On the Γ-factors attached to motives
On the Γ-factors attached to motives
复制标题
论与动机相关的 Γ 因素
DOI：
10.1007/bf01245075
复制
发表时间：
1991
期刊：
影响因子：
--
通讯作者：
C. Deninger
中科院分区：
文献类型：
--
作者：
C. Deninger
考虑数域k上的光滑射影簇X。在[Se2]中，Serre对ka的每个位置v定义了“动机”HW(X)的局部欧拉因子，其中0<；w<；2dimX。对于有限位置，这本质上是作用于L上同调的Frobenius的特征多项式。对于无穷多处v，局部因子由一个公式给出，该公式涉及F-函数和kw上X=X·kkv的Hodge结构的某些数值不变量。我们猜想，在代换S~-，w+1-S下，取k的所有地方的局部因子的乘积推广到~上的亚纯函数和一个简单的函数方程。所有可以证明这一点的情况都使用了自同构L函数理论。目前还看不到一般函数方程的证明。作为最低限度的先决条件，我似乎有必要以概念性的方式理解无限大的局部因素的性质。利用Fontaine环Bor的一个简单的阿基米德类似物Bo，我们定义了一个新的上同调理论H~*，称为R或r上光滑射影簇的阿基米德上同调.向量空间H~在IR上是无限维且带有自然自同态.我们证明了在无限行列式的适当定义下，逆“特征多项式”
Consider a smooth and projective variety X over a number field k. In [Se2] Serre defines for every place v of ka local Euler factor of the" motive" Hw (X) where 0< w< 2dimX. For the finite places this is essentially a characteristic polynomial of Frobenius acting on l-adic cohomology. For the infinite places v the local factor is given by a formula involving F-functions and certain numerical invariants of the Hodge structure of X~= X• kkv over k w. There is no obvious analogy to the local factors at the finite places. It is conjectured that the product of the local factors taken over all places of k extends to a meromorphic function on~ with a simple functional equation under the substitution s~--, w+ 1-s. All cases where this can be verified use the theory of automorphic L-functions. A proof of the general functional equation is not in sight. As a minimal prerequisite it seems necessary to me to understand the nature of the local factors at infinity in a conceptual way. The present note is adressed to this problem.Using a simple archimedian analogue Bo, of Fontaine's ring BoR we define a new cohomology theory H* called archimedian cohomology for smooth projective varieties over R or r The vector spaces H~ are infinite dimensional over IR and carry a natural endomorphism@. We show that with a suitable definition of an infinite determinant the inverse" characteristic polynomial"