Fiber products and class groups of hyperelliptic curves

Abstract

Abstract. Let F q be a finite field of odd characteristic, and let N be an odd natural number. An explicit fiber product construction shows that if N divides the class number of some quadratic function field over F q , then it does so for infinitely many such function fields.The asymptotic behavior of -torsion in class groups cl(K) of quadratic function fields K over finite fields F q is now well understood [EVW09], and to a lesser extent it has been apprehended for two decades [Ach06,FW89]. 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. Theorem 1 flows readily from a special case of a fiber product construction in [GP05], as follows.Let F be a field in which 2 is invertible. A quadratic function field K/F is a quadratic extension of the rational field F(t). Equivalently, it is the field of rational functions on a hyperelliptic curve, i.e., K = F(X) for some smooth, projective curve X/F equipped with an involution ι such that X/ ι is isomorphic to P 1 . Let g be the genus of X. Then X is determined by its branch locus B ⊂ P 1 , a reduced divisor of degree 2g + 2. Although B is necessarily defined over F, it may not admit any F-rational points.Let t be a coordinate on P 1 , and suppose that B is disjoint from {0, ∞} ⊂ P 1 t . Then B is the vanishing locus of a squarefree monic polynomial h(t) ∈ F[t] of degree 2g + 2 such that h(0) = 0, andt , if B is defined over F, then so is √ B. The field K/F is called imaginary if it admits a model F(u)[ f (u)] with deg f odd; equivalently, the branch locus B contains an F-rational point.