A study of arithmetic varieties

首席调查员以管状邻域和上同调下降为关键词，研究了具有正特征的品种的上同调理论。具体内容如下：(1)单纯变种的管状邻域的构造，(2)刚性上同调中的上同调下降(部分与Chiarellotto合作)，(3)适当且光滑态射的相对刚性上同调的一般有限性，(4)de Rham上同调与Hodge滤子中的上同调下降(与Chiarellotto合作)，(5)单纯族的刚性与De Rham上同调之间的比较映射，以及(6)超收敛同晶的纯净定理。Kato研究了(1)Mumford曲线和具有覆盖的p-adadyorforold(与Cornelissen合作)，以及(2)刚性解析空间的基础(与Fujiwara合作)。木村研究了动机的有限维度。伊藤与首席研究员一起研究了计算有限域上变元的有理点个数的算法。Sugiyama研究了(1)三维流形的Selberg Zeta函数中的BSD型问题和(2)Langland对应的几何模拟。Shiho发展了(1)晶体基团理论和(2)利用对数晶体上同调的重量理论。Sumida研究了数域的几个岩泽不变量。田口在某些条件下的Galois表示的有限性方面得到了结果(与Moon合作)，。他还研究了计算有限域上椭圆曲线有理点个数的快速算法(与佐藤合作)，。

The head investigator studied the cohomology theories for varieties of positive characteristic under the key words : tubular neighborhoods and cohomological descent. The contents are as follow ; (1)Construction of tubular neighborhoods of simplicial varieties, (2)Cohomological descent in rigid cohomology (partially collaborated with Chiarellotto), (3)Generic finiteness of relative rigid cohomology for proper and smooth morphisms, (4)Cohomological descent in de Rham cohomology and Hodge filtrations (collaborated with Chiarellotto), (5)Comparison map between rigid and de Rham cohomologies for simplicial families, and (6)Purity theorem for overconvergent isocrystals.Kato studied (1)Mumford curves and p-adic orbifolds with a covering which are Mumford curves (collaborated with Cornelissen), and (2)the foundations of rigid analytic spaces (collaborated with Fujiwara). Kimura studied the finite dimensionality of motives. Ito worked on algorithms of computing numbers of rational points of varieties over finite fields with the head investigator. Sugiyama studied (1)the BSD type problem in the Selberg Zeta functions for 3-dimentional manifolds and (2)a geometric analogue of Langlands correspondence. Shiho developed (1)the theory of crystalline fundamental group and (2)the theory of weights using logarithmic crystalline cohomology. Sumida studied several Iwasawa invariants for number fields. Taguchi had results on the finiteness of Galois representations with certain conditions (collaborated with Moon),. He also worked on the fast algorithm for computing numbers of rational points of elliptic curves over finite fields (collaborated with Sato),.