On some questions related to the Gauss conjecture for function fields

Yves Aubry (1, 2) , Régis Blache (3)

1 IMATH - Institut de Mathématiques de Toulon - EA 2134
2 IML - Institut de mathématiques de Luminy
3 LAMIA - Laboratoire de Mathématiques Informatique et Applications

We show that, for any finite field Fq , there exist infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is a separable polynomial. As pointed out by Anglès, this is a necessary condition for the existence, for any finite field Fq, of infinitely many real function fields over Fq with ideal class number one (the so-called Gauss conjecture for function fields). We also show conditionally the existence of infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is an irreducible polynomial.

Mots clés: Functions fields; Gauss conjecture; Zeta functions; Jacobian; Hyperelliptic curves; Finite fields

Domaines: Géométrie algébrique [math.AG]; Théorie des nombres [math.NT]

Yves Aubry, Régis Blache. On some questions related to the Gauss conjecture for function fields. Journal of Number Theory, 2008, 128 (7), pp.2053--2062. ⟨10.1016/j.jnt.2007.10.014⟩. ⟨hal-00978911⟩