Class numbers of cyclotomic function fields
Class numbers of cyclotomic function fields
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DOI：
10.4064/aa102-3-4
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发表时间：
2002
期刊：
影响因子：
0.7
通讯作者：
S. Bae;P. Kang
中科院分区：
文献类型：
--
作者：
S. Bae;P. Kang
0. Introduction. Let A = Fq[T ] be the ring of polynomials over a finite field Fq with q elements, and k = F(T ). We assume that q > 2. For each M ∈ A one uses the Carlitz module to construct a field extension k(M), called the Mth cyclotomic function field , and its real subfield k+(M). It is well known that the divisor class number h(M) of k(M) is divisible by the class number h+(M) of k+(M). Write h−(M) = h(M)/h+(M). We call h−(M) the first factor or relative class number , and h+(M) the second factor or real class number of k(M). In a recent paper Guo and Shu [GS] studied h−(Pn) and h+(Pn), where P is an irreducible polynomial in A. Shu obtained very useful formulas for these factors as products of character sums. We use these formulas to obtain upper bounds for these factors in the case of prime cyclotomic function fields, modifying the method of Feng [F]. In the classical case there are certain matrices, called the Maillet matrix and Dem’yanenko matrix, whose determinants are very closely related to the first factor of the class number. We define some matrices analogous to those matrices, which are easy to define and easy to handle. Adopting the ideas of Wang [W] and Hazama [H] and using the formulas of Shu we can express h−(Pn) and h+(Pn) as the determinants of those matrices. The matrices are useful to compute the class numbers in many cases. We use them to improve the lower bound of [GS, Theorem 3.1]. In the final section we find all the possible polynomials M with h−(M) = 1 when q is odd. The cases of h(M) and h+(M) are given in [KM].
