IwasawaL-functions of varieties over algebraic number fields
IwasawaL-functions of varieties over algebraic number fields
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DOI：
10.1007/bf01389099
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发表时间：
1983-06
影响因子：
3.1
通讯作者：
P. Schneider
中科院分区：
文献类型：
--
作者：
P. Schneider
A fascinating task in algebraic number theory is the study of the values of various complex L-functions (Dedekind zeta function, Artin L-function, Hasse-Weil L-function...) at integer points. It has often turned out that these values are essentially rational numbers (see [7]). Therefore it is of course a fundamental problem to give arithmetic interpretations of these numbers. One possibility of attack on this problem seems to be the following: First interpolate these rational numbers by a p-adic L-function and then relate this function to the characteristic power series of an" Iwasawa module" which is naturally associated with the underlying arithmetic problem. But this program is extremely difficult and has been fully established only in special cases (recent work of Mazur/Wiles concerning the" main conjecture" in cyclotomic Iwasawa theory). For this reason, we pursue in this paper the much simpler problem of calculating, up to a p-adic unit, the values of the above mentioned characteristic power series at integer points. A considerable body of work has already been done in this direction. Of course, the results we obtain contain much of this earlier work, and also are compatible with known conjectures about the corresponding values of the complex L-functions. Let X be a proper smooth scheme over an algebraic number field k; let [r be an algebraic closure of k,);:= X xk, and Gk:= Gal (k/k) the absolute Galois group of k. Obviously H~ 2g) is a Gk-mOdule finitely generated and free over Z. It defines by duality an algebraic torus T (X) over k (for example, T (Spec (k)) is the multiplicative group G m over k). In Part II of this paper we shall define and study the Iwasawa L-functions of an arbitrary algebraic torus T over k. In the case T= T (X) they should be viewed as the 0-dimensional Iwasawa L-functions of X, because they depend only on the 0-cohomology of X. Their complex analogue is the 0-dimensional L-function of X in the sense of Serre [32], which is nothing else but the Artin L-function (in the sense of [6] which differs slightly from the original one) associated with the representation of G k on H~()~,@).
A fascinating task in algebraic number theory is the study of the values of various complex L-functions (Dedekind zeta function, Artin L-function, Hasse-Weil L-function...) at integer points. It has often turned out that these values are essentially rational numbers (see [7]). Therefore it is of course a fundamental problem to give arithmetic interpretations of these numbers. One possibility of attack on this problem seems to be the following: First interpolate these rational numbers by a p-adic L-function and then relate this function to the characteristic power series of an" Iwasawa module" which is naturally associated with the underlying arithmetic problem. But this program is extremely difficult and has been fully established only in special cases (recent work of Mazur/Wiles concerning the" main conjecture" in cyclotomic Iwasawa theory). For this reason, we pursue in this paper the much simpler problem of calculating, up to a p-adic unit, the values of the above mentioned characteristic power series at integer points. A considerable body of work has already been done in this direction. Of course, the results we obtain contain much of this earlier work, and also are compatible with known conjectures about the corresponding values of the complex L-functions. Let X be a proper smooth scheme over an algebraic number field k; let [r be an algebraic closure of k,);:= X xk, and Gk:= Gal (k/k) the absolute Galois group of k. Obviously H~ 2g) is a Gk-mOdule finitely generated and free over Z. It defines by duality an algebraic torus T (X) over k (for example, T (Spec (k)) is the multiplicative group G m over k). In Part II of this paper we shall define and study the Iwasawa L-functions of an arbitrary algebraic torus T over k. In the case T= T (X) they should be viewed as the 0-dimensional Iwasawa L-functions of X, because they depend only on the 0-cohomology of X. Their complex analogue is the 0-dimensional L-function of X in the sense of Serre [32], which is nothing else but the Artin L-function (in the sense of [6] which differs slightly from the original one) associated with the representation of G k on H~()~,@).