 
Summary: PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 00, Number 0, Pages 000000
S 00029939(XX)00000
FIBER PRODUCTS AND CLASS GROUPS OF HYPERELLIPTIC
CURVES
JEFFREY D. ACHTER
Abstract. Let Fq 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 Fq, 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 Fq is now well understood [EVW09], and to a lesser
extent 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 references therein). In view of this, the
following observation, whose proof is quite explicit, may be of some interest:
Theorem 1. Let Fq be a finite field of odd characteristic. Let K/Fq be a quadratic
function field whose genus is at most (q  3)/2. Let A be a finite abelian group
of odd order. If A is a subgroup of cl(K) (and if K is imaginary), then there are
