Mordell–Weil lattices for higher genus fibration over a curve

Summary

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 Néron-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).