CLOSED POINTS ON CURVES OVER FINITE FIELDS

Yves Aubry (1, 2) , Fabien Herbaut (3, 1) , Julien Monaldi (1)

1 IMATH - Institut de Mathématiques de Toulon - EA 2134
2 I2M - Institut de Mathématiques de Marseille
3 UCA INSPE - Institut national supérieur du professorat et de l'éducation - Académie de Nice

We are interested in the quantity ρ(q, g) defined as the smallest positive integer such that r ≥ ρ(q, g) implies that any absolutely irreducible smooth projective algebraic curve defined over F q of genus g has a closed point of degree r. We provide general upper bounds for this number and its exact value for g = 1, 2 and 3. We also improve the known upper bounds on the number of closed points of degree 2 on a curve.

Mots clés: Algebraic curves; Finite fields; Closed points; Diophantine stability

Domaines: Géométrie algébrique [math.AG]

Yves Aubry, Fabien Herbaut, Julien Monaldi. CLOSED POINTS ON CURVES OVER FINITE FIELDS. 2023. ⟨hal-04245190⟩