Class Number Growth of a Family of Z -Extensions p over Global Function Fields

Let F be a finite field with q elements and of characteristic p. In this q paper, we construct a family of geometric Z -extensions over global p function field k of transcendence degree one over F and study the q asymptotic behavior of class numbers in such Z -extensions. By the analog p of the Brauer]Siegel theorem in function fields, it suffices to investigate w x the genus of each layer of such Z -extensions. In 5 , Gold and Kisilevsky p gave a lower bound of the genus for all geometric Z -extensions of k. They p also constructed Z -extensions whose class numbers grow arbitrarily fast p Ž . see Remark 3, Section 1 of their paper . We want to reverse the direction of investigation and try to construct Z -extensions such that the growth p rate is close to the lower bound. It is known that very typical geometric Z -extensions arise from cyclop tomic extensions, both in the number field case and in the function field case. The first systematic study of cyclotomic function fields was carried Ž w x. out in 1930s by L. Carlitz see 1]4 . Its application in class field theory over rational function fields was found by his student, D. Hayes, in 1974 Ž w x. see 8 . For generalization to any global function field, we refer the w x reader to 9 . First we want to fix some notation. Let ` be a fixed prime of k and Ž . d s deg ` . Let A be the ring of elements of k holomorphic away from ` `. We denote by k the completion of k at ` and V the completion of a ` fixed algebraic closure of k . Let F s F d be the constant field of k and ` ` ` q `