Higher rank subgroups in the class groups of imaginary function fields

Abstract

Let F be a finite field and T a transcendental element over F. In this paper, we construct, for integers m and n relatively prime to the characteristic of F(T ), infinitely many imaginary function fields K of degree m over F(T ) whose class groups contain subgroups isomorphic to (Z/nZ) m . This increases the previous rank of m − 1 found by the authors in [Y. Lee, A. Pacelli, Class groups of imaginary function fields: The inert case, Proc. Amer. Math. Soc. 133 (2005) 2883-2889].

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

“…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.…”

Surface modifying oligomers used to functionalize polymeric surfaces: Consideration of blood contact applications

“…Surface modification methods can be divided into three different categories: grafting, noncovalent coating, and blending strategies. Surface coating modifications include physical adsorption via intermolecular interactions or dip‐coating, self‐assembled monolayers technology, plasma deposition, Langmuir‐Blodgett techniques, layer by layer, and more recently mussel‐inspired surface modification strategies . Grafting technologies are classified into two categories referred to as “grafting to” and “grafting from” and include covalent reactions such as ozone‐induced grafting, chain growth grafting via surface initiated atom transfer radical polymerization (ATRP), graft‐to‐surface via dipping, crosslinking, or reaction of the specific groups of polymers with the substrate, and plasma treatments .…”

The ℓ-rank structure of a global function field

“…A number of constructions for hyperelliptic curves of high 3-rank were presented in [2]; see also Chapter 7 of [14] for a somewhat different approach to generating such fields via cubic extensions. In a sequence of papers, Pacelli et al found infinite families of quadratic [8,9] and higher degree [11,12] function fields of large 3-rank, and more generally, n-rank.…”