The prime at infinity and the rank of the class group in global function fields

Abstract

In this paper we construct, for any integers m and n, and 2 g m − 1, infinitely many function fields K of degree m over F(T ) such that the prime at infinity splits into exactly g primes in K and the ideal class group of K contains a subgroup isomorphic to (Z/nZ) m−g . This extends previous results of the author and Lee for the cases g = 1 and g = m.

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

“…The same type of polynomial f (X) was also used in [5,6,11,12]. If θ is a root of f (X), then we will show that K = k(θ) satisfies the conditions of Theorem 1.1 and Theorem 1.2.…”

“…Recently, more general function field analogues of these results developed in number fields have been proved by Pacelli and the author in several papers such as [5,6,7,11,12]. In detail, [11] works on the cases where the prime at infinity splits completely (with the guaranteed class group n-rank 1) or is totally ramified (with the guaranteed class group n-rank m − 1), [6] works on the case in which the prime at infinity is inert (also with the guaranteed class group n-rank m − 1), and this result is improved in [7] by increasing the guaranteed class group n-rank from m − 1 to m. The results in [6,7,11] are the unit rank 0 (minimum possible unit rank) or the unit rank m − 1 (maximum possible unit rank).…”

“…In fact, given an integer n, infinitely many number fields and function fields have class number divisible by n (see for example Nagell [8] for imaginary quadratic number fields, Yamamoto [14] for real quadratic number fields, and Friesen [2] for real quadratic function fields). It is known that given integers m and n, infinitely many number fields and function fields of fixed degree m have class number divisible by n (see for example Azuhata and Ichimura [1] and Nakano [9] for number fields, and the author and Pacelli [5,6,7,11,12] for function fields).…”

“…In detail, [11] works on the cases where the prime at infinity splits completely (with the guaranteed class group n-rank 1) or is totally ramified (with the guaranteed class group n-rank m − 1), [6] works on the case in which the prime at infinity is inert (also with the guaranteed class group n-rank m − 1), and this result is improved in [7] by increasing the guaranteed class group n-rank from m − 1 to m. The results in [6,7,11] are the unit rank 0 (minimum possible unit rank) or the unit rank m − 1 (maximum possible unit rank). The arbitrary unit rank case is proved in [12] for the guaranteed class group rank m − r − 1, and this is improved in [5] by increasing the guaranteed class group n-rank from m − r − 1 to m − r. However, these two results in [5,12] assume specific splitting behaviors of the prime at infinity. Therefore, other cases of splitting behaviors of the prime at infinity are still missing.…”

“…Substituting XT β for X in f (X) = 0 and then dividing both sides of Eq. (12) by T mβ , we have that…”