A FAMILY OF ARITHMETIC SURFACES OF GENUS 3 Jordi

The study of a curve from the arithmetical or the Arakelov viewpoints is a hard task, since it involves a very good knowledge of its geometry (differential forms, periods), its arithmetic (locus of bad reduction, stable models) and its analysis (Green function). Classically, two families of curves have been extensively studied: Fermat curves and modular curves. The study of these curves is feasible because they have a large automorphism group. On the other hand, the curves in these families have variable genus. If one wants to study the behaviour of some arithmetical or Arakelov invariants on the moduli space of curves of a given genus, these families are not useful. The Arakelov invariants of elliptic curves are completely determined ([Fa84]). Some concrete examples of Arakelov invariants for curves of genus 2 were provided by Bost, Mestre and Moret-Bailly in [Bo-M-M90]. We present here the study of a family of curves of genus 3. Let n ∈ N, n ≥ 2 be a natural number such that n ≡ 2 (mod 3) and n ≡ 0, 1 (mod 25), and consider the projective curve Cn given by the equation Y 4 = X − (4n− 2)XZ + Z. We have studied the geometry of the curves Cn in [Gu01]. They are nonsingular curves of genus 3. They have a large group of automorphisms, which gives the chance of performing a great deal of calculations on them. In this article we study the curves Cn from the arithmetical and Arakelov viewpoints. We find the stable models of the arithmetic surfaces given by them. Combining both the geometric and the arithmetic information compiled about the curves Cn, we initiate the study of their Arakelov invariants: Their modular height and the self-intersection of their canonical sheaf.