Uniform Boundedness Conjecture for Rational Points on Curves

Uniform Boundedness Conjecture for Rational Points on Curves

Definitions and mathematical statement

Fix a number field $F$ and an integer $g\ge2$. In the usual curve convention, $C/F$ is smooth, projective, and geometrically connected, and its genus is $g=\dim_F H^0(C,\Omega_C^1)$. The requested uniform bound is \[ \exists c(g,F)<\infty\quad\forall C/F\text{ of genus }g, \qquad\#C(F)\le c(g,F). \] For an affine smooth curve, genus refers to its smooth projective completion and its rational points form a subset of those of that completion. The constant must not depend on coefficients, heights, or the isomorphism class of $C$. The target does not ask for a formula or an effective algorithm for the bound.

Formulation and concept references: [S1].

Short English statement

With the number field and genus fixed, is there one finite bound for the number of rational points on every curve of that genus?

Research attempt

Current frontier.

Uniform Mordell--Lang bounds for a curve embedded in its Jacobian have the form $c_1(g)c_2(g)^r$, where $r$ is the rank of the subgroup being intersected. They do not remove rank from the bound [S1]. Yu--Yuan--Zhou's February 2026 manuscript makes the constants explicit, still with Mordell--Weil rank in the exponent [E1], Theorems 1.1--1.3. This recent theorem statement is not being certified by an independent audit of its 178-page proof here; in any event its rank dependence does not settle the printed target. The classical conditional route via Lang's conjecture is different from an unconditional genus-and-field bound [E2]. The investigation below asks whether the geometry of short sums of curve points can avoid counting the entire Mordell--Weil lattice.

Attack route.

One route is to bound the ranks of all relevant Jacobians, then use the known rank-dependent theorem. That would require another major unresolved uniformity assertion and may be much stronger than needed. A second is to combine good reductions to a small finite group; the difficulty is lifting modular collisions to genuine divisor equivalence. A third is to use the large number of distinct low-degree effective divisors on a curve together with a height-packing argument. I pursue the third and quantify exactly which nonuniform parameters remain.

Attempt: short-sum rigidity from gonality.

Assume $C(F)\ne\varnothing$, choose $P_0\in C(F)$, and let $i(P)=[P-P_0]\in J(F)$. This map is injective for genus at least one: a principal divisor $P-Q$ with $P\ne Q$ would give a degree-one map to ${\mathbb{P}}^1$. For a finite subset $\mathcal P\subset C(F)$ of size $N$, put $A=i(\mathcal P)$.

Let $\operatorname{gon}_{\overline F}(C)$ be the least degree of a nonconstant morphism $C_{\overline F}\to{\mathbb{P}}^1$. If $1\le d<\operatorname{gon}_{\overline F}(C)$, then \begin{equation}\label{wsref-0-33-30-30-30-36-37-36-33-3a-73-75-6d-73-65-74} |dA|=\binom{N+d-1}{d}, \qquad dA=\{a_1+\cdots+a_d:a_i\in A\}. \end{equation} To prove it, two distinct effective degree-$d$ divisors supported on $\mathcal P$ cannot be linearly equivalent: cancelling their common part would produce the zero and pole divisors of a nonconstant rational function of degree at most $d$. Thus the map from size-$d$ multisets of $\mathcal P$ into $\operatorname{Pic}^d(C)$ is injective. Counting those multisets proves (30006763:sumset). Equality of $F$-rational Jacobian points implies the geometric divisor equivalence used in this argument, so no descent hypothesis is hidden here.

In particular, for a nonhyperelliptic curve the additive energy is \begin{equation}\label{wsref-0-33-30-30-30-36-37-36-33-3a-73-69-64-6f-6e-2d-65-6e-65-72-67-79} \mathcal E(A):=\#\{(a,b,c,d)\in A^4:a+b=c+d\}=2N^2-N. \end{equation} There are $N$ diagonal sums with one ordered representation and $\binom N2$ other sums with two ordered representations. This proves (30006763:sidon-energy) exactly, not just up to an implicit constant.

Attempt: the hyperelliptic correction.

The preceding degree-two assertion is false for hyperelliptic curves, including every genus-two curve. Let $\iota$ be the hyperelliptic involution. Suppose $\mathcal P$ contains $h$ complete unordered pairs $\{P,\iota P\}$ with $P\ne\iota P$, and $w$ fixed points of $\iota$. The unique degree-two pencil contributes $t=2h+w$ ordered pairs to one divisor class. Every other effective degree-two divisor still has its unique unordered representation. Consequently \begin{equation}\label{wsref-0-33-30-30-30-36-37-36-33-3a-68-79-70-65-72-2d-65-6e-65-72-67-79} \mathcal E(A)=2N^2-N+t^2-4h-w =2N^2-N+t^2-2t+w\le3N^2. \end{equation} For the inequality use $0\le w\le t\le N$. The uniqueness of the hyperelliptic pencil follows because two distinct degree-two maps would make $C$ birational to a curve of bidegree $(2,2)$ in ${\mathbb{P}}^1\times{\mathbb{P}}^1$, whose arithmetic genus is one; if the product map has degree two, the maps have the same pencil. Either alternative rules out distinct pencils in genus at least two.

The formula includes a subtle multiplicity check. A nonfixed conjugate pair previously contributed $2^2=4$ to the energy, while a fixed point previously contributed $1$. Replacing these separate contributions by the single $t^2$ gives exactly (30006763:hyper-energy), not a correction of merely $t^2-t$. The script checks the combinatorics on formal sets $\{\pm e_1,\ldots,\pm e_h\}$; those are tests of the identity, not constructed counterexamples or actual curves over a number field.

Attempt: combining divisor rigidity with a lattice bound.

Choose a symmetric ample height on $J$, with canonical quadratic height $\widehat h$. Let \[ r=\operatorname{rank}J(F),\quad \tau=|J(F)_{\mathrm{tors}}|,\quad H=\max_{a\in A}\widehat h(a). \] For $r>0$ let $\mu>0$ be the minimum positive squared norm in the Mordell--Weil lattice $J(F)/J(F)_{\mathrm{tors}}$ with this height. Positivity follows from finite generation and positive definiteness of the canonical height pairing. The triangle inequality for its Euclidean norm puts $dA$ in the ball of radius $d\sqrt H$ modulo torsion.

Distinct lattice points have distance at least $\sqrt\mu$. The disjoint open balls of radius $\sqrt\mu/2$ around lattice points in a ball of radius $d\sqrt H$ lie in a ball of radius $d\sqrt H+\sqrt\mu/2$. Comparing $r$-dimensional volumes gives \begin{equation}\label{wsref-0-33-30-30-30-36-37-36-33-3a-70-61-63-6b-69-6e-67} |dA|\le\tau\left(1+2d\sqrt{H/\mu}\right)^r. \end{equation} There are at most $\tau$ lifts of any lattice point, which accounts for the torsion factor. Combining (30006763:sumset) and (30006763:packing) yields the proved finite-set bound \begin{equation}\label{wsref-0-33-30-30-30-36-37-36-33-3a-64-65-72-69-76-65-64-2d-62-6f-75-6e-64} \boxed{\displaystyle \binom{N+d-1}{d}\le \tau\left(1+2d\sqrt{H/\mu}\right)^r \quad\text{for }d<\operatorname{gon}_{\overline F}(C).} \end{equation} If $r=0$, the simpler statement $N\le\tau$ applies; no nonexistent minimum positive lattice length is introduced. Empty $C(F)$ needs no argument. The quantities $H,\mu$ and $r$ belong to the chosen curve and height; (30006763:derived-bound) is not uniform in those quantities.

For fixed $d$ this gives a bound of scale $\tau^{1/d}(1+2d\sqrt{H/\mu})^{r/d}$ up to a factor depending on $d$. It suggests why using divisors rather than points may help a quantitative argument, but $d$ is bounded by the gonality of a fixed-genus curve and cannot absorb arbitrary rank. No improvement over the strongest known uniform Mordell--Lang bounds is claimed.

Stress test: why reductions do not close the gap.

Suppose $C$ and $J$ have good reduction at finitely many places and put $G_0=\prod_v J(k_v)$. Distinct sums may have the same image in $G_0$. The finite-group energy inequality $\mathcal E(B)\ge |B|^4/|G_0|$ therefore cannot be combined with (30006763:sidon-energy) for the original set without a collision-lifting lemma. This is not a technicality: on the free part of $J(F)$, the kernel of the map to any finite group contains $|G_0|J(F)$ and hence has positive rank whenever $r>0$. Reduction information alone cannot imply that a difference in this kernel is zero. Special geometry of curve points could still control these differences, but that is an additional arithmetic theorem, not a property of the reduction map.

Likewise, low additive energy by itself does not bound cardinality in a rank-free ambient group. The sets $\{e_1,\ldots,e_N\}\subset{\mathbb{Z}}^N$ satisfy (30006763:sidon-energy) for every $N$, and arbitrarily large finite Sidon sets also exist in ${\mathbb{Z}}$ if no height restriction is imposed. Thus the geometric rigidity calculation cannot alone eliminate either rank or the height scale. A putative compactness argument over the moduli space must also contend with its noncompact degenerations and unbounded arithmetic heights; none is suppressed here.

If one assumes a bound $r\le R(g,F)$ for all these Jacobians, the established rank-dependent theorem immediately supplies the requested $c(g,F)$. That is a valid conditional route, not a theorem established in this entry. The desired bound on curve points need not logically require a bound on all Jacobian ranks; treating that stronger conjecture as necessary would also overstate the reduction.

Outcome.

Outcome: Exploratory approach with unresolved gap.

The exact short-sum count, the hyperelliptic energy correction, and the height-sensitive packing inequality are proved as notebook lemmas, with no claim of literature priority. They identify what geometry contributes and which arithmetic parameters it fails to control. A new argument uniformly controlling rational points in high-rank or high-height situations is still required. Neither a counterexample family with unbounded numbers of $F$-points nor the genus-and-field-only bound has been obtained.

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Sources

• S1 Kuhne, The number of rational points on a curve of genus at least two, EMS Press, 2023. https://ems.press/content/book-chapter-files/33181. Catalogue formulation source.
• E1 Jiawei Yu, Xinyi Yuan and Shengxuan Zhou, "Quantitativity on the number of rational points in the Mordell conjecture," arXiv:2602.01820v1 (2 February 2026), 178 pages. Theorems 1.1--1.3. https://arxiv.org/pdf/2602.01820. Explicit rank-dependent bounds in a recent manuscript; no independent audit of the complete proof is claimed.
• E2 Lucia Caporaso, Joe Harris and Barry Mazur, "Uniformity of rational points," J. Amer. Math. Soc. 10 $1997$, 1--35, Theorem 1.1. See also the authors' "Corrections to Uniformity of rational points and further comments" (arXiv title), Tunisian J. Math. 4 $2022$, 183--201; arXiv:2012.14461, Introduction. https://doi.org/10.1090/S0894-0347-97-00195-1. https://arxiv.org/pdf/2012.14461. Conditional uniformity via Lang; the later correction concerns a stronger generic-bound formulation.