Fact-checked by Grok 2 weeks ago

Faltings's theorem

Faltings's theorem states that a smooth projective of g \geq 2 defined over a number field K has only finitely many rational points in K. This result, proved by German mathematician in 1983, resolves the Mordell conjecture, which Louis Mordell had proposed in 1922 as a generalization of his earlier work on the finiteness of rational points on elliptic curves (genus 1). The proof, published in the paper Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, actually establishes a more general finiteness theorem for isomorphism classes of abelian varieties over number fields with bounded dimension and good reduction outside a fixed of places, from which the curve case follows as a special instance. The theorem represents a profound advance in arithmetic geometry, demonstrating how geometric invariants like constrain the arithmetic of rational solutions to equations. For curves of 0 or 1, the rational points can be infinite or form a (by results of Lüroth and the , respectively), but Faltings's theorem imposes strict finiteness for higher , ruling out infinite families of solutions in many Diophantine problems. A notable consequence is that equations of the form y^2 = f(x) where f is a of at least 5, or more generally superelliptic equations f(x,y) = 0 with sufficiently high , admit only finitely many solutions in rational numbers. This finiteness directly applies to Fermat's equation x^n + y^n = z^n for n > 2, implying only finitely many integer solutions (up to scaling), though it does not resolve the full conjecture of no nontrivial solutions—that required further breakthroughs building on Faltings's ideas, culminating in Andrew Wiles's 1995 proof. Faltings's work earned him the in 1986, the highest honor for mathematicians under 40, awarded by the for his pioneering use of arithmetic algebraic geometry to tackle long-open problems in . The theorem has since inspired extensive research into effective versions, uniform bounds on the number of points, and generalizations to higher-dimensional varieties, though many such extensions remain conjectural.

Preliminaries

Algebraic Curves and Genus

In algebraic geometry, an algebraic curve over the rational numbers \mathbb{Q} is the zero set of a polynomial f(x,y) \in \mathbb{Q}[x,y] in the affine plane \mathbb{A}^2_{\mathbb{Q}}, forming a one-dimensional affine variety that is integral and of finite type over \mathbb{Q}. For global study, the affine curve is compactified via its projective closure in the projective plane \mathbb{P}^2_{\mathbb{Q}}, which embeds the curve as a closed subscheme and adds points at infinity to ensure properness. A non-singular projective curve is then a smooth, proper, one-dimensional scheme over \mathbb{Q} with no singular points, birational to the original affine model and serving as its nonsingular projective model. The genus g of a smooth projective C over \mathbb{Q} (or \mathbb{C}) is a birational invariant that topologically classifies the curve; over \mathbb{C}, C is a compact homeomorphic to an orientable surface of g, a with g handles. For a smooth of degree d in \mathbb{P}^2, the genus is computed by the degree-genus : g = \frac{(d-1)(d-2)}{2}, derived via the Riemann-Hurwitz formula from the degree-d projection to \mathbb{P}^1. Elliptic curves provide the basic example of genus g=1, realized as smooth projective models of Weierstrass equations y^2 = x^3 + ax + b with discriminant nonzero. Hyperelliptic curves of genus g > 1 are smooth projective curves given by y^2 + h(x)y = f(x) with \deg f = 2g+1 or $2g+2 and h of degree at most g, often arising as double covers of \mathbb{P}^1. The genus quantifies the curve's complexity, equaling the dimension of the space of global holomorphic differentials on the . The concept of genus originated in Bernhard Riemann's 1857 work on Abelian functions, where he defined it for algebraic s via the of associated Riemann surfaces, linking it to the order p of independent first-kind integrals and the equation degree through p = (n-1)(m-1) - r for a F(s,z)=0 of degrees n,m.

Rational Points and

In the arithmetic study of algebraic curves defined over the rational numbers \mathbb{Q}, the set C(\mathbb{Q}) consists of rational points, which are pairs (x, y) \in \mathbb{Q}^2 satisfying the polynomial equation f(x, y) = 0 defining the curve C, including points at infinity in the projective closure. Height functions provide a measure of the arithmetic size or complexity of these rational points, facilitating bounds on their number and distribution. For a x = a/b expressed in lowest terms with a, b \in \mathbb{Z} and b > 0, the naive (multiplicative) is H(x) = \max(|a|, b). For an affine point P = (x, y) on a , the naive is H(P) = \max(H(x), H(y)); in \mathbb{P}^n(\mathbb{Q}), for a point P = [x_0 : \cdots : x_n] with coordinates \gcd(x_0, \dots, x_n) = 1, it is H(P) = \max_i |x_i|. On the of a , particularly for elliptic curves where the Jacobian is the curve itself, the (Néron-Tate) \hat{h}: E(\mathbb{Q}) \to \mathbb{R} is a positive that vanishes precisely on the and satisfies \hat{h}(nP) = n^2 \hat{h}(P) for n, enabling the detection of independent points via over \mathbb{Z}. The Mordell-Weil theorem exemplifies the structure of rational points on , which are smooth projective curves of 1 with a specified base point. It states that for an elliptic curve E over \mathbb{Q}, the E(\mathbb{Q}) of rational points is finitely generated: E(\mathbb{Q}) \cong \mathbb{Z}^r \oplus T, where r \geq 0 is the Mordell-Weil rank and T is the finite torsion subgroup. Diophantine geometry examines rational solutions to polynomial equations through geometric lenses, where the genus—a measure of curve complexity—influences whether C(\mathbb{Q}) is finite or infinite, contrasting with , which addresses the undecidability of integer solutions but is less pertinent to bounding rational point counts on curves. For genus 0 curves, such as conics (e.g., ax^2 + bxy + cy^2 + dx + ey + f = 0 with rational coefficients), the Hasse principle governs solubility, and if one exists, then infinitely many do, arising from rational parametrizations like from that point. By contrast, higher-genus curves often exhibit finiteness; for instance, the y^2 = x^3 + 1 ( 1, 0) has exactly six s: (-1, 0), (0, 1), (0, -1), (2, 3), (2, -3), and the point at infinity.

Historical Development

Mordell's Conjecture

In 1922, Louis Mordell formulated a conjecture asserting that for an algebraic curve C of genus g \geq 2 defined over the rational numbers \mathbb{Q}, the set C(\mathbb{Q}) of rational points on C is finite. This statement appeared as a remark in his paper proving the finite generation of rational points on elliptic curves (genus 1). The conjecture was motivated by the desire to extend the Mordell-Weil theorem, which shows that the group of rational points on an over \mathbb{Q} is finitely generated and thus can be infinite if the rank is positive. Mordell sought a sharper finiteness result for curves of higher , where the geometry prevents the infinite generation seen in genus 1 cases. Examples from Diophantine problems, such as those arising in the study of via Frey curves, highlighted the need to understand rational points on such curves, though these specific instances involve genus 1; the conjecture provided a framework for anticipating finiteness in more complex geometric settings. Early efforts verified the for specific families of curves using explicit methods in the mid-20th century. The remained open until , despite significant partial progress. A key related result was Siegel's 1929 theorem, which establishes the finiteness of integral points (points with integer coordinates) on affine curves of at least 1 over the integers, using effective ; this provided insight into boundedness but did not directly address rational points on the full projective curve. Historically, Mordell's conjecture emerged within the broader landscape of Hilbert's tenth problem from 1900, which sought algorithms to solve general Diophantine equations, though the focus here was narrowed to the arithmetic of algebraic curves and their rational points.

Pre-Faltings Results and Attempts

One of the earliest significant partial results toward understanding the finiteness of points on algebraic curves came from Carl Siegel in 1929, who proved that an affine algebraic curve of genus at least 1 defined over a number field admits only finitely many integral points. This theorem relied on diophantine approximation techniques and the introduction of heights to bound the growth of solutions, providing a foundation for later work on rational points by distinguishing integral points on affine models from the full set of rational points on the projective curve. In the 1940s, developed the theory of heights on , which became essential for bounding the size of rational points and influenced subsequent approaches to Diophantine problems on curves. In 1957, Kurt Mahler advanced the study of Diophantine equations on curves through p-adic methods, developing techniques that yielded effective bounds for solutions to specific equations, such as those arising from superelliptic curves or Thue equations, though these were restricted to particular families and did not achieve general finiteness for rational points on higher-genus curves. Mahler's approach extended classical theory into the p-adic setting, offering tools for explicit computations but highlighting the limitations in handling the full . During the 1960s, efforts to leverage complex multiplication and modular forms, notably by Goro Shimura, aimed to exploit symmetries in abelian varieties associated to curves with extra endomorphisms, yielding insights into rational points for special cases like elliptic curves with complex multiplication. However, these methods proved insufficient for the general case, as they depended heavily on the presence of such special structure and failed to provide uniform finiteness results across arbitrary curves of greater than 1. Serge Lang's conjectures from the further shaped the landscape by proposing that rational points on subvarieties of symmetric spaces—such as those arising from Shimura varieties—should be finite unless the subvariety contains a positive-dimensional special subset, influencing by generalizing Mordell's ideas to higher dimensions but stopping short of proving the core finiteness for curves. These conjectures underscored the role of geometric invariants like the in controlling point distribution. Key obstacles to proving Mordell's conjecture in this era included the absence of uniform height bounds for rational points on curves, which prevented effective control over infinite families, and the escalating complexity of varieties for genera greater than 1, where the lack of a simple rank structure akin to the Mordell-Weil theorem for elliptic curves impeded analogous finiteness arguments.

Statement of the Theorem

Precise Formulation

Faltings's theorem asserts that if C is a smooth projective curve of g \geq 2 defined over a number field K, then the set C(K) of K-rational points on C is finite. This result generalizes Mordell's conjecture from 1922, which posited the same finiteness for curves over the rational numbers \mathbb{Q}. Since \mathbb{Q} is a number field, Faltings's theorem immediately implies Mordell's conjecture, establishing that |C(\mathbb{Q})| < \infty for any such curve over \mathbb{Q}. The theorem extends to finite extensions L of K, as L is also a number field, yielding finiteness of C(L) and ensuring that rational points are integral at all but finitely many places of K. A reformulation arises via the Jacobian variety \mathrm{Jac}(C), an abelian variety of dimension g, where K-rational points on C correspond to K-rational points of degree 1 on C, embedding as a curve in \mathrm{Jac}(C) that contains no positive-dimensional abelian subvariety translate, thereby inheriting finiteness from the general case for subvarieties of abelian varieties.

Examples of Application

One illustrative example of Faltings's theorem is the hyperelliptic curve defined by the equation y^2 = x^5 + 1, which has genus 2. The obvious rational points on this curve are (0,1), (0,-1), and (-1,0). Faltings's theorem implies that there are only finitely many rational points on this curve. In practice, computational methods confirm that these are the only rational points. Another example is the Fermat curve given by the equation x^n + y^n = 1 for integers n \ge 3. For n = 3, the genus is 1, but for n \ge 4, the genus is (n-1)(n-2)/2 > 1. Faltings's theorem implies that there are only finitely many rational points on these curves. For small n \ge 4, the only rational points are the trivial ones on the axes: (\pm 1, 0) and (0, \pm 1). The theorem excludes curves of genus 1, where the Mordell-Weil theorem states that the group of rational points is finitely generated and thus infinite if the rank is positive. For instance, elliptic curves like y^2 = x^3 - 2 have infinitely many rational points when the rank is positive. Post-Faltings, explicit finiteness proofs for specific curves often use the Chabauty method or 2-descent on the to verify the complete set of rational points. These p-adic techniques bound the rational points when the rank of the is less than the . Prior to Faltings's proof, computations of rational points on low- curves relied on methods and searches for small-height points, with tables compiled for hyperelliptic curves of up to certain bounds. For example, 2 curves with small discriminants were enumerated, revealing sporadic cases with multiple rational points, though finiteness was conjectural.

Proofs

Faltings's Original Approach

Faltings announced his proof of the Mordell conjecture in 1983 through the seminal paper "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern," published in Inventiones Mathematicae volume 73, pages 349–366, with an erratum in volume 75. The core strategy establishes the finiteness of isomorphism classes of abelian varieties over a fixed number field with bounded Faltings height within the A_g of principally polarized abelian varieties of dimension g \geq 2. This finiteness result directly implies the Mordell conjecture by associating curves to their Jacobians and bounding the heights of rational points via pairings on these varieties. A pivotal tool in the proof is Arakelov intersection theory applied to the moduli stack of stable curves \overline{\mathcal{M}}_g, where metrics are introduced on line bundles to incorporate archimedean places. Faltings employs Green's theorem, which relates the Arakelov degree of the canonical bundle to the geometry of the moduli space, enabling the definition of intersection numbers that control heights. Specifically, the theory provides a way to metrize the Hodge bundle over the moduli space, allowing for the computation of degrees that bound the complexity of families of curves and their Jacobians. This arithmetic intersection theory extends classical algebraic geometry to number fields, ensuring that only finitely many such objects exist with controlled invariants. The plays a central role by embedding the rational points of the curve into the via the Abel-Jacobi map, where the number of points corresponds to the of the Mordell-Weil group. Height pairings, defined using the Néron-Tate height on the , pair divisors or points to produce bilinear forms that bound the canonical heights of rational points. If there were infinitely many rational points, their heights would grow unbounded, contradicting the finiteness in A_g after base change to achieve semistable reduction. This mapping reduces the problem of rational points on curves to arithmetic properties of abelian varieties. Central to the height bounds is the lambda invariant, defined as the determinant line bundle \lambda = \det \mathbb{E}, where \mathbb{E} is the Hodge bundle on \overline{\mathcal{M}}_g. Faltings demonstrates the positivity of the Arakelov degree of \lambda, computed via intersections involving the canonical bundle and Green's functions, which ensures that the Faltings height h_F(A) for an abelian variety A satisfies h_F(A) = \frac{1}{g} \deg_{\Ar} (\lambda) + O(1), with the leading term positive. This positivity implies that abelian varieties with bounded height form a compact subset in the Arakelov metric, yielding only finitely many up to isomorphism over the number field. The determinant bundles thus provide the arithmetic positivity needed to conclude finiteness without enumerating all cases.

Subsequent Simplifications and Alternatives

In the years following Gerd Faltings's 1983 proof, mathematicians sought simpler and alternative routes to establish the finiteness of rational points on curves of greater than 1 over number fields, often avoiding the full machinery of Arakelov geometry central to the original approach. A key development came from Paul Vojta's program in 1991, which provided an analytic proof relying on techniques, particularly Schmidt's subspace theorem. This method reduces the problem to showing finiteness on certain subvarieties by bounding heights of rational points through logarithmic forms and value distribution theory, yielding an ineffective but conceptually distinct argument. A 2025 paper reorganized Vojta's proof in terms of Arakelov geometry, introducing new ingredients to replace certain estimates. Enrico Bombieri built on Vojta's ideas in 1990, offering a more elementary simplification that streamlines the arguments while preserving the analytic flavor. By substituting refined estimates from height theory and avoiding some of Vojta's more abstract value distribution tools, Bombieri's variant proves the theorem for curves in a self-contained manner, emphasizing explicit bounds on indices and proximity functions. This approach highlights the interplay between arithmetic heights and geometric invariants without invoking arithmetic . p-adic methods emerged as another alternative, particularly effective for obtaining bounds in specific cases. In the 1990s, Victor Abrashkin explored p-adic cohomology structures inspired by Faltings's own comparison theorems, providing tools for analyzing rational points via rigid analytic spaces and period mappings over p-adic fields. These techniques, later refined, allow for effective finiteness results when combined with local-global principles, though they remain partial for general curves. A more comprehensive p-adic proof was given by Brian Lawrence and in 2018, using to construct period mappings that bound the dimension of spaces of rational points, reducing the problem to finite fibers over Shimura varieties. This approach integrates Galois representations and to yield an effective version in low-genus cases, bypassing much of the original proof's complexity. Recent work up to 2025 has incorporated modular methods via Galois representations, leveraging the resolution of Serre's modularity conjecture to derive effective bounds for rational points on certain curves. For instance, deformations of residual Galois representations attached to Jacobians allow control over classes, implying finiteness through level-lowering and Langlands techniques. These approaches, while not fully alternative proofs, simplify applications of Faltings's theorem to explicit Diophantine problems by providing quantitative estimates tied to modular forms. Compared to Faltings's original proof, which spans about 18 pages in the main publication plus an , these subsequent simplifications and alternatives are notably more concise in some cases, such as Bombieri's 26-page paper, by focusing on targeted analytic or p-adic tools.

Consequences and Applications

Finiteness of Rational Points

Faltings's theorem implies that for any projective C of g \geq 2 defined over a number field K, the set C(K) of K-rational points is finite. More precisely, there exists a constant N = N(g, [K:\mathbb{Q}]) such that |C(K)| \leq N, where the bound depends only on the genus and the degree of the extension; however, this bound is ineffective, as Faltings's original proof does not provide an to compute it or explicit estimates for N. A stronger uniform boundedness conjecture posits that for fixed g \geq 2, there exists a constant N(g) independent of K such that |C(K)| \leq N(g) for any such over any number field K. Caporaso, Harris, and Mazur proved that this holds conditionally on the Bombieri-Lang on rational points on varieties of general type. Partial effective versions of the finiteness result have been obtained using heights on the . For instance, when the Mordell-Weil rank of the is less than the , the Chabauty-Coleman provides an explicit upper bound on |C(K)| in terms of the and p-adic data at suitable primes, allowing computational determination of all points; this bound can be refined using the conductor of the to control local contributions. Szpiro's conjecture on the relation between the conductor and discriminant of elliptic curves extends to higher genus via minimal models of curves, where bounding the Faltings height of the Jacobian in terms of the conductor would yield effective height bounds for rational points on C, making Faltings's theorem effective in those cases. In the 2020s, computational databases like the L-functions and Modular Forms Database (LMFDB) have verified the finiteness predicted by Faltings's theorem for thousands of explicit curves of 2 and 3 over \mathbb{Q}, with absolute discriminants up to $10^6, by enumerating all rational points using methods such as Chabauty-Coleman and ; for example, 66,158 genus-2 curves have been classified with their complete sets of rational points.

Links to Other Diophantine Problems

Faltings's theorem establishes the finiteness of rational points on curves of genus greater than 1, which directly implies that for any fixed n \geq 3, the superelliptic curve x^n + y^n = 1 in has only finitely many rational points, corresponding to finitely many solutions to the a^n + b^n = c^n. This provides a weak form of , reducing the problem to showing that there are no nontrivial solutions among these finitely many points. The full proof of for n \geq 3, establishing the absence of such solutions, was achieved by in 1995 using the for semistable s, building on Gerhard Frey's 1986 construction of an (now called a Frey curve) associated to a hypothetical solution and Ken Ribet's 1986 level-lowering theorem, which links the non-modularity of this curve to the Taniyama-Shimura conjecture (now theorem). The theorem also connects to the ABC conjecture through its implications for superelliptic equations. Specifically, Faltings's finiteness result applies to equations of the form y^k = f(x), where k \geq 3 is fixed and f is a fixed of degree at least 3, yielding only finitely many solutions (x, y). This finiteness supports the radical power bounds predicted by the , as the latter would provide effective estimates on the size of solutions, refining the non-effective finiteness from Faltings via descent methods akin to those of Kummer. In particular, the ABC conjecture implies effective versions of Mordell's conjecture (proved non-effectively by Faltings), strengthening the control over integral points on these curves. Faltings's proof extends to a generalization of the Shafarevich theorem, establishing the finiteness of classes of abelian varieties (including curves) over a number field K with good reduction outside a fixed of places S and bounded dimension. For curves specifically, this means there are only finitely many classes of curves of fixed g \geq 2 over K with good reduction outside S and at most a bounded number of rational points. This finiteness of families complements the point finiteness on individual curves, providing a uniform bound on the geometry of solutions to Diophantine problems within bounded . Beyond these, Faltings's theorem implies the finiteness of primitive algebraic points on curves of sufficiently high . Recent developments in inter-universal Teichmüller theory (IUT), proposed by since 2012 and further explored in 2024 preprints by Kirti Joshi, aim to prove the using arithmetic Teichmüller spaces, which would yield effective refinements and broader implications for finiteness results like those of Faltings, though the approach remains controversial due to ongoing debates over its validity and .

Generalizations and Extensions

To Abelian Varieties

Faltings's methods from the proof of the Mordell conjecture extend to broader finiteness results for over number fields. In a companion paper to his 1983 work on the Mordell conjecture, Faltings established what is known as Arakelov's theorem for . This theorem states that for a fixed number field K, there are only finitely many classes of over K of a given with Faltings bounded above by a fixed constant. The Faltings h_F(A) of an A over K is an absolute, archimedean invariant defined via the Arakelov degree of the of the relative canonical sheaf under the Abel-Jacobi map, providing a measure of the complexity of A. This height bound has significant applications to the arithmetic of polarized abelian varieties. In particular, Faltings proved that for a fixed number field K, finite set of places S, and fixed degree d of polarization, there are only finitely many isomorphism classes of abelian varieties over K of dimension g equipped with a polarization of degree d that have good reduction outside S. For principally polarized abelian varieties (where d=1), this yields the finiteness of the number of such varieties over K with good reduction outside S, corresponding to the rational points on the moduli space \mathcal{A}_g lying in the open subset with good reduction outside S. When the abelian variety is simple (i.e., admits no proper abelian subvariety), the result holds unconditionally for dimension g \geq 2, as the simplicity condition aligns with the boundedness in the isogeny classes via the endomorphism algebra structure. A related development concerns the distribution of torsion points on . In the , Michel Raynaud proved the Manin-Mumford conjecture in the context of over number fields, establishing that if V is a proper subvariety of an A over a number field K, then the intersection V(K)_{\tors} \cap A(K)_{\tors} is finite unless V is a torsion translate of an of A. This provides a strong finiteness statement for torsion points avoiding certain subvarieties, complementing the always-finite torsion A(K)_{\tors} from the Mordell-Weil . Raynaud's proof relies on p-adic methods and uniformization, extending earlier work on the complex case. Despite these advances, significant limitations remain. The full Shafarevich conjecture for unpolarized abelian varieties—finiteness of isomorphism classes with good reduction outside S—remains open for dimension g \geq 2, as Faltings's result only establishes finiteness up to . Without the polarizing structure to distinguish classes, the number of non-isomorphic varieties in a single class can be infinite, even for simple cases. Recent progress toward resolving these open questions has leveraged connections to the . Conditional on automorphic forms and Galois representations associated to abelian varieties (as anticipated by the arithmetic Langlands correspondence), results such as those by Kisin and others provide effective bounds on the conductor and thus support finiteness for polarized cases with additional level structures. For instance, in 2021, effective versions of the Shafarevich conjecture were established for abelian varieties of GL_2-type over Q, relying on modularity theorems.

Over Function Fields and Other Settings

The analog of Faltings's theorem over function fields states that a projective curve of genus at least 2 defined over the function field of a projective curve over an of characteristic zero has only finitely many rational points, where "rational points" refer to places of degree one. This result was established independently by Yuri I. Manin in 1963 and Hans Grauert in 1965, predating Faltings's work on number fields by two decades. For the special case of elliptic curves (genus 1) over function fields like \mathbb{C}(t), the Mordell-Weil group is finitely generated, as proved using height functions adapted to the geometric setting by John Tate and others in the . These proofs exploit the geometric nature of function fields, where the "infinite places" correspond to points at infinity on the base , allowing for effective arguments. Faltings's methods from the number field case have been adapted to function fields through Arakelov theory over curves, which equips bundles with metrics at the infinite places to mimic the . In this setting, Arakelov theory naturally extends classical on the base , enabling proofs of finiteness results for bundles of fixed and on arithmetic surfaces arising from relative curves over function fields. Specifically, Faltings established that there are only finitely many isomorphism classes of semistable Arakelov bundles of given and degree on such surfaces, a key ingredient in bounding the number of rational points via the geometry of moduli spaces. This adaptation highlights how function fields provide a geometric for the challenges in Faltings's original proof, where the lack of a natural at infinite places required innovative constructions. In positive characteristic, analogues of the theorem for curves over function fields of curves over finite fields face additional challenges due to the Frobenius and potential wild ramification in extensions. The finiteness of rational points in this setting was proved by A. N. Parshin in 1967, with refinements by Lucien Szpiro in 1979 incorporating the action of Frobenius on the base and controlling ramification through cohomological bounds. However, wild ramification can lead to infinite families of covers, complicating uniform effective versions. Victor Abrashkin developed further analogues in the using strict modules over rings of positive characteristic, providing a framework for finite flat group schemes that parallels and addresses ramification filtrations via Galois representations. These tools have enabled partial extensions to abelian varieties but highlight ongoing difficulties with unbounded genera in covers over the base. Broader settings include local fields like p-adic fields, where the analog of finiteness for rational points on curves of at least 2 is trivial in the sense that the local-global principle does not apply, and the set of p-adic points is often infinite due to the non-archimedean topology allowing dense subgroups. For relative curves over a Dedekind base (such as the in a number field), Faltings's theorem implies finiteness of rational points on each smooth fiber of genus at least 2, but without uniform bounds across the family.