Class groups of global function fields with certain splitting behaviors of the infinite prime

Abstract

Abstract. For certain two cases of splitting behaviors of the prime at infinity with unit rank r, given positive integers m, n, we construct infinitely many global function fields K such that the ideal class group of K of degree m over F(T ) has n-rank at least m − r − 1 and the prime at infinity splits in K as given, where F denotes a finite field and T a transcendental element over F. In detail, for positive integers m, n and r with 0 ≤ r ≤ m − 1 and a given signature (e i , f i ), 1 ≤ i ≤ r + 1, such that r+1 i=1 e i f i = m, in the following two cases where e i is arbitrary and f i = 1 for each i, or e i = 1 and f i 's are the same for each i, we construct infinitely many global function fields K of degree m over F(T ) such that the ideal class group of K contains a subgroup isomorphic to (Z/nZ) m−r−1 and the prime at infinity ℘ ∞ splits into r + 1 primes P 1 , P 2 , · · · , P r+1 in K with e(P i /℘ ∞ ) = e i and f(P i /℘ ∞ ) = f i for 1 ≤ i ≤ r + 1 (so, K is of unit rank r).