Mordell–Weil Groups: Structure & Applications

• Mordell–Weil groups are finitely generated groups of rational points on abelian varieties, decomposing into a finite torsion subgroup and a free rank part.
• Descent methods, including Selmer and signed variants, alongside control theorems in Iwasawa theory, are key techniques for determining their structure and rank.
• Their explicit structure underpins advances in arithmetic geometry by linking rational points with Galois representations, L-functions, and topological invariants.

A Mordell–Weil group is the group of rational points of an abelian variety (most classically, an elliptic curve) defined over a field, taken with respect to addition on the variety. The foundational Mordell–Weil theorem asserts that for an abelian variety $A$ over a number field (or more generally, for an abelian variety over a finitely generated field over its prime field), the group $A(K)$ of $K$-rational points is finitely generated, i.e., $$A(K) \simeq A(K)_{\mathrm{tors}} \oplus \mathbb{Z}^r$$ for some integer rank $r$. The structure, arithmetic, and geometric properties of Mordell–Weil groups are central in modern algebraic and arithmetic geometry, with profound connections to Galois representations, L-functions, Selmer and Tate–Shafarevich groups, and topological invariants of algebraic varieties.

1. Definitions and Basic Structure

Given a base field $K$, an abelian variety $A$ over $K$ defines a group $A(K)$ under its group law. The Mordell–Weil group $A(K)$ is the group of $K$-rational points: $$A(K) = \{ P \in A(\overline{K}) \mid P \text{ is fixed by } \operatorname{Gal}(\overline{K}/K) \}$$ For a number field $K$ (or function field of finite type), the Mordell–Weil theorem implies $A(K)$ is finitely generated. The structure theorem provides a decomposition: $A(K) \simeq A(K)_{\text{tors}} \oplus \mathbb{Z}^r$ where $A(K)_{\text{tors}}$ is the finite torsion subgroup and $r \in \mathbb{Z}_{\geq 0}$ is the Mordell–Weil rank. The determination of $r$ and the explicit structure of $A(K)/A(K)_{\text{tors}}$ are central challenges in arithmetic geometry.

For Jacobians of higher genus curves over function fields or over number fields, the Mordell–Weil group refers similarly to $J(K)$, where $J$ is the Jacobian.

2. Methods of Analysis: Descent, Selmer, and Control Theories

Determining the structure of the Mordell–Weil group involves several interlocking techniques:

• Descent via Isogeny/Selmer Groups:

One classical approach uses the exact sequence:

$0 \to A(K)/nA(K) \to \mathrm{Sel}^n(A/K) \to \Sha(A/K)[n] \to 0$

where $\mathrm{Sel}^n(A/K)$ denotes the $n$-Selmer group and $\Sha(A/K)$ the Tate–Shafarevich group. The computation of the Selmer group provides finite group approximations to $A(K)/nA(K)$; bounding its size, together with torsion, yields bounds or the exact value of the rank $r$ (Behrens, 2018).

• Control Theorems in Iwasawa Theory:

For a $\mathbb{Z}_p$-extension $K_\infty/K$, Iwasawa module theory allows for the paper of the growth and structure of Mordell–Weil and fine Mordell–Weil groups via the Pontryagin duals and their characteristic ideals over the Iwasawa algebra $\Lambda = \mathbb{Z}_p[[\Gamma]]$ (Lei, 2023, Gambheera et al., 27 Jul 2025).

• Selmer group refinements:

For curves with supersingular reduction, "signed" (plus/minus) variants of Mordell–Weil and Selmer groups can be defined, leading to a layered structure reflected in signed $p$-adic $L$-function divisibility and congruences (Lei, 2023, Gambheera et al., 27 Jul 2025).

• Topological and Geometric Inputs:

For Jacobians of curves over function fields, methods express the Mordell–Weil rank in terms of homomorphism groups between model Jacobians over the constant field, using adaptations of the Shioda–Tate formula and the geometry of fibered surfaces (Ulmer, 2010). Relationships with Alexander polynomials and cyclic covers appear in the context of elliptic threefolds (Cogolludo-Agustin et al., 2010, Libgober, 2012).

3. Explicit Structure and Refined Invariants

Recent work has focused on explicit structural decompositions of Mordell–Weil and fine Mordell–Weil groups:

• Cyclotomic Decomposition and Characteristic Ideals:

For an elliptic curve $E$ over $K$ in a cyclotomic $\mathbb{Z}_p$-extension $K_\mathrm{cyc}$, under certain finiteness assumptions on fine Tate–Shafarevich groups, the Pontryagin dual of the fine Mordell–Weil group admits an explicit decomposition:

$$M(E/K_\mathrm{cyc})^\vee \sim \bigoplus_n \left(\Lambda/\langle \Phi_n(x) \rangle\right)^{e_n - \theta_n}$$

where $e_n$ is the rank jump at the $n$-th layer, $\Phi_n(x)$ is the $p^n$-th cyclotomic polynomial, and $\theta_n$ quantifies growth in the relevant $p$-adic image (Gambheera et al., 27 Jul 2025).

• Plus/Minus Mordell–Weil Groups:

For supersingular reduction and $a_p(E)=0$, the plus and minus Mordell–Weil groups $M^\pm(E/K_\mathrm{cyc})$ have characteristic ideals given by

$$\operatorname{char}_\Lambda(M^+(E/K_\mathrm{cyc})^\vee) = \left\langle x^{e_0} \prod_{n > 0,\, n \text{ odd}} \Phi_n(x)^{e_n - \theta_n} \prod_{n > 0,\, n \text{ even}} \Phi_n(x)^{e_n}\right\rangle$$

and analogously interchanging odd/even for $M^-$ (Gambheera et al., 27 Jul 2025).

• Equivariant Structures:

Extensions to equivariant settings analyze the action of Galois groups and express the dual fine Mordell–Weil group as a $\Lambda[G]$-module, yielding decompositions indexed by irreducible representations of $G$ and associated arithmetic invariants (Gambheera et al., 27 Jul 2025).

4. Mordell–Weil Groups in Function Field and Geometric Settings

The behavior over function fields and in geometric families displays a variety of phenomena:

• Explicit Rank Formulas in Towers:

For Jacobians of certain pencils on fibered surfaces (e.g., Berger's construction), the Mordell–Weil rank over $k(t^{1/d})$ is governed by the group of homomorphisms between model Jacobians over the base $k$, up to correction terms from the fiber data. For example,

$$\operatorname{Rank} \mathrm{MW}(J_{X_d}) = \operatorname{Rank} \operatorname{Hom}_{k-\mathrm{av}}(J_{C_d}, J_{D_d})^{\mu_d} - c_1(d) + c_2(d)$$

enabling explicit construction of explicit points of large rank in positive characteristic and moderately high rank in characteristic zero (Ulmer, 2010).

• Topological Interpretation for Isotrivial Abelian Varieties:

For isotrivial abelian varieties with cyclic holonomy over function fields $K = \mathbb{C}(x,y)$, the Mordell–Weil rank is determined by the cyclotomic factors in the Alexander polynomial of the discriminant complement, connecting algebraic and topological invariants (Libgober, 2012).

• Kummer-Faithful Fields and Infinite Rank:

Over certain large algebraic extensions (e.g., those fixed by finitely many automorphisms or generated by Mordell–Weil type points), the divisibility or freeness of $A(K)/A(K)_\text{tors}$ can display infinite rank, but the divisible part may vanish; in certain cases the group is $\aleph_0$-free, in others not free at all (Asayama et al., 7 Aug 2024).

5. Applications in Arithmetic, Geometry, and Physics

• Arithmetic Applications:

The explicit structure of Mordell–Weil groups underlies algorithms for effective determination of rational points on elliptic curves and higher abelian varieties, control theorems in Iwasawa theory, and significant progress on the Birch and Swinnerton-Dyer conjecture in both analytic and equivariant settings (Burns et al., 2015, Shimada, 2022).

• Automorphism Groups and Lattice Theory:

For elliptic K3 surfaces, the action of the Mordell–Weil group on the Néron–Severi lattice is computable and plays a central role in determining the automorphism group of the surface; such computations give insight into rational curve configurations and moduli (Shimada, 2022).

• F-Theory and Physics:

In string-theoretic models, the torsion in the Mordell–Weil group encodes refined discrete data about the global gauge group topology and constraints on the matter spectrum of F-theory compactifications. For elliptic fibrations with nontrivial Mordell–Weil torsion, the torsion subgroup directly enters the structure of the coweight and weight lattices and produces non-simply-connected gauge groups (Mayrhofer et al., 2014, Esole et al., 2017).

6. Connections to Selmer, Fine, and Signed Mordell–Weil Groups

A significant focus has emerged on fine Selmer and fine Mordell–Weil groups—subgroups defined by imposing stricter local conditions at $p$-adic places, especially in the Iwasawa-theoretic context. The structure of these groups, including the plus/minus variants in the supersingular setting, is given by explicit formulas in terms of cyclotomic polynomials; their characteristic ideals match analytic invariants arising from $p$-adic $L$-functions for the curve (Lei, 2023, Lim, 2023, Gambheera et al., 27 Jul 2025). The greatest common divisor of their characteristic ideals matches predicted divisors of the analytic $p$-adic $L$-functions (Kurihara–Pollack Conjecture).

These refinements provide tools to investigate deep relationships between arithmetic invariants (Selmer, Mordell–Weil, and Tate–Shafarevich groups), topological invariants (Alexander polynomials, fundamental groups), and analytic invariants (special $L$-values and $p$-adic $L$-functions) across both number fields and function fields.

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In summary, the theory of Mordell–Weil groups lies at the arithmetic–geometric interface, with explicit module structures determined by descent, cohomological, and Iwasawa-theoretic methods, and deep interrelations to topology, geometry, and arithmetic of abelian varieties and their moduli. The recursive structure of these groups—refined through signed and fine variants, and made explicit via characteristic ideals and equivariant decompositions—continues to drive research at the confluence of number theory, algebraic geometry, and mathematical physics.