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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
 

Summary: PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 00, Number 0, Pages 000000
S 0002-9939(XX)0000-0
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

  

Source: Achter, Jeff - Department of Mathematics, Colorado State University

 

Collections: Environmental Sciences and Ecology; Mathematics