A Smooth Shioda-Tate Isomorphism for Elliptically Fibered 4-Manifolds

Introduction

We start by giving our main theorem and then explain the objects involved in it. For an elliptic fibration $\pi:M\to B$ with fixed zero-section $\sigma_{0}$, the smooth Mordell–Weil group is

The following is a smooth analog of the classical Shioda–Tate isomorphism between the holomorphic Mordell–Weil group $\operatorname{MW}^{\mathrm{hol}}(\pi)$ of an elliptic surface and the quotient of its Néron–Severi lattice by the trivial lattice ([SS19, Theorem 6.5]).

We apply Theorem 1.1 in concrete examples below. One useful consequence is that for any elliptic surface over a closed base, $H^{2}(M):=H^{2}(M;\mathbb{Z})$ is generated by horizontal elements (elements corresponding to sections) and vertical elements (ones of the trivial lattice). As a key application of Theorem 1.1 and the Shioda–Tate isomorphism, we show the following corollary.

Main Objects. We introduce the main objects of this paper. An elliptic fibration is a triple $(M, B, \pi)$, where $M$ is a smooth complex surface, $B$ is a smooth complex curve and $\pi: M \to B$ is a proper, surjective, holomorphic map whose general fiber is a smooth genus 1 curve. In this paper, we further assume that an elliptic fibration comes equipped with a smooth zero-section $\sigma_0 : B \to M$, that $\pi$ has at least one critical point, and that $\pi$ is relatively minimal (i.e. no fiber contains a complex curve of self-intersection $-1$). The choice of zero-section equips each smooth fiber with a distinguished point, which gives it an abelian group structure. This group structure extends to the smooth locus of the singular fibers, as explained in § 2.2.

Let $\Gamma^{\mathrm{sm}}(\pi)$ denote the group of smooth sections of $\pi: M \to B$, where sections are added pointwise and the identity is the distinguished section $\sigma_0$. The (smooth) Mordell–Weil group of an elliptic fibration $\pi: M \to B$ is

Replacing smooth sections by holomorphic sections of $\pi$ and stipulating that the zero-section is holomorphic yields the classical (holomorphic) Mordell–Weil group $\operatorname{MW}^{\mathrm{hol}}(\pi)$. If the base $B$ is closed, both $\operatorname{MW}(\pi)$ and $\operatorname{MW}^{\mathrm{hol}}(\pi)$ are finitely generated abelian groups (see [4 Theorem 3.4] and [12 Theorem 6.6]).

The trivial lattice associated to an elliptic fibration $\pi: M \to B$ is the subgroup of $H^2(M)$ generated by the Poincaré duals of the zero-section and the irreducible components of all fibers of $\pi$.

Using Theorem 1.1, we compute the Mordell–Weil group of two elliptic K3 surfaces. We fill in the details of these examples in § 5.3.

The elliptic surfaces in Example 1.3 and Example 1.4 are both K3 surfaces and thus diffeomorphic, as shown by Kodaira. The two examples show that the Mordell–Weil group detects the difference between the two elliptic fibration structures.

Proof Strategy

The main idea in proving Theorem 1.1 is to give a cohomological characterization for $\operatorname{MW}(\pi)$ in terms of sheaf cohomology, similarly to [4 Proposition 3.3] and [10 Proposition 3.7]. Such a characterization is made difficult by the possibility of a nontrivial component group of the relative smooth locus of reducible singular fibers, so we proceed by removing the singular fibers.

For an elliptic fibration $\pi:M\to B$, let $\Sigma\subset B$ be the set of critical values of $\pi$. For each $t\in\Sigma$, consider a small open disk $D_t$ around $t$ and assume that the disks $D_t$ are pairwise disjoint. Define

and the compact 4-manifold with boundary

Denote by $\pi^\circ:M^\circ\to B^\circ$ the restriction of $\pi$ (see Figure 2). Any section $\sigma:B^\circ\to M^\circ$ naturally represents an element $[\sigma]\in H_2(M^\circ,\partial M^\circ)$.

Step 1. Construct an isomorphism between $\operatorname{MW}(\pi^\circ)$, the group of isotopy classes of sections of $\pi^\circ$, and the annihilator of the fiber class in $H^2(M^\circ)$. We first express $\operatorname{MW}(\pi^\circ)$ in terms of sheaf cohomology (Proposition 3.1). Using a spectral sequence argument, similar to the one in [4] and [10], we construct the desired isomorphism (Proposition 3.2).

Step 2. Relate the groups $\operatorname{MW}(\pi)$ and $\operatorname{MW}(\pi^\circ)$. For $t\in B$, let $M_t:=\pi^{-1}(t)$. We construct a map $\Psi:\operatorname{MW}(\pi^\circ)\to\bigoplus_{t\in\Sigma}H_1(M_t)$, which can be described geometrically as in § 5.1. The following theorem shows that $\Psi$ is the only obstruction to extending sections of $\pi^\circ$ to sections of $\pi$ up to isotopy.

Step 3. Conclusion Using Step 1, we show that the kernel of $\Psi$ is isomorphic to $H^2(M)/\operatorname{Triv}(\pi)$, which by Step 2 proves Theorem 1.1.

MW($\pi$) as a Subgroup of the Mapping Class Group

The Mordell–Weil group can be rephrased in terms of diffeomorphisms as

Indeed, a diffeomorphism $f\in\operatorname{Diff}(M)$ that acts by translation on each smooth fiber determines a smooth section $f\circ\sigma_0$ of $\pi$. Conversely, any smooth section $\sigma$ determines a unique diffeomorphism $f_\sigma\in\operatorname{Diff}(M)$, that acts by translation by $\sigma$ on the smooth locus of $M$ (see [10 Lemma 3.1]).

A consequence of Theorem 1.1 is that $\operatorname{MW}(\pi)$ acts faithfully on $H^2(M)$. This action factors as

and injectivity of the composition implies that the map $\operatorname{MW}(\pi)\to\pi_0(\operatorname{Diff}(M))$ is injective.

A strong version of Nielsen realization holds for Mordell–Weil elements: every torsion element of the Mordell–Weil group has a holomorphic representative that is torsion.

Theorem 1.6 provides a practical method for computing $\operatorname{MW}(\pi)$. Let $g$ be the genus of $B$. Compute the number of additive and multiplicative singular fibers of $\pi$, denoted $a$ and $m$. [10 Corollary 1.4] implies that the rank of $\operatorname{MW}(\pi)$ is $4g+2a+m-4$. Then determine the torsion part using classical methods.

Related Work

This paper extends previous work of the author [10]. The previous paper computed the narrow Mordell–Weil group, a finite-index subgroup of $\operatorname{MW}(\pi)$ of an elliptic fibration. The smooth Mordell–Weil group was first introduced by Farb and Looijenga [4]. They proved finite generation and computed the smooth Mordell–Weil group of a generic (i.e. assuming that all fibers are nodal) elliptic fibration $\pi: M \to\mathbb{P}^1$. A symplectic version of the Mordell–Weil group under the assumption that all singular fibers are nodal was computed by Hacking-Keating [6]. Related ideas were used by Cox and Zucker [3], who used an injection of the tensor of holomorphic Mordell–Weil group with $\mathbb{Q}$ into $H^1(B,R^1\pi_*\mathbb{Q})$ for computations.

Outline

In Section 2, we collect some background on elliptic surfaces that is used throughout the paper. In Section 3, we give a cohomological characterization for $\operatorname{MW}(\pi^\circ)$ and express this group as the annihilator of the fiber class. Section 4 contains an extensive analysis of an elliptic fibration over a disk with only one singular fiber at the origin, with explicit computations of $\operatorname{MW}(\pi)$ and $\operatorname{MW}(\pi^\circ)$. Section 5.1 establishes a complete obstruction to extending isotopy classes of sections of $\pi^\circ$ to isotopy classes of sections of $\pi$ up to isotopy. In Section 5.2, we prove Theorem 1.1. In Section 5.3, we apply Theorem 1.1 in some concrete examples.

Computational Resource Disclosure. Gemini Pro 3.1 and Claude Opus 5.5 were used in the creation of pictures. GPT-5.6 Terra was involved in checking the proofs and literature search. The only mathematical use of artificial intelligence in this paper was in Example 5.6, which was computed in conversation with GPT-6.1 Sol.

Acknowledgements. I am very grateful to my advisor Benson Farb for suggesting this project, for his constant support, and for extensive comments on earlier drafts. I am thankful to Eduard Looijenga for many useful discussions. I would like to thank Eduard Looijenga, Dan Margalit and Katherine Booth for helpful comments.

Background on Elliptic Surfaces

We record the background which is necessary for the paper.

Notation and Conventions: For an elliptic fibration $\pi: M \to B$, and $b \in B$, we denote $M_t := \pi^{-1}(t)$. All (co)homology is taken with $\mathbb{Z}$-coefficients, unless otherwise specified. For any closed $n$-manifold $X$, we denote by $\operatorname{PD}_X : H_k(X) \to H^{n-k}(X)$ the Poincaré duality isomorphism in $X$. For a compact $n$-manifold $X$ with boundary, we denote by $\operatorname{PD}_{[X,\partial X]} : H^k(X) \to H_{n-k}(X,\partial X)$ and $\operatorname{PD}_{[X,\partial X]} : H^k(X,\partial X) \to H_{n-k}(X)$ the isomorphisms given by Poincaré–Lefschetz duality. We assume that the zero-section $\sigma_0$ is holomorphic in a neighborhood of each critical value$^{1}$. The topology on $\operatorname{Diff}(M)$ and $\Gamma^{\mathrm{sm}}(\pi)$ is the Whitney $C^\infty$ topology.