Mordell–Weil lattices for higher genus fibration over a curve

Abstract

Introduction, notation Let K = k(C) be the function field of an algebraic curve C over an algebraically closed ground field k. Let Γ/K be a smooth projective curve of genus g > 0 with a k -rational point O ∈ Γ(K), and let J/K denote the Jacobian variety of Γ/K . Further let (τ, B) be the K/k :-trace of J (see §2 below and). Then the Mordell-Weil theorem (in the function field case) states that the group of K -rational points J(K) modulo the subgroup τB(k) is a finitely generated Abelian group. Now, given Γ/K, there is a smooth projective algebraic surface S with genus g fibration f : S → C which has Γ as its generic fibre and which is relatively minimal in the sense that no fibres contain an exceptional curve of the first kind (−1-curve). It is known that the correspondence Γ/K ↔ ( S, f ) is bijective up to isomorphisms (cf.). The main purpose of this paper is to give the Mordell-Weil group M = J(K)/τB(k) (modulo torsion) the structure of Euclidean lattice via intersection theory on the algebraic surface S. The resulting lattice is the Mordell- Weil lattice (MWL) of the Jacobian variety J/K, which we sometimes call MWL of the curve Γ/K or of the fibration f : S → C . For this, we first establish the relationship between the Mordell-Weil group and the Neron-Severi group NS( S ) of S (Theorem 1, stated in §2 and proved in §3). Then (in §4) we introduce the structure of lattice on the Mordell-Weil group by defining a natural pairing in terms of the intersection pairing on NS( S ).