21.09.02 · number-theory / arakelov-faltings

Faltings / Mordell theorem

shipped3 tiersLean: none

Anchor (Master): Mordell 1922 *Proceedings of the Cambridge Philosophical Society* 21 (3), 179-192 (originator of the conjecture for curves of genus $\geq 2$ over $\mathbb{Q}$, via the proof of the Mordell-Weil theorem for elliptic curves); Faltings 1983 *Inventiones Mathematicae* 73 (3), 349-366 *Endlichkeitssätze für abelsche Varietäten über Zahlkörpern* (originator paper proving all three finiteness theorems together: Mordell, Shafarevich, Tate); Parshin 1968 *Mathematics of the USSR-Izvestiya* 2 (5), 1145-1170 (the Kodaira-Parshin reduction from Mordell to Shafarevich-type finiteness for families over the curve); Arakelov 1971 *Mathematics of the USSR-Izvestiya* 5 (6), 1277-1302 (Arakelov height theory underlying Faltings' height function); Faltings 1984 in *Arithmetic Geometry* (Cornell-Silverman eds., Springer) (English exposition); Szpiro (ed.) 1985 *Astérisque* 127 (the Paris seminar on Faltings' proof); Bombieri 1990 *Annali della Scuola Normale Superiore di Pisa* (4) 17 (4), 615-640 (simplification via the product-theorem method); Vojta 1991 *Annals of Mathematics* (2) 133 (3), 509-548 *Siegel's theorem in the compact case* (second proof via arithmetic intersection on $C \times C$); Faltings 1991 *Annals of Mathematics* 133 (3), 549-576 *Diophantine approximation on abelian varieties* (Vojta-method extension to subvarieties of abelian varieties, Lang's conjecture); Hindry-Silverman *Diophantine Geometry* (GTM 201, 2000) Ch. E (textbook account); Bombieri-Gubler *Heights in Diophantine Geometry* (Cambridge New Math. Mono. 4, 2006) Ch. 11; Dimitrov-Gao-Habegger 2021 *Annals of Mathematics* (2) 194 (1), 237-298 *Uniformity in Mordell-Lang for curves* (uniform Mordell via the new gap principle, after Mazur-Caporaso-Harris)

Intuition Beginner

A polynomial equation in two variables defines a curve in the plane. If you ask for solutions in integers or rationals, the answer depends sharply on a single number attached to the curve: its genus, an invariant of the topological shape of the complex points. A line has genus zero. A smooth cubic equation like $y^{2} = x^{3} + a x + b$ defines a curve of genus one, the elliptic curve. A smooth quartic plane curve has genus three. Higher-degree curves have higher genus.

For genus zero, rational points are typically infinite — once you have one, you can parameterise the rest by a rational map. For genus one, rational points form a finitely generated group (Mordell-Weil 1922), so there are infinitely many if and only if the rank is positive. For genus two and higher, something dramatic happens: rational points become finite in number. That is the Mordell conjecture, posed in 1922, and Faltings proved it in 1983.

The Fermat equation $x^{n} + y^{n} = z^{n}$ for $n \geq 4$ cuts out a curve of genus at least three on the projective plane. Faltings' theorem gives, for each $n$ , finitely many primitive solutions $(x, y, z)$ . Wiles later showed there are no nontrivial solutions at all, but Faltings was the first to bound the count.

Visual Beginner

A schematic showing three curves stacked by genus: a sphere (genus zero) covered densely by rational points, a torus (genus one) showing a finitely-generated lattice of rational points, and a double-torus (genus two) marked with a small finite set of dots. The image conveys the principle that higher topology forces fewer rational points.

The picture captures the central dichotomy: the source of finiteness is the topology of the curve. Genus zero is too simple to constrain points; genus one allows infinitely many by group-law iteration; genus two and beyond is rigid enough that the rational points cannot proliferate.

Worked example Beginner

Consider the Fermat curve $C_{n} : x^{n} + y^{n} = z^{n}$ as a smooth projective curve in $P^{2}$ over $Q$ . We compute its genus and apply Faltings' theorem.

Step 1. The degree of $C_{n}$ is $n$ . A smooth projective plane curve of degree $d$ has genus $g = (d - 1) (d - 2) /2$ by the genus-degree formula. For $C_{n}$ with $d = n$ , this gives $g = (n - 1) (n - 2) /2$ .

Step 2. For $n = 2$ : $g = 0$ . The conic $x^{2} + y^{2} = z^{2}$ has infinitely many rational points, the Pythagorean triples $(3, 4, 5)$ , $(5, 12, 13)$ , $(8, 15, 17)$ , and so on, parameterised by $(m^{2} - n^{2}, 2 mn, m^{2} + n^{2})$ for coprime $m, n$ . Genus zero matches infinite rational points.

Step 3. For $n = 3$ : $g = 1$ . The cubic $x^{3} + y^{3} = z^{3}$ has finitely many rational solutions in fact (Euler proved no nontrivial ones), but the genus-one machinery alone gives finite generation of the Mordell-Weil group, not finiteness.

Step 4. For $n = 4$ : $g = 3$ . The Fermat quartic has genus three, which exceeds two. Faltings' theorem applies and gives finitely many primitive rational solutions $(x, y, z)$ with $g cd (x, y, z) = 1$ .

Step 5. For $n \geq 4$ in general: $g = (n - 1) (n - 2) /2 \geq 3$ . The hypothesis of Faltings' theorem is met, so the Fermat curve $C_{n}$ has finitely many primitive rational solutions for every $n \geq 4$ . Wiles's 1995 modularity argument later upgraded "finitely many" to "none nontrivial" for $n \geq 3$ .

What this tells us: Faltings' theorem does not solve Fermat directly, since it does not give zero solutions and does not give an effective bound on the size of solutions. It does prove that the count is finite for every $n \geq 4$ , which was the first general result of its kind and the precursor to the modularity proof.

Check your understanding Beginner

Exercise (easy, multiple choice).

Faltings' theorem applies to which class of curves?

A. Every algebraic curve over $Q$ B. Smooth projective curves of genus $0$ over $Q$ C. Smooth projective curves of genus $1$ over $Q$ D. Smooth projective curves of genus $\geq 2$ over a number field

Hint

The Mordell conjecture covers curves where the topology is rigid enough to force finiteness, and the natural setting is curves over number fields, not only over the rationals.

Answer

D. Smooth projective curves of genus $\geq 2$ over a number field.

The Mordell conjecture is stated for curves of genus at least two, over any number field $K$ . Genus zero curves have infinitely many rational points whenever they have one. Genus one curves are elliptic curves and obey the Mordell-Weil theorem, which gives finite generation but not finiteness of rational points.

Formal definition Intermediate+

Let $K$ be a number field, meaning a finite field extension of $Q$ , with ring of integers $O_{K}$ . Let $C$ be a smooth projective curve over $K$ of genus $g$ , where the genus is the dimension of the global differentials $g = dim_{K} H^{0} (C, Ω_{C / K}^{1})$ or equivalently $g = dim_{K} H^{1} (C, O_{C})$ . The set of $K$ -rational points $C (K)$ is the set of morphisms $Spec (K) \to C$ over $Spec (K)$ , which is in bijection with the $K$ -valued points of any chosen affine model.

An abelian variety $A$ over $K$ is a smooth projective group scheme over $K$ of dimension $dim A \geq 1$ . Equivalently $A (\overset{ˉ}{K})$ is a complex torus carrying an algebraic structure. The Jacobian $Jac (C)$ of a smooth projective curve of genus $g$ is a principally polarised abelian variety of dimension $g$ , defined as $Pic^{0} (C)$ ; for $g = 1$ , $Jac (C) = C$ once a base point is chosen.

Let $S$ be a finite set of places of $K$ (a set of nonzero prime ideals of $O_{K}$ together with the archimedean places). An abelian variety $A / K$ has good reduction outside $S$ when, for every place $v \in / S$ , the Néron model of $A$ over the local ring $O_{K, v}$ has smooth proper special fibre. Equivalently the $ℓ$ -adic Galois representation $ρ_{A, ℓ} : G_{K} \to GL (T_{ℓ} A)$ is unramified at every $v \in / S \cup {v : v ∣ ℓ}$ (Néron-Ogg-Shafarevich criterion).

The Tate module $T_{ℓ} A = lim_{n} A [ℓ^{n}] (\overset{ˉ}{K})$ is a free $Z_{ℓ}$ -module of rank $2 dim A$ carrying a continuous $G_{K}$ -action $ρ_{A, ℓ}$ . Two abelian varieties $A, B$ over $K$ are isogenous over $K$ when there exists a surjective $K$ -rational homomorphism $A \to B$ with finite kernel. Isogeny is an equivalence relation. The hom-set $Hom_{K} (A, B)$ is a finitely generated free $Z$ -module.

The Faltings height $h_{F} (A)$ of an abelian variety $A / K$ of dimension $g$ with everywhere semistable reduction is defined by the Arakelov intersection number $h_{F} (A) = de g \overline{ω}_{A / O_{K}} / [K : Q]$ , where $\overline{ω}_{A / O_{K}}$ is the metrised top exterior power of the relative differentials on the Néron model, equipped with the Faltings metric on the archimedean fibres. For curves, $h_{F} (Jac (C))$ is the Faltings height of $C$ .

Counterexamples to common slips

The genus assumption is sharp. A smooth projective curve of genus zero with at least one $K$ -rational point is isomorphic to $P_{K}^{1}$ , which has $∣ C (K) ∣ = \infty$ since $P^{1} (K)$ contains $K \cup {\infty}$ , infinite for any number field $K$ . So the genus- $\geq 2$ hypothesis cannot be relaxed to genus $\geq 1$ .
The smoothness assumption matters. A singular plane curve of high degree can have infinitely many rational points coming from the singular locus. Faltings' theorem is about the smooth projective model of an irreducible curve, not about plane equations that happen to have high degree.
The base field must be a number field. Over an algebraically closed field, every curve has infinitely many rational points. Over a function field $k (B)$ for a curve $B$ over $F_{q}$ , the analogous statement is the Mordell conjecture for function fields, proved earlier by Grauert 1965 and Manin 1963 by entirely different methods.
Effectivity fails in Faltings' original proof. The proof argues by contradiction inside a sequence of isogenies; it gives no constructive procedure to enumerate the rational points. This is independent of the truth of the finiteness statement.

Key theorem with proof Intermediate+

Theorem (Faltings 1983; Mordell conjecture). Let $K$ be a number field, and let $C$ be a smooth projective geometrically irreducible curve over $K$ of genus $g \geq 2$ . Then $C (K)$ is finite.

The proof goes through three statements that Faltings established in the same paper, of which Mordell is a corollary of the second.

Theorem (Faltings 1983; Shafarevich finiteness). Let $K$ be a number field, let $S$ be a finite set of places of $K$ including the archimedean places, and fix an integer $g \geq 1$ . The set of $K$ -isomorphism classes of $g$ -dimensional principally polarised abelian varieties over $K$ with good reduction outside $S$ is finite.

Theorem (Faltings 1983; Tate isogeny conjecture). For $K$ a number field and $A, B$ abelian varieties over $K$ , the natural map $$ \mathrm{Hom}K(A, B) \otimes\mathbb{Z} \mathbb{Z}\ell \longrightarrow \mathrm{Hom}{G_K}(T_\ell A, T_\ell B) $$ is an isomorphism for every prime $ℓ$ . In particular two abelian varieties are isogenous over $K$ if and only if their $ℓ$ -adic Galois representations $V_{ℓ} A = T_{ℓ} A \otimes Q_{ℓ}$ and $V_{ℓ} B$ are isomorphic as $G_{K}$ -representations.

Proof outline (the three statements together). The argument is a tournament of finitenesses in which each step bounds a count by the next. The pivot is the Faltings height $h_{F} (A)$ on the moduli space of principally polarised abelian varieties.

Step 1 (Faltings height is bounded under isogeny). For an isogeny $φ : A \to B$ of abelian varieties over $K$ , Faltings establishes the inequality $$ |h_F(B) - h_F(A)| \leq C_1(K, g) \log(\deg \varphi), $$ together with a stronger statement when the isogeny is induced by an $ℓ$ -power torsion subgroup: among the abelian varieties $A^{'}$ isogenous to a fixed $A$ via $ℓ$ -power isogenies, the height $h_{F} (A^{'})$ takes finitely many values. The proof uses Arakelov intersection theory on the Néron model and a comparison of metrised differentials before and after the isogeny.

Step 2 (finitely many abelian varieties with bounded height). Northcott's finiteness theorem for heights on quasi-projective varieties applies to the moduli space $A_{g}$ of principally polarised abelian varieties of dimension $g$ , suitably compactified. With the Faltings height as a Weil height on $A_{g}$ , the set of $K$ -points of bounded height is finite once the degree $[K : Q]$ is fixed. So a bound on $h_{F}$ produces a finite set of isomorphism classes.

Step 3 (Shafarevich from height-control of isogenies plus Néron-Ogg-Shafarevich). Fix $g$ and $S$ . An abelian variety $A / K$ with good reduction outside $S$ has Tate module $T_{ℓ} A$ unramified outside $S \cup {v : v ∣ ℓ}$ . By Hermite-Minkowski there are only finitely many number-field extensions of $K$ of bounded degree ramified inside any finite set. Faltings combines this with Step 1: the isogeny class of $A$ contains finitely many height-distinct members, the moduli space has finitely many points of any given height bound, and the constraints on Tate-module ramification trap the isogeny class within a finite list. This establishes the Shafarevich statement.

Step 4 (Tate from Shafarevich). Given two abelian varieties $A, B$ with $V_{ℓ} A ≅ V_{ℓ} B$ as $G_{K}$ -representations, the family of $G_{K}$ -stable $Z_{ℓ}$ -lattices in $V_{ℓ} A$ produces a family of abelian varieties $A_{n}$ all isogenous to $A$ and all with $T_{ℓ} A_{n} ≅ T_{ℓ} B$ as $G_{K}$ -modules. The good-reduction set $S$ is the same for every $A_{n}$ since the Tate module determines ramification (Néron-Ogg-Shafarevich). By Shafarevich finiteness, the family ${A_{n}}$ contains only finitely many isomorphism classes, so among the $A_{n}$ there is a pair $A_{n} ≅ A_{m}$ matching by a $G_{K}$ -equivariant isomorphism that descends to an isogeny $A \to B$ . The Tate isogeny statement follows.

Step 5 (Mordell from Shafarevich, via Kodaira-Parshin). Parshin's construction (Parshin 1968): given a curve $C$ of genus $g \geq 2$ over $K$ and a rational point $P \in C (K)$ , construct a finite étale cover $\tilde{C}_{P} \to C$ of bounded degree (in terms of $g$ alone) and form the Jacobian $J_{P} := Jac (\tilde{C}_{P})$ . The covering is set up so that the discriminant of $\tilde{C}_{P} / C$ is supported on $P$ , and as $P$ varies over $C (K)$ the family ${J_{P}}$ has good reduction outside a fixed finite set $S$ determined by $C$ alone. Each $J_{P}$ is an abelian variety of dimension uniformly bounded by a constant $g^{'} (g)$ . By Shafarevich there are only finitely many $K$ -isomorphism classes among the $J_{P}$ . By a separate finiteness statement (the Torelli theorem combined with the Mordell-Weil theorem for the cover), each isomorphism class of $J_{P}$ corresponds to finitely many points $P \in C (K)$ . Therefore $C (K)$ is finite. $□$

Bridge. The Faltings proof builds toward every modern result that controls rational points on higher-genus curves through their Jacobians. The foundational reason finiteness holds is exactly that the Faltings height, an Arakelov invariant of an abelian variety, is bounded along an isogeny class and that Northcott's theorem then forces the isogeny class to contain finitely many isomorphism classes. The central insight is that putting these together with the Kodaira-Parshin construction identifies the set of $K$ -rational points on the curve with a subset of an isogeny class of abelian varieties whose ambient moduli space has the Northcott property. This is exactly the same height-finiteness mechanism that appears again in 21.06.01 (Modularity Theorem and BSD) where the Néron-Tate height on an elliptic curve controls the rank of $E (Q)$ and generalises to the higher-dimensional setting of Heegner-point heights. The bridge is that the Faltings height on $A_{g}$ and the Néron-Tate height on $E (Q)$ are both Arakelov-theoretic Weil heights pulled back from compactified moduli spaces, and both obey the Northcott finiteness property. Putting these together, one Arakelov-height framework runs through Mordell, Shafarevich, Tate, Mordell-Weil, BSD, and Iwasawa theory, identifying height growth with arithmetic complexity across all of these settings.

Exercises Intermediate+

Exercise 4 (medium, short-answer).

State the Néron-Ogg-Shafarevich criterion and explain how it enters the Shafarevich finiteness proof.

Hint

The criterion relates good reduction of an abelian variety at a place $v$ to the ramification of its $ℓ$ -adic Galois representation at $v$ .

Answer

The Néron-Ogg-Shafarevich criterion (1968): for an abelian variety $A / K$ and a place $v$ of $K$ with $v ∤ ℓ$ , $A$ has good reduction at $v$ if and only if the action of the inertia subgroup $I_{v} \subset G_{K}$ on the Tate module $T_{ℓ} A$ is the identity action — equivalently, $ρ_{A, ℓ}$ is unramified at $v$ . In the Shafarevich finiteness proof, this is the bridge from the geometric condition "good reduction outside $S$ " to the Galois-theoretic condition " $ρ_{A, ℓ}$ unramified outside $S \cup {v : v ∣ ℓ}$ ". By Hermite-Minkowski, the latter condition cuts out finitely many extensions of $K$ of bounded degree, controlling the possible Tate modules. Combined with Faltings' height-bounded-under-isogeny inequality and Northcott's theorem on heights, the bound finishes the proof. Rubric: full credit for the statement plus the explicit role in the Shafarevich step.

Exercise 5 (medium, short-answer).

Explain why the Kodaira-Parshin construction reduces Mordell's conjecture to Shafarevich finiteness.

Hint

For each rational point $P$ , build a finite étale cover of $C$ whose Jacobian is an abelian variety of bounded dimension with good reduction outside a fixed $S$ .

Answer

The Kodaira-Parshin construction associates to each $P \in C (K)$ a finite étale cover $\tilde{C}_{P} \to C$ whose ramification is supported on $P$ and whose degree is bounded by a constant $d (g)$ . The Jacobian $J_{P} = Jac (\tilde{C}_{P})$ is a principally polarised abelian variety of dimension $g^{'} := g^{'} (g) \leq d (g) \cdot g$ over $K$ (or a quadratic extension), and the good-reduction set of $J_{P}$ is contained in a fixed finite set $S$ that depends only on $C$ and not on $P$ (the discriminant of the cover lives over $P$ , but the bad reduction of the Jacobian is controlled by the bad reduction of the base curve). By Shafarevich finiteness, the set of $K$ -isomorphism classes ${[J_{P}] : P \in C (K)}$ is finite. A separate counting argument using Torelli plus Mordell-Weil for the cover shows that each isomorphism class corresponds to finitely many $P$ . So $C (K)$ is finite. Rubric: full credit for the bounded-degree-cover step plus the iteration into Shafarevich plus the Torelli closing.

Exercise 6 (medium, symbolic).

For two abelian varieties $A, B$ over a number field $K$ , state precisely the Tate isogeny conjecture (proved by Faltings) and derive the corollary that $A$ is isogenous to $B$ if and only if $V_{ℓ} A ≅ V_{ℓ} B$ as $G_{K}$ -representations.

Hint

Apply the isomorphism $Hom_{K} (A, B) \otimes Z_{ℓ} = Hom_{G_{K}} (T_{ℓ} A, T_{ℓ} B)$ to extract a nonzero $K$ -rational morphism from a $G_{K}$ -equivariant Tate-module isomorphism.

Answer

Tate isogeny conjecture (Faltings 1983). The natural map $Hom_{K} (A, B) \otimes_{Z} Z_{ℓ} \to Hom_{G_{K}} (T_{ℓ} A, T_{ℓ} B)$ is an isomorphism for every prime $ℓ$ .

Derivation of the isogeny corollary. Suppose $V_{ℓ} A ≅ V_{ℓ} B$ as $G_{K}$ -representations via some $G_{K}$ -equivariant isomorphism $ψ$ . Then $ψ \in Hom_{G_{K}} (V_{ℓ} A, V_{ℓ} B)$ , and after rescaling by a power of $ℓ$ we may assume $ψ$ sends $T_{ℓ} A$ into $T_{ℓ} B$ . By the Tate isogeny statement, $ψ$ comes from an element $φ \in Hom_{K} (A, B) \otimes Z_{ℓ}$ . Clearing denominators in the $Z_{ℓ}$ -factor produces a nonzero $Z$ -rational element $φ_{0} \in Hom_{K} (A, B)$ . Since $ψ$ is an isomorphism of $V_{ℓ}$ -spaces, $φ_{0}$ has finite kernel, hence is an isogeny. The converse is immediate: an isogeny $φ : A \to B$ induces an isomorphism $V_{ℓ} A \to V_{ℓ} B$ . Rubric: full credit for the statement plus the rescaling argument plus the converse.

Exercise 7 (hard, short-answer).

Outline Vojta's 1991 proof of Mordell using arithmetic intersection theory on $C \times C$ , identifying where it differs from Faltings' original argument.

Hint

Vojta's method uses Mumford's gap inequality between the Néron-Tate height of two points on the Jacobian, the geometry of the diagonal on $C \times C$ , and an arithmetic Bezout / product theorem in place of the moduli-of-abelian-varieties step.

Answer

Vojta 1991 strategy. Fix infinitely many distinct rational points $P_{1}, P_{2}, \dots \in C (K)$ for contradiction. By the Mordell-Weil theorem applied to $Jac (C)$ , the Néron-Tate height $\hat{h}$ on $Jac (C) (K)$ has Mumford's gap property: for $P, Q \in C (K) \subset Jac (C) (K)$ with $\hat{h} (P), \hat{h} (Q)$ large enough, the angle between $P$ and $Q$ in the height-norm is bounded below. Vojta constructs, on the arithmetic surface $C \times C$ over $Spec O_{K}$ , a global section $s$ of a high tensor power of an ample line bundle that vanishes to high order along the diagonal $Δ$ . Estimating the height of $(P_{i}, P_{j}) \in (C \times C) (K)$ from above (via $s$ ) and from below (via the arithmetic Bezout / product theorem of Faltings 1991) produces an inequality that an infinite sequence of points cannot satisfy. The contradiction forces $C (K)$ to be finite.

Difference from Faltings 1983. Faltings' first proof routes through the moduli space $A_{g}$ of abelian varieties and the Faltings height there; Vojta's second proof works directly on the curve via arithmetic intersection on $C \times C$ and replaces the moduli-space step with an explicit auxiliary-section / Bezout argument. Bombieri 1990 streamlined Vojta's proof further using Mumford's gap inequality plus the product theorem. Both proofs remain ineffective; effective Mordell is open. Rubric: full credit for the gap-inequality input, the diagonal section, the arithmetic Bezout / product theorem, and the comparison to the 1983 proof.

Exercise 8 (hard, short-answer).

Compare Faltings' Mordell theorem with the function-field analogue (Grauert 1965, Manin 1963) and state precisely the difference in effectivity.

Hint

The function-field setting replaces the number field $K$ by the function field $k (B)$ of a curve $B$ over a finite or algebraically closed field. The proof goes via the height-jet bundle of the curve and admits an effective version.

Answer

Function-field Mordell (Grauert 1965; Manin 1963). Let $B$ be a smooth projective curve over an algebraically closed field $k$ of characteristic zero, and let $C \to B$ be a non-isotrivial smooth proper family of curves of genus $g \geq 2$ . Then the set of sections $B \to C$ (the function-field analogue of $C_{η} (k (B))$ ) is finite. The proof uses the geometry of the jet bundle of $C$ over $B$ and yields an effective bound on the number of sections in terms of the geometric invariants of the family.

Comparison to Faltings 1983. The number-field statement is finiteness of $C (K)$ ; the function-field statement is finiteness of $C (B)$ . The number-field proof yields no effective bound on the heights of the points in $C (K)$ , hence no algorithm to list them. The function-field proof yields effective height bounds via the explicit jet-bundle computation. The two proofs share the moral structure (a height-bound argument on the Jacobian) but diverge sharply on effectivity: the absence of an archimedean "infinity" place in the function-field setting eliminates the analytic obstruction that defeats effectivity over number fields. Rubric: full credit for the statements plus the explicit identification of the effectivity gap.

Advanced results Master

Theorem (Faltings 1991; Lang's conjecture for subvarieties of abelian varieties). Let $A$ be an abelian variety over a number field $K$ , let $X \subseteq A$ be a Zariski-closed subvariety, and let $Γ \subseteq A (K)$ be a finitely generated subgroup. Then $X (K) \cap Γ$ is contained in a finite union of translates of abelian subvarieties of $A$ contained in $X$ .

In the special case where $X$ contains no positive-dimensional translate of an abelian subvariety, this gives $∣ X (K) \cap Γ∣ < \infty$ . Applied with $Γ = A (K) = Jac (C) (K)$ via Mordell-Weil and $X = C ↪ Jac (C)$ for a curve of genus $\geq 2$ embedded by a fixed base point, this recovers Mordell as a special case. The Vojta-Faltings 1991 proof uses arithmetic intersection theory on a high product $A^{N}$ together with the product theorem of Faltings.

Theorem (Uniform Mordell-Lang; Dimitrov-Gao-Habegger 2021). Let $K$ be a number field. There exists a constant $c (g, d, r)$ depending only on $g \geq 2$ , $d = [K : Q]$ , and $r = rank Jac (C) (K)$ such that $$ #C(K) \leq c(g, d, r) $$ for every smooth projective curve $C / K$ of genus $g$ with Mordell-Weil rank of its Jacobian equal to $r$ .

The bound is uniform: it does not depend on $C$ beyond the listed invariants. The proof builds on Caporaso-Harris-Mazur's 1997 conditional result (under Lang's conjecture, $# C (K) \leq c (g, d)$ depending only on $g$ and $d$ ), Stoll's gap-principle work in the 2000s and 2010s, and the new gap principle of Gao-Habegger 2019, with the Pila-Zannier strategy applied to the Faltings-Zhang functional transcendence on $A_{g}$ providing the inductive engine.

Theorem (Vojta 1991; arithmetic intersection proof of Mordell). The Mordell conjecture admits a second proof via arithmetic intersection theory on the surface $C \times C / Spec O_{K}$ , using an auxiliary section of a high tensor power of an ample line bundle that vanishes along the diagonal, an arithmetic Bezout-type product theorem to estimate the heights of $(P_{i}, P_{j})$ , and Mumford's gap inequality for the Néron-Tate height on $Jac (C) (K)$ to force the sequence $P_{1}, P_{2}, \dots$ to be finite.

Bombieri's 1990 Annali Pisa exposition simplifies Vojta's proof using the Faltings product theorem, and is the most accessible modern account. The Vojta method later generalises to Faltings 1991 for subvarieties of abelian varieties, completing Lang's conjecture in its formulation as the Mordell-Lang conjecture for finitely generated subgroups.

Theorem (effective Mordell remains open). No proof of Faltings' theorem produces, given a curve $C / K$ of genus $\geq 2$ , an effective upper bound $H$ on the heights of the points in $C (K)$ depending computably on $C$ and $K$ .

Effective Mordell is a longstanding open problem. Partial effective bounds exist in special cases (the Chabauty-Coleman method when $rank Jac (C) (K) < g$ ; Kim's nonabelian Chabauty refinements 2005 onwards) but the general case is open. An effective bound would yield an algorithm to enumerate $C (K)$ and would settle longstanding Diophantine questions including the explicit list of rational points on individual curves of arithmetic interest. Coleman-Chabauty produces effective bounds for $r < g$ ; Coleman 1985 and Kim 2005-2020 expand the technique. The full effective Mordell program is consonant with the still-conjectural ABC conjecture: a proof of ABC would yield effective Mordell.

Theorem (Coleman-Chabauty effective method for $r < g$ ). Let $C$ be a smooth projective curve over $Q$ of genus $g \geq 2$ with $Mordell - Weil$ rank $r = rank Jac (C) (Q) < g$ . For each prime $p$ of good reduction, there is an explicit upper bound $$ #C(\mathbb{Q}) \leq #C(\mathbb{F}_p) + 2r, $$ and an explicit algorithm to list $C (Q)$ as the zero set of finitely many $p$ -adic analytic functions on $C (Q_{p})$ obtained by integrating logarithm-of-Coleman-integral 1-forms on $Jac (C)$ that vanish on the image of $Jac (C) (Q)$ .

This is the effective half of Mordell that is currently available. The $r < g$ assumption is essential to the method, since it ensures the existence of nontrivial vanishing differentials on the Jacobian. Coleman 1985 Duke Mathematical Journal 52 gave the foundational construction; Kim 2005 onward extended to nonabelian Chabauty via the unipotent fundamental group.

Theorem (Tate isogeny conjecture, Faltings 1983). For abelian varieties $A, B$ over a number field $K$ and any prime $ℓ$ , $Hom_{K} (A, B) \otimes_{Z} Z_{ℓ} = Hom_{G_{K}} (T_{ℓ} A, T_{ℓ} B)$ . Equivalently the functor $A \mapsto V_{ℓ} A$ from abelian varieties over $K$ up to isogeny to $ℓ$ -adic $G_{K}$ -representations is fully faithful.

The fully-faithful formulation packages the Tate conjecture as a categorical statement: the isogeny category of abelian varieties over $K$ embeds into the category of continuous $ℓ$ -adic Galois representations of $G_{K}$ . Tate 1966 proved the analogous statement over finite fields. Zarhin 1974 proved it for function fields. Faltings 1983 closed the number-field case and the Tate-Faltings theorem is the bridge that connects modularity, $ℓ$ -adic Galois representations, and isogeny classifications.

Synthesis. The Faltings-Mordell theorem is the foundational reason every rational-point problem on a higher-genus curve reduces, at the cost of effectivity, to a height-finiteness statement on the moduli space of abelian varieties. The central insight is that the Faltings height $h_{F}$ on $A_{g}$ is the load-bearing invariant: it is bounded along an isogeny class, controlled by Northcott's theorem on points of bounded height in a quasi-projective variety, and pulled back from the Jacobian via the Kodaira-Parshin construction to constrain the points of $C (K)$ . Putting these together, one Arakelov-height framework yields Mordell, Shafarevich, and Tate simultaneously, with each of the three theorems leaning on the same height-bounded-under-isogeny inequality. The bridge between the topological hypothesis (genus $\geq 2$ ) and the arithmetic conclusion (finiteness of $C (K)$ ) is exactly the Kodaira-Parshin construction: high genus on the base produces high-dimensional abelian varieties on the cover, and high-dimensional abelian-variety moduli has the Northcott property. This is exactly the same height-finiteness principle that appears again in 21.05.01 ( $ℓ$ -adic Galois representations) where the Tate isogeny theorem identifies the isogeny category of abelian varieties with a category of Galois representations, and in 21.06.01 (Modularity Theorem and BSD) where the modularity bridge is itself an instance of the Tate-Faltings identification applied to elliptic curves and their associated modular abelian varieties.

The two proofs of Mordell that exist as of 2026 — the Faltings 1983 argument via the moduli space of abelian varieties and the Vojta 1991 argument via arithmetic intersection on $C \times C$ — share their moral structure and diverge in technique. The Faltings argument is global-modular: it routes the finiteness through the Northcott property on $A_{g}$ . The Vojta-Bombieri argument is local-Diophantine: it uses Mumford's gap inequality on $Jac (C) (K)$ , an auxiliary section vanishing to high order along the diagonal of $C \times C$ , and an arithmetic Bezout-type product theorem. Both arguments are ineffective, and the effective Mordell problem is open. Conditional approaches include Chabauty-Coleman for $r < g$ , Kim's nonabelian extension, and the ABC conjecture which would imply effective Mordell uniformly. The uniform Mordell conjecture of Caporaso-Harris-Mazur, partially resolved by Dimitrov-Gao-Habegger 2021 in the form $# C (K) \leq c (g, d, r)$ , identifies the bound with a function of the Mordell-Weil rank rather than the curve itself, generalises the absolute finiteness to a uniformity statement, and builds toward the still-conjectural Lang and Vojta principles for varieties of general type.

Full proof set Master

Proposition 1 (Faltings height bounded under isogeny). Let $φ : A \to B$ be an isogeny of abelian varieties over a number field $K$ , both with everywhere semistable reduction. Then $$ h_F(B) = h_F(A) - \tfrac{1}{2} \log \deg \varphi + \tfrac{1}{2} \sum_{v \nmid \infty} (\text{local terms}) $$ with the local terms vanishing at primes of good reduction for both $A$ and $B$ and bounded in absolute value at the others by a constant depending on $dim A$ and the residue characteristic. In particular $∣ h_{F} (B) - h_{F} (A) ∣ \leq C_{1} (g, K) lo g de g φ$ along the isogeny.

Proof. The Faltings height is defined by $h_{F} (A) = (de g \overline{ω}_{A / O_{K}}) / [K : Q]$ , where $\overline{ω}_{A / O_{K}}$ is the metrised top exterior power of the relative differentials on the Néron model of $A$ , equipped with the Faltings $L^{2}$ -metric on the archimedean fibres. An isogeny $φ : A \to B$ induces a comparison of Néron models away from primes where the isogeny ramifies, and a comparison of differentials $φ^{*} ω_{B} \to ω_{A}$ . The induced map on metrised line bundles is an isomorphism away from a finite set, and at each prime of the finite set the change in Arakelov degree is controlled by the local invariant cycles theorem and an explicit calculation. The archimedean contribution uses the Faltings $L^{2}$ -metric and Mumford-Tate's archimedean comparison, with the metric change bounded by the standard volume distortion under an isogeny of complex tori. Adding up the local and archimedean contributions yields the displayed formula. $□$

Proposition 2 (Northcott finiteness on $A_{g}$ ). For each $H, g, d \geq 1$ , the set of $K$ -points of the moduli space $A_{g}$ (of principally polarised abelian varieties of dimension $g$ ) with $h_{F} \leq H$ and $[K : Q] \leq d$ is finite.

Proof. The moduli space $A_{g}$ admits a compactification (Baily-Borel or toroidal) on which the Faltings height extends to a Weil height attached to a fixed ample line bundle. Northcott's theorem (1949): on a quasi-projective variety, the set of points of bounded Weil height defined over fields of bounded degree is finite. Apply this to the compactified $A_{g}$ . The boundary of the compactification consists of degenerations, where $h_{F}$ blows up by the Faltings height theorem, so any point with $h_{F} \leq H$ stays in the interior. The Northcott theorem applied to the line bundle pulled back from the compactification yields the displayed finiteness. $□$

Proposition 3 (Shafarevich finiteness, Faltings 1983). For $K$ a number field, $S$ a finite set of places, and $g \geq 1$ , the set of $K$ -isomorphism classes of $g$ -dimensional principally polarised abelian varieties over $K$ with good reduction outside $S$ is finite.

Proof. Fix an abelian variety $A_{0}$ in the set. The isogeny class of $A_{0}$ over $K$ is parameterised by $G_{K}$ -stable $Z_{ℓ}$ -lattices in $V_{ℓ} A_{0}$ . Each such lattice produces an abelian variety isogenous to $A_{0}$ ; the corresponding $T_{ℓ}$ -module determines the ramification, and by the Néron-Ogg-Shafarevich criterion the ramification is contained in $S \cup {v : v ∣ ℓ}$ . Proposition 1 plus Faltings' isogeny-class height theorem (which strengthens Proposition 1 to a bound on the height that depends only on $A_{0}$ , not on the isogeny) bounds $h_{F}$ across the isogeny class. By Proposition 2 this isogeny class contains finitely many isomorphism classes. To handle the union over isogeny classes: a $g$ -dimensional abelian variety over $K$ with good reduction outside $S$ has Tate module unramified outside $S \cup {v : v ∣ ℓ}$ , so by Hermite-Minkowski plus the bound on dimension of $V_{ℓ} A$ , there are finitely many possible isomorphism classes of $V_{ℓ} A$ as $G_{K}$ -representations. Each such class corresponds to one isogeny class via Tate (Proposition 4 below), and each isogeny class contributes finitely many isomorphism classes by the argument above. The total is finite. $□$

Proposition 4 (Tate isogeny conjecture). For $A, B$ abelian varieties over a number field $K$ and any prime $ℓ$ , $Hom_{K} (A, B) \otimes Z_{ℓ} \to Hom_{G_{K}} (T_{ℓ} A, T_{ℓ} B)$ is an isomorphism.

Proof (assuming Proposition 3). Injectivity is classical (Weil): a $K$ -rational morphism that vanishes on $T_{ℓ} A$ vanishes on all torsion and hence is zero. Surjectivity goes as follows. Given $ψ : T_{ℓ} A \to T_{ℓ} B$ a $G_{K}$ -equivariant homomorphism, the family of $G_{K}$ -stable $Z_{ℓ}$ -lattices $L_{n} = im (ψ) + ℓ^{n} T_{ℓ} B \subset V_{ℓ} B$ produces a family of abelian varieties $B_{n}$ isogenous to $B$ with $T_{ℓ} B_{n} = L_{n}$ . The good-reduction set of every $B_{n}$ equals that of $B$ (Néron-Ogg-Shafarevich, since the Tate-module ramification is preserved). By Proposition 3 there are only finitely many isomorphism classes among the $B_{n}$ , so two of them are equal: $B_{n} ≅ B_{m}$ for some $n \neq = m$ . The induced $G_{K}$ -equivariant map $T_{ℓ} B_{n} \to T_{ℓ} B_{m}$ corresponds to a $K$ -rational endomorphism of $B$ (up to isogeny), and tracing through the construction yields $ψ$ as the $ℓ$ -adic image of a $K$ -rational homomorphism $φ : A \to B$ . $□$

Proposition 5 (Kodaira-Parshin reduction). Let $C$ be a smooth projective curve of genus $g \geq 2$ over $K$ . There exist constants $d (g)$ , $g^{'} (g)$ , a finite extension $K^{'} / K$ , and a finite set $S^{'} = S^{'} (C)$ of places of $K^{'}$ , such that for every $P \in C (K)$ there is a finite étale cover $\tilde{C}_{P} \to C$ of degree dividing $d (g)$ , defined over $K^{'}$ , whose Jacobian $J_{P} = Jac (\tilde{C}_{P})$ is a principally polarised abelian variety of dimension $\leq g^{'} (g)$ over $K^{'}$ with good reduction outside $S^{'}$ .

Proof. Parshin's 1968 construction: fix an integer $n \geq 2$ . For each rational point $P \in C (K)$ , the function field $K (C)$ admits a quadratic extension $K (C) (f_{P})$ ramified exactly at $P$ and at a fixed auxiliary divisor of degree depending only on $g$ and $n$ , where $f_{P} \in K (C)$ is constructed to have divisor $(f_{P}) = n \cdot P - n \cdot Q_{0}$ for a fixed base point $Q_{0} \in C (K^{'})$ . Apply this to a base extension $K^{'} / K$ large enough to contain a fixed $n$ -th root of $f_{P}$ . The resulting double cover $\tilde{C}_{P} \to C$ has degree $2$ , is ramified at $P$ and at the auxiliary divisor, and has genus $\tilde{g}$ determined by Riemann-Hurwitz: $\tilde{g} = 1 + 2 (g - 1) + (de g ramification) /2$ , bounded by a constant $g^{'} (g)$ . The Jacobian $J_{P} = Jac (\tilde{C}_{P})$ is principally polarised of dimension $\tilde{g}$ . The good-reduction set of $J_{P}$ equals the bad-reduction set of $\tilde{C}_{P}$ , which is contained in the bad-reduction set of $C$ together with the places lying over the auxiliary divisor — a finite set $S^{'}$ depending only on $C$ , not on $P$ , because the ramification at $P$ alone does not contribute to bad reduction of the Jacobian after passing to the integral model (the local monodromy at $P$ acts as the identity on the relevant $H^{1}$ ). $□$

Proposition 6 (Mordell from Shafarevich plus Kodaira-Parshin). Let $C$ be a smooth projective curve of genus $g \geq 2$ over a number field $K$ . Then $C (K)$ is finite.

Proof. Apply Proposition 5: each $P \in C (K)$ produces a principally polarised abelian variety $J_{P}$ over $K^{'}$ of dimension $\leq g^{'} (g)$ with good reduction outside a fixed finite set $S^{'} = S^{'} (C)$ . By Proposition 3 (Shafarevich), the set of $K^{'}$ -isomorphism classes of such $J_{P}$ is finite. It remains to show that each isomorphism class corresponds to finitely many $P$ . This uses the Torelli theorem (the Jacobian $Jac (\tilde{C}_{P})$ together with its principal polarisation determines $\tilde{C}_{P}$ up to isomorphism over $\overset{ˉ}{K}$ ) combined with the Mordell-Weil theorem for the cover $\tilde{C}_{P}$ . Given $[J_{P}] = [J_{P^{'}}]$ , Torelli yields an isomorphism $\tilde{C}_{P} ≅ \tilde{C}_{P^{'}}$ , and the construction $P \mapsto \tilde{C}_{P}$ is sufficiently rigid (the auxiliary divisor and the base point are fixed) that $P = P^{'}$ or $P, P^{'}$ lie in a single orbit of a bounded automorphism group of $\tilde{C}_{P}$ . The set of $P$ producing a given isomorphism class $[J_{P}]$ is therefore finite. Combining: $C (K)$ is the preimage under $P \mapsto [J_{P}]$ of a finite set, with each fibre finite, so $C (K)$ is finite. $□$

Proposition 7 (Vojta-Bombieri arithmetic intersection proof, sketch). Mordell's conjecture admits a second proof, due to Vojta 1991 and simplified by Bombieri 1990, using arithmetic intersection theory on $C \times C$ , Mumford's gap inequality for the Néron-Tate height on $Jac (C) (K)$ , and Faltings' product theorem.

Proof sketch. Suppose for contradiction $C (K)$ is infinite. Order the points $P_{1}, P_{2}, \dots$ by Néron-Tate height on $Jac (C) (K)$ , embedded by a fixed base point. Mumford's gap inequality (Mumford 1965 Inst. Hautes Études Sci. Publ. Math. 25) states: there exists a constant $γ > 0$ such that for any two distinct $P, Q \in C (K)$ with sufficiently large heights, the angle $θ_{P Q}$ between $P$ and $Q$ in the Mordell-Weil lattice satisfies $cos θ_{P Q} \leq 1 - γ$ . Construct, on the arithmetic surface $X = C \times C$ over $Spec O_{K}$ , an auxiliary global section $s$ of a high tensor power $L^{\otimes N}$ of an ample line bundle, with the property that $s$ vanishes to order $\geq d \cdot N$ along the diagonal $Δ \subset C \times C$ , for $d$ chosen large enough. The arithmetic Hilbert-Samuel theorem (Gillet-Soulé) produces such a section. Estimate the value of $s$ at $(P_{i}, P_{j}) \in (C \times C) (K)$ : an upper bound from the size of the coefficients of $s$ (Northcott on $X$ ) and a lower bound from the order of vanishing along $Δ$ (which forces a large value off the diagonal). The product theorem of Faltings 1991 provides a quantitative form of the arithmetic Bezout inequality on $X$ . Combining the upper and lower bounds with Mumford's gap inequality produces an inequality that any infinite sequence of $P_{i}$ violates after finitely many terms; this is the contradiction. $□$

Connections Master

Elliptic curves 04.04.03. The Mordell-Weil theorem of 1922 — that $E (K)$ is finitely generated for an elliptic curve over a number field — is Mordell's foundational result and the seed of the conjecture for higher genus. The genus-one case is excluded from Faltings' theorem because $E (K)$ is allowed to be infinite (rank $\geq 1$ ), but the same originator (Mordell), the same year (1922), and the same height-descent technique seed the entire program. Faltings' theorem is the genus- $\geq 2$ analogue: where Mordell-Weil gives finite generation, Faltings gives outright finiteness.
$ℓ$ -adic Galois representations 21.05.01. The Tate isogeny conjecture, one of the three theorems Faltings proved together with Mordell and Shafarevich, identifies the isogeny category of abelian varieties over a number field with a full subcategory of $ℓ$ -adic Galois representations. This is the foundational reason every modern arithmetic statement about abelian varieties — modularity, BSD, Iwasawa theory — can be packaged as a statement about Galois representations. Faltings' theorem is the bridge from abelian-variety geometry to Galois-representation arithmetic.
Modularity theorem and BSD conjecture 21.06.01. The modularity bridge $ρ_{E, ℓ} ≅ ρ_{f_{E}, ℓ}$ between an elliptic curve and a weight- $2$ cusp newform uses Faltings' Tate isogeny theorem as one of its ingredients: an isomorphism of $ℓ$ -adic Galois representations on Tate modules upgrades to an isogeny of abelian varieties (in the modular case, $E$ is isogenous over $Q$ to $A_{f_{E}}$ ). Without Faltings 1983 the modularity bridge would not be possible at the level of isogenies.
Hecke operators and Hecke algebra 21.04.02. The Eichler-Shimura relations $T_{p} = Frob_{p} + p \cdot Ver_{p}$ on the Jacobian $J_{0} (N) (mod p)$ are statements about Hecke operators on the Jacobian of the modular curve, an abelian variety to which Faltings' theorems apply. The good-reduction-outside- $S$ analysis underlying the Eichler-Shimura congruence is exactly the kind of Shafarevich-finiteness input that Faltings supplies.
$Z_{p}$ -extensions and Iwasawa theory 21.07.01. The Iwasawa-theoretic study of the Selmer group $Sel_{p^{\infty}} (E / K_{\infty})$ uses Faltings' Tate isogeny theorem to identify automorphisms of the Tate module with $K$ -rational endomorphisms, a key input to the structure theory of Iwasawa modules over the cyclotomic $Z_{p}$ -extension.

Historical & philosophical context Master

Mordell stated the conjecture that bears his name in 1922 in On the rational solutions of the indeterminate equations of the third and fourth degrees (Proceedings of the Cambridge Philosophical Society 21, 179-192) ^{[source pending]}. The paper's main result was the Mordell-Weil theorem for elliptic curves over $Q$ : $E (Q)$ is finitely generated, proved by Mordell's 2-descent argument. At the close of the paper Mordell conjectured the higher-genus analogue: every smooth projective curve over $Q$ of genus $\geq 2$ has only finitely many rational points. The conjecture was open for sixty-one years.

Partial results accumulated in the intervening period. Weil 1929 generalised Mordell-Weil from $Q$ to arbitrary number fields. Siegel 1929 Abh. Preuss. Akad. Wiss. proved finiteness of integral points on affine curves of positive genus, a different but related Diophantine finiteness statement. Mahler 1933 extended Siegel's theorem to $S$ -integers. Parshin 1968 (Math. USSR-Izv. 2, 1145-1170) ^{[source pending]} introduced the Kodaira-Parshin construction reducing Mordell to a Shafarevich-type finiteness for families of abelian varieties over the curve, a reduction that turned out to be the structural core of the eventual proof. Arakelov 1971 (Math. USSR-Izv. 5, 1277-1302) ^{[source pending]} developed Arakelov intersection theory on arithmetic surfaces, providing the Faltings-height machinery. Grauert 1965 and Manin 1963 proved the function-field analogue of Mordell, by entirely different geometric methods that yielded effective bounds.

Faltings 1983 (Inventiones Mathematicae 73, 349-366) ^{[source pending]} Endlichkeitssätze für abelsche Varietäten über Zahlkörpern proved Mordell together with the Shafarevich conjecture and the Tate isogeny conjecture, in a single paper of seventeen pages. The strategy: bound the Faltings height of an abelian variety along an isogeny class, deduce Shafarevich from Northcott on the moduli space $A_{g}$ , deduce Tate from Shafarevich plus Néron-Ogg-Shafarevich, deduce Mordell from Shafarevich via Kodaira-Parshin. Faltings received the Fields Medal in 1986 for this work. An English translation by Edward Shipz appears in Cornell-Silverman Arithmetic Geometry (Springer 1986). The Storrs conference of 1984 and the subsequent Paris seminar of 1985 (Szpiro ed., Astérisque 127) walked through every step of the proof.

Vojta 1991 (Annals of Mathematics 133, 509-548) ^{[source pending]} Siegel's theorem in the compact case gave a second proof using arithmetic intersection theory on $C \times C$ — a Diophantine-approximation argument parallel to Roth's theorem rather than a moduli-theoretic one. Bombieri 1990 (Annali Pisa 17, 615-640) ^{[source pending]} simplified Vojta's proof using Mumford's 1965 gap inequality and Faltings' product theorem. Faltings 1991 (Annals 133, 549-576) extended the Vojta method to subvarieties of abelian varieties, proving Lang's conjecture. The Mordell-Lang conjecture for finitely generated subgroups of abelian varieties became the standard formulation of the result, with Mordell as the curve-in-Jacobian special case.

Effective Mordell — producing an algorithm to enumerate $C (K)$ — has remained open through every proof. Coleman 1985 and Chabauty 1941 give effective methods when $rank Jac (C) (Q) < g$ , extended by Kim 2005 onward via the unipotent fundamental group. The ABC conjecture would imply effective Mordell uniformly. Caporaso-Harris-Mazur 1997 (Journal of the AMS 10, 1-35) ^{[source pending]} showed that Lang's conjecture for varieties of general type would yield a uniform bound $# C (K) \leq c (g, d)$ depending only on the genus and the degree of $K / Q$ . Dimitrov-Gao-Habegger 2021 (Annals 194, 237-298) ^{[source pending]} proved a uniform Mordell-Lang for curves, bounding $# C (K)$ in terms of $g$ , $d = [K : Q]$ , and the Mordell-Weil rank — partial resolution of the Caporaso-Harris-Mazur conjecture, using the Pila-Zannier strategy and the Gao-Habegger gap principle.

Bibliography Master

@article{Mordell1922,
  author  = {Mordell, L. J.},
  title   = {On the rational solutions of the indeterminate equations of the third and fourth degrees},
  journal = {Proceedings of the Cambridge Philosophical Society},
  volume  = {21},
  number  = {3},
  year    = {1922},
  pages   = {179--192}
}

@article{Faltings1983,
  author  = {Faltings, Gerd},
  title   = {Endlichkeitss{\"a}tze f{\"u}r abelsche Variet{\"a}ten {\"u}ber Zahlk{\"o}rpern},
  journal = {Inventiones Mathematicae},
  volume  = {73},
  number  = {3},
  year    = {1983},
  pages   = {349--366}
}

@article{Parshin1968,
  author  = {Parshin, A. N.},
  title   = {Algebraic curves over function fields {I}},
  journal = {Mathematics of the USSR-Izvestiya},
  volume  = {2},
  number  = {5},
  year    = {1968},
  pages   = {1145--1170}
}

@article{Arakelov1971,
  author  = {Arakelov, S. Yu.},
  title   = {Families of algebraic curves with fixed degeneracies},
  journal = {Mathematics of the USSR-Izvestiya},
  volume  = {5},
  number  = {6},
  year    = {1971},
  pages   = {1277--1302}
}

@article{Vojta1991,
  author  = {Vojta, Paul},
  title   = {Siegel's theorem in the compact case},
  journal = {Annals of Mathematics (2)},
  volume  = {133},
  number  = {3},
  year    = {1991},
  pages   = {509--548}
}

@article{Bombieri1990,
  author  = {Bombieri, Enrico},
  title   = {The {M}ordell conjecture revisited},
  journal = {Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (4)},
  volume  = {17},
  number  = {4},
  year    = {1990},
  pages   = {615--640}
}

@article{Faltings1991,
  author  = {Faltings, Gerd},
  title   = {Diophantine approximation on abelian varieties},
  journal = {Annals of Mathematics (2)},
  volume  = {133},
  number  = {3},
  year    = {1991},
  pages   = {549--576}
}

@book{CornellSilverman1986,
  editor    = {Cornell, Gary and Silverman, Joseph H.},
  title     = {Arithmetic Geometry},
  publisher = {Springer-Verlag},
  year      = {1986}
}

@article{CaporasoHarrisMazur1997,
  author  = {Caporaso, Lucia and Harris, Joe and Mazur, Barry},
  title   = {Uniformity of rational points},
  journal = {Journal of the American Mathematical Society},
  volume  = {10},
  number  = {1},
  year    = {1997},
  pages   = {1--35}
}

@article{DimitrovGaoHabegger2021,
  author  = {Dimitrov, Vesselin and Gao, Ziyang and Habegger, Philipp},
  title   = {Uniformity in {M}ordell-{L}ang for curves},
  journal = {Annals of Mathematics (2)},
  volume  = {194},
  number  = {1},
  year    = {2021},
  pages   = {237--298}
}

@book{HindrySilverman2000,
  author    = {Hindry, Marc and Silverman, Joseph H.},
  title     = {Diophantine Geometry: An Introduction},
  publisher = {Springer-Verlag},
  series    = {Graduate Texts in Mathematics},
  volume    = {201},
  year      = {2000}
}

@book{BombieriGubler2006,
  author    = {Bombieri, Enrico and Gubler, Walter},
  title     = {Heights in Diophantine Geometry},
  publisher = {Cambridge University Press},
  series    = {New Mathematical Monographs},
  volume    = {4},
  year      = {2006}
}

@book{ManinPanchishkin2005,
  author    = {Manin, Yuri I. and Panchishkin, Alexei A.},
  title     = {Introduction to Modern Number Theory},
  publisher = {Springer-Verlag},
  series    = {Encyclopaedia of Mathematical Sciences},
  volume    = {49},
  edition   = {2nd},
  year      = {2005}
}

@article{Mazur1986Arithmetic,
  author  = {Mazur, Barry},
  title   = {Arithmetic on curves},
  journal = {Bulletin of the American Mathematical Society},
  volume  = {14},
  number  = {2},
  year    = {1986},
  pages   = {207--259}
}

@article{Coleman1985,
  author  = {Coleman, Robert F.},
  title   = {Effective {C}habauty},
  journal = {Duke Mathematical Journal},
  volume  = {52},
  number  = {3},
  year    = {1985},
  pages   = {765--770}
}

@article{Kim2005,
  author  = {Kim, Minhyong},
  title   = {The motivic fundamental group of ${\mathbb P}^1 \setminus \{0, 1, \infty\}$ and the theorem of {S}iegel},
  journal = {Inventiones Mathematicae},
  volume  = {161},
  number  = {3},
  year    = {2005},
  pages   = {629--656}
}

@article{Mumford1965,
  author  = {Mumford, David},
  title   = {A remark on {M}ordell's conjecture},
  journal = {American Journal of Mathematics},
  volume  = {87},
  number  = {4},
  year    = {1965},
  pages   = {1007--1016}
}

@article{Grauert1965,
  author  = {Grauert, Hans},
  title   = {Mordells {V}ermutung {\"u}ber rationale {P}unkte auf algebraischen {K}urven und {F}unktionenk{\"o}rper},
  journal = {Publications Math{\'e}matiques de l'IH{\'E}S},
  volume  = {25},
  year    = {1965},
  pages   = {131--149}
}

@article{Manin1963,
  author  = {Manin, Yu. I.},
  title   = {Rational points of algebraic curves over function fields},
  journal = {Izvestiya Akad. Nauk SSSR Ser. Mat.},
  volume  = {27},
  year    = {1963},
  pages   = {1395--1440}
}

Prerequisites

04.04.03
21.04.01
21.05.01

Tier anchors

beginner: Manin-Panchishkin *Introduction to Modern Number Theory* (Springer EMS 49, 2nd ed. 2005) Ch. 6 §3.5 informal opening — every curve of genus at least two has only finitely many rational points, the Mordell conjecture proved by Faltings in 1983
intermediate: Manin-Panchishkin *Introduction to Modern Number Theory* (Springer EMS 49, 2nd ed. 2005) Ch. 6 §3.5; Cornell-Silverman (eds.) *Arithmetic Geometry* (Springer 1986) Chs. II-VI (the Storrs conference proceedings: Faltings' three finiteness theorems unpacked); Hindry-Silverman *Diophantine Geometry* (GTM 201, 2000) Ch. E
master: Mordell 1922 *Proceedings of the Cambridge Philosophical Society* 21 (3), 179-192 (originator of the conjecture for curves of genus $\geq 2$ over $\mathbb{Q}$, via the proof of the Mordell-Weil theorem for elliptic curves); Faltings 1983 *Inventiones Mathematicae* 73 (3), 349-366 *Endlichkeitssätze für abelsche Varietäten über Zahlkörpern* (originator paper proving all three finiteness theorems together: Mordell, Shafarevich, Tate); Parshin 1968 *Mathematics of the USSR-Izvestiya* 2 (5), 1145-1170 (the Kodaira-Parshin reduction from Mordell to Shafarevich-type finiteness for families over the curve); Arakelov 1971 *Mathematics of the USSR-Izvestiya* 5 (6), 1277-1302 (Arakelov height theory underlying Faltings' height function); Faltings 1984 in *Arithmetic Geometry* (Cornell-Silverman eds., Springer) (English exposition); Szpiro (ed.) 1985 *Astérisque* 127 (the Paris seminar on Faltings' proof); Bombieri 1990 *Annali della Scuola Normale Superiore di Pisa* (4) 17 (4), 615-640 (simplification via the product-theorem method); Vojta 1991 *Annals of Mathematics* (2) 133 (3), 509-548 *Siegel's theorem in the compact case* (second proof via arithmetic intersection on $C \times C$); Faltings 1991 *Annals of Mathematics* 133 (3), 549-576 *Diophantine approximation on abelian varieties* (Vojta-method extension to subvarieties of abelian varieties, Lang's conjecture); Hindry-Silverman *Diophantine Geometry* (GTM 201, 2000) Ch. E (textbook account); Bombieri-Gubler *Heights in Diophantine Geometry* (Cambridge New Math. Mono. 4, 2006) Ch. 11; Dimitrov-Gao-Habegger 2021 *Annals of Mathematics* (2) 194 (1), 237-298 *Uniformity in Mordell-Lang for curves* (uniform Mordell via the new gap principle, after Mazur-Caporaso-Harris)

References

Mordell, L. J. — On the rational solutions of the indeterminate equations of the third and fourth degrees · *Proceedings of the Cambridge Philosophical Society* 21 (3), 179-192 (1922). Originator paper of the Mordell conjecture: every curve of genus $\geq 2$ over $\mathbb{Q}$ has only finitely many rational points. Proves the finite-generation half of the Mordell-Weil theorem for elliptic curves $E/\mathbb{Q}$ in the same paper, by 2-descent.
Faltings, G. — Endlichkeitssätze für abelsche Varietäten über Zahlkörpern · *Inventiones Mathematicae* 73 (3), 349-366 (1983). The originator paper proving Mordell's conjecture together with the Shafarevich conjecture (finiteness of abelian varieties of bounded dimension with good reduction outside $S$) and the Tate conjecture on homomorphisms of abelian varieties; English translation by Edward Shipz in Cornell-Silverman *Arithmetic Geometry* (Springer 1986), 9-27.
Parshin, A. N. — Algebraic curves over function fields I · *Mathematics of the USSR-Izvestiya* 2 (5), 1145-1170 (1968); Russian original *Izvestiya Akad. Nauk SSSR Ser. Mat.* 32, 1191-1219. Introduces the Kodaira-Parshin construction: from a rational point $P \in C(K)$ on a curve of genus $\geq 2$, build a family of abelian varieties (a covering construction over $C$) whose isomorphism class is controlled by $P$; reduces Mordell's conjecture to a Shafarevich-type finiteness theorem for abelian varieties.
Arakelov, S. Yu. — Families of algebraic curves with fixed degeneracies · *Mathematics of the USSR-Izvestiya* 5 (6), 1277-1302 (1971). Originator of Arakelov intersection theory on arithmetic surfaces: extends Hodge-index and intersection-pairing machinery to integral models of curves over the ring of integers $\mathcal{O}_K$, with archimedean Green-function contributions at infinity. Underpins the Faltings height of an abelian variety.
Vojta, P. — Siegel's theorem in the compact case · *Annals of Mathematics* (2) 133 (3), 509-548 (1991). Vojta's second proof of Mordell's conjecture using arithmetic intersection theory on the product $C \times C$ and a refined geometric-of-numbers argument; recasts the Faltings argument into a Diophantine-approximation framework parallel to Roth and Siegel.
Bombieri, E. — The Mordell conjecture revisited · *Annali della Scuola Normale Superiore di Pisa, Classe di Scienze* (4) 17 (4), 615-640 (1990). Bombieri's simplification of Vojta's proof using Mumford's gap inequality and the product theorem of Faltings; the cleanest available exposition of the Vojta-Faltings strategy.
Faltings, G. — Diophantine approximation on abelian varieties · *Annals of Mathematics* (2) 133 (3), 549-576 (1991). Extends the Vojta method to subvarieties of abelian varieties, proving Lang's conjecture: a subvariety of an abelian variety contains only finitely many rational points outside a finite union of translates of abelian subvarieties contained in the subvariety.
Dimitrov, V., Gao, Z. and Habegger, P. — Uniformity in Mordell-Lang for curves · *Annals of Mathematics* (2) 194 (1), 237-298 (2021). Proves the uniform Mordell-Lang conjecture for curves of genus $\geq 2$ over a number field: $\#C(K) \leq c(g, [K : \mathbb{Q}], \mathrm{rank}\, \mathrm{Jac}(C)(K))$ with the bound depending only on the indicated invariants. Uses Gao-Habegger's new gap principle and the Pila-Zannier strategy.
Manin, Yu. I. and Panchishkin, A. A. — Introduction to Modern Number Theory · Encyclopaedia of Mathematical Sciences 49, 2nd ed., Springer (2005). Ch. 6 §3.5 (Mordell-Faltings theorem statement, the three companion finiteness theorems, the Kodaira-Parshin reduction, Vojta's second proof).
Hindry, M. and Silverman, J. H. — Diophantine Geometry · Graduate Texts in Mathematics 201, Springer (2000). Ch. E: Mordell-Faltings theorem, Vojta's proof, the heights underlying it; canonical English textbook account.
Cornell, G. and Silverman, J. H. (eds.) — Arithmetic Geometry · Springer (1986). Proceedings of the Storrs conference on Faltings' theorem; Chs. II-VI walk through the proof step by step (Faltings height, semistable reduction, Shafarevich-Tate, Mordell).
Bombieri, E. and Gubler, W. — Heights in Diophantine Geometry · Cambridge New Mathematical Monographs 4, Cambridge University Press (2006). Ch. 11: heights on abelian varieties, the Faltings height, the Vojta proof of Mordell; canonical reference on heights.
Mazur, B. — Arithmetic on curves · *Bulletin of the American Mathematical Society* 14 (2), 207-259 (1986). Survey of the state of Diophantine geometry immediately after Faltings, with emphasis on the still-open effectivity problem and the uniform Mordell conjecture.
Caporaso, L., Harris, J. and Mazur, B. — Uniformity of rational points · *Journal of the American Mathematical Society* 10 (1), 1-35 (1997). The uniform Mordell conjecture: conditional on Lang's conjecture, $\#C(K)$ is bounded by a constant depending only on $g$ and $[K : \mathbb{Q}]$, not on the curve $C$.

Estimated time

beginner: 25m
intermediate: 55m
master: 90m