Capitulation problem for global function fields

Abstract

Let q be a power of an odd prime number p, k = Fq(t) be the rational function field over the finite field Fq. In this paper, we construct infinitely many real (resp. imaginary) quadratic extensions K over k whose ideal class group capitulates in a proper subfield of the Hilbert class field of K. The proof of the infinity of such fields K relies on an estimation of certain character sum over finite fields. Mathematics Subject Classification (2010). Primary 11R58;Secondary 11A15, 11R37.

Twists of Elliptic Curves

“…In other words, when does E tw correspond to the trivial element in H 1 G K/K , Aut K (E) , under the assumption that it comes from a non-trivial element in the group H 1 G K/K , −1 = Hom G K/K , Z/2Z . This question is analogous to a similar question in algebraic number theory and in function field arithmetic, namely the so-called capitulation (or principalization) problem for ideals, see, e.g., [2,3,10,16].…”

“…This question is analogous to a similar question in algebraic number theory and in function field arithmetic, namely the so-called capitulation (or principalization) problem for ideals, see, e.g., [2,3,10,16].…”

GENUS FIELDS OF CYCLIC l-EXTENSIONS OF RATIONAL FUNCTION FIELDS

“…B. Anglès and J.-F. Jaulent [1] used narrow S-class groups to establish the fundamental results of genus theory for finite extensions of global fields, where S is an arbitrary finite set of places. Using the genus theory for quadratic function fields, Y. Li and S. Hu [11] obtained an analogue in the function field framework of the number field case by constructing infinitely many real (resp. imaginary) quadratic extensions K over F q (T ) whose ideal class group capitulates in a proper subfield of the Hilbert class field of K. G. Peng [13] explicitly described the genus theory for Kummer function fields.…”