Function fields with 3-rank at least 2

Abstract

1. Introduction. It is well known that there are infinitely many quadratic number fields and function fields with class number divisible by a given integer n (see Nagell [15] (1922) for imaginary number fields, Yamamoto [22] (1969) and Weinberger [21] (1973) for real number fields, and Friesen [6] (1990) for function fields). A related question concerns the n-rank of the field, that is, the greatest integer r for which the class group contains a subgroup isomorphic to (Z/nZ) r . In [22], Yamamoto showed that infinitely many imaginary quadratic number fields have n-rank at least 2 for any positive integer n ≥ 2. In 1978, Diaz y Diaz [3] developed an algorithm for generating imaginary quadratic fields with 3-rank 2, and Craig [2] showed in 1973 that there are infinitely many real quadratic number fields with 3-rank at least 2 and infinitely many imaginary quadratic number fields with 3-rank at least 3. A few examples of higher 3-rank have also been found (see for instance Llorente and Quer [14,18] who found three imaginary quadratic number fields with 3-rank 6 in 1987/1988). In a recent paper [4], Erickson, Kaplan, Mendoza, Shayler, and the author gave infinite, simply parameterized families of real and imaginary quadratic fields with 3-rank 2. Here we give a function field analogue.Note that Bauer, Jacobson, Lee, and Scheidler [1] have given algorithms which yield imaginary quadratic function fields with 3-rank at least 2 and a possibly empty set of imaginary quadratic function fields with 3-rank at least 3. The construction below yields infinitely many quadratic function fields, of any given signature, with 3-rank at least 2. See [9], [11], [12], [16], and [17] for constructions of function fields of arbitrary degree m with large n-rank for general n.Throughout we let q be a power of an odd prime, q ≡ 1 (mod 3). We use sgn(f ) to denote the leading coefficient of a polynomial f ∈ F q [T ], and we let |f | = q deg(f ) for f ∈ F q [T ]. The main result is as follows.

Fiber products and class groups of hyperelliptic curves

“…Nonetheless, there is persistent interest in constructing quadratic function fields with control over the class group (e.g., [BJLS08,Pac09] and the references therein). In view of this, the following observation, whose proof is quite explicit, may be of some interest: Note that the field of constants F q is preserved.…”