institute of Mathematics and Interdisciplinary Sciences at Xidian University（西安电子科技大学数学与交叉科学研究院）；Institute of Mathematical Sciences, ShanghaiTech University（上海科技大学数学科学研究所）
Let $A$ be an abelian variety over the function field of a smooth projective curve $C$ over an algebraically closed field $k$. We compute the $l$-adic cohomology groups of $C$ with coefficients in the locally constant sheaf associated to $H^1(\bar A,\mathbf{Q}_l)$ in terms of arithmetico-geometric invariants of $A$. We apply this, when $k$ is the algebraic closure of a finite field, to a motivic computation of the $L$-function of $A$.
