ON SOME EXPONENTIAL SUMS
ON SOME EXPONENTIAL SUMS
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DOI：
10.1073/pnas.34.5.204
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发表时间：
1948-01-01
影响因子：
11.1
通讯作者：
WEIL, A
中科院分区：
文献类型：
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作者：
WEIL, A
数学：a.从度量学的角度讲，这是一条射影直线在地面k上的函数场。在这条直线上，我们考虑因子，即具有积分(正或负)系数的点的形式和；我们将自己一劳永逸地限制在k上有理的因子，即使得k上的共轭点具有相同的系数。如果一个除数不包含系数非零的无穷远点，则称它为有限的。除符号外，有限正因子本质上与环k[t]中的理想相同；对于每个这样的因子a，我们附加多项式PA(T)=tn+altn-1+…+an，它产生相应的理想，即，其零点是a中的点，其重数分别等于它们在a中的系数；假设a在k上是有理的，则pa(T)的系数以k为单位；n是a的次数。每个有限因子m可记为m=a-b，其中a，b是有限正因子；对于m，我们附加函数Rm(T)=Pa(T)/Pb(T)；我们有m-0当且仅当a和b，即Pa(T)和Pn(T)是同次的，则k(T)中有且只有一个函数以m为其因子，并且取值为1，即Rm(T)本身。设x是k中非零元素乘法群k*的特征，b是由点t组成的有限因子，系数a，如果R(T)在k(T)中，当R(%)中没有一个是0或co时，我们记R(B)=h1，R(，)Ap；因为b在k上有理，所以R(B)在k中。我们将假设没有a，是x的阶数的倍数。此外，设w是系数在k中的不定T中幂级数的乘群的一个特征；我们假设对于每一个化为单项CTZ的级数，w都有值1。根据通常的定义，如果C0对每个=1mod的级数有1的值，我们就说Co有导体(Tn)。TN，如果N是具有该属性的最小整数。则X的值是PS次单位根，如果p是k的特征，且S使得PS)N。我们写道，对xEk，X(X)=w(1-xt)。对于k(T)中的每个函数R(T)，我们可以附加一个幂函数R(1/T)，它是有理函数R(1/T)按T的幂递增展开而产生的；这正是R(T)在无穷远处的通常展开式。然后定义了w[R(1/T)]；特别地，当co(T)=1时，我们有co[(I-XT)/T]=X(X)。现在，对于每个与b没有共同点的有限除数m，我们写道
MATHEMATICS: A. WEIL metrically, this is the function-field, over the ground-field k, of a projective straight line. On that straight line, we consider divisors, ie, formal sums of points with integral (positive or negative) coefficients; and we limit ourselves, once for all, to divisors which are rational over k, ie, such that conjugate points over k have the same coefficient. A divisor is called finite if it does not contain the point at infinity with a non-zero coefficient. Except for the notation, finite positive divisors are essentially the same as ideals in the ring k [t]; to every such divisor a, we attach the polynomial Pa (t)= tn+ altn-1+...+ an which generates the corresponding ideal, ie, whose zeros are the points in a, with multiplicities respectively equal to their coefficients in a; as a is assumed to be rational over k, Pa (t) has its coefficients in k; and n is the degree of a. Every finite divisor m can be written as m= a-b, where a, b are finite positive divisors; to m, we attach the function Rm (t)= Pa (t)/Pb (t); we have m-0 if and only if a and b, ie, Pa (t) and PN (t), are ofthe same degree, and then there is one and only one function in k (t) having m as its divisor and taking the value 1 at infinity, viz., Rm (t) itself.Let x be a character of the multiplicative group k* of the non-zero elements in k. Let b be a finite divisor, consisting of the points t, with the coefficients a,; if R (t) is in k (t), we shall write R (b)= H1, R (,) ap whenever none of the R (%) is 0 or co; as b is rational over k, R (b) is in k. We shall assume that no a, is a multiple of the order of x. Furthermore, let w be a character of the multiplicative group of power series in an indeterminate T with coefficients in k; we assume that w has the value 1 for every series reduced to a monomial cTz. According to the usual definition, we say that co has the conductor (TN) if it has the value 1 for every power-series which is= 1 mod. TN, and if N is the smallest integer with that property. Then the values of X are ps-th roots of unity, if p is the characteristic of k, and s is such that ps) N. We shallwrite, for xe k, X (x)= w (1-xT). To every function R (t) in k (t), we can attach a power-series R (1/T), arising from the expansion of the rationalfunction R (1/T) according to increasing powers of T; this is no other than the usual expansion of R (t) at infinity. Then w [R (1/T)] is defined; in particular, as co (T)= 1, we have co [(I-xT)/T]= X (x). Now, for every finite divisor m with no point in common with b, we write