A study of arithmetic varieties

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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),.