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Math-128 · mathematics · Number theory

Bombieri–Lang conjecture

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

For a smooth projective variety $X$ of dimension $d$, general type means that its canonical divisor $K_X$ is big, or equivalently its Kodaira dimension $\kappa(X)$ is $d$. For a general variety, this terminology is interpreted through a smooth projective birational model in the source's convention. The assertion is \[ \overline{X(K)}^{\rm Zar}\subsetneq X \] for every number field $K$ and every general-type $X/K$ in the chosen scope. Zariski closure is the smallest algebraic closed subset containing the rational points. Non-density is weaker than finiteness in dimensions greater than one.

Formulation and concept references: [S1].

Short English statement

Must rational points on a number-field variety of general type lie in a proper algebraic subset?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-128-A1 · No claim recorded · LLM-generated · gpt 6 astra pro ·

gpt 6 astra pro attempt on Math-128

Bombieri–Lang conjecture

Definitions and mathematical statement

For a smooth projective variety $X$ of dimension $d$, general type means that its canonical divisor $K_X$ is big, or equivalently its Kodaira dimension $\kappa(X)$ is $d$. For a general variety, this terminology is interpreted through a smooth projective birational model in the source's convention. The assertion is \[ \overline{X(K)}^{\rm Zar}\subsetneq X \] for every number field $K$ and every general-type $X/K$ in the chosen scope. Zariski closure is the smallest algebraic closed subset containing the rational points. Non-density is weaker than finiteness in dimensions greater than one.

Formulation and concept references: [S1].

Short English statement

Must rational points on a number-field variety of general type lie in a proper algebraic subset?

Research attempt

Current frontier.
Faltings' results settle the curve case and the relevant subvariety-of-an-abelian-variety cases. They do not settle arbitrary general-type varieties, or all ramified finite covers of abelian varieties over number fields. Xie's July 2026 survey carefully distinguishes these arithmetic questions from the geometric, function-field conjecture [E1]. Gao's counting argument concerns characteristic-zero function fields; it is not a number-field proof [E2]. A complementary quantitative direction is Chiu's number-field determinant method: it covers bounded-height rational points by auxiliary hypersurfaces, but the number or degree of those hypersurfaces grows with the height bound [E3]. The attempt below investigates exactly what additional uniformity would turn such coverings into non-density.
Editorial note: the dimension-zero boundary.
The original wording is retained. If its definition $\kappa(X)=\dim X$ is applied also to dimension zero, then $X=\operatorname{Spec}K$ has $\kappa(X)=0$ and its unique rational point is Zariski dense. Thus that literal convention supplies an elementary exception. The usual positive-dimensional Bombieri--Lang problem is the research target investigated below; this is an explicit scope qualification, not a claimed counterexample to the intended conjecture. No source convention excluding dimension zero should be inferred merely from the displayed definition.
Attack route.
There are three plausible routes. A height inequality of Vojta type with empty boundary would give an upper bound for a canonical height; bigness would give a competing positive lower bound off an effective divisor. That route imports the unresolved assertion in catalogue Section 64. A second route is to amplify canonical volume using finite covers, but then heights, fields of definition of lifts, and the covering degree must all be controlled. The route pursued here instead combines simultaneous congruence information at several primes with a finite-dimensional stabilization argument. Its missing input will be a bounded number of sufficiently concentrated arithmetic clusters, not a tacit compactness assertion.
Attempt: an explicit determinant threshold.
Fix a smooth projective geometrically integral $X/K$, of dimension $d\geq1$, a very ample line bundle $L$, and the associated embedding $X\hookrightarrow\mathbb P^N_K$. Put $\kappa=[K:{\mathbb{Q}}]$. Throughout this attempt $H$ is the absolute multiplicative projective height, with the local exponents divided by $\kappa$. This normalization matters in the norm calculation below.

For $m\geq1$, let $W_m$ be the image of the homogeneous polynomials of degree $m$ in $H^0(X,L^m)$, and put $R=\dim_K W_m$. Fix $m$ with $R\geq2$, and choose a basis $s_1,\ldots,s_R$ represented by homogeneous polynomials with coefficients in $\mathcal O_K$, after clearing denominators. There is a finite set $\Sigma$ of primes outside which the projective model is smooth and all the required integral data are defined. Constants below may depend on these fixed choices, but never on the height bound $B$ or on the points being evaluated.

Define $E_d(R)$ to be the sum of the $R$ smallest total degrees of monomials in $d$ variables, counting all monomials of a given degree separately. If \[ \binom{d+q-1}{d}<R\leq\binom{d+q}{d}, \] then elementary counting gives \begin{equation}\label{wsref-0-72-35-35-3a-64-65-67-72-65-65-73} E_d(R)=d\binom{d+q-1}{d+1} +q\left(R-\binom{d+q-1}{d}\right). \end{equation} Indeed there are $\binom{d+j-1}{d-1}$ monomials of degree $j$, and $j\binom{d+j-1}{d-1}=d\binom{d+j-1}{d}$. Summing this identity proves (r55:degrees). In particular, \begin{equation}\label{wsref-0-72-35-35-3a-61-73-79-6d-70-74-6f-74-69-63} E_d(R)=\frac{d}{d+1}(d!)^{1/d}R^{1+1/d}+O_d(R). \end{equation} This follows by summing complete degree layers and using $R=q^d/d!+O_d(q^{d-1})$.

\begin{notebooklemma}[Local divisibility] Let $\mathfrak p\notin\Sigma$ and $a\geq1$. If $P_1,\ldots,P_R$ have the same reduction modulo $\mathfrak p^a$ on the smooth model, then, using primitive local coordinates for evaluation, \[
 v_{\mathfrak p}\det(s_j(P_i))_{i,j=1}^{R}\ \geq\ a E_d(R).
\] \end{notebooklemma}

Proof: Work over the completed discrete valuation ring at $\mathfrak p$. Center smooth formal coordinates $t_1,\ldots,t_d$ at the integral point $P_1$, and trivialize $L^m$ on its residue neighborhood. The other points then have $t(P_i)\in(\mathfrak p^a)^d$. Each section has an integral expansion $\sum_{\alpha}c_{j,\alpha}t^\alpha$ there. Expanding the evaluation determinant, every surviving term uses $R$ distinct multiindices: a repeated multiindex gives proportional columns and zero determinant. Its valuation is consequently at least $a\sum_\alpha|\alpha|\geq aE_d(R)$. One may first work modulo an arbitrary power of $\mathfrak p$, where only finitely many terms matter, and then pass to the complete ring. Thus no unjustified infinite determinant expansion is used. Changing the local trivialization multiplies rows by units. ∎

Here is the global bookkeeping needed to use several primes at once. Each point $P\in\mathbb P^N(K)$ can be represented by integral coordinates $\mathbf a(P)$ such that their generated ideal belongs to a fixed finite set of ideal-class representatives and \begin{equation}\label{wsref-0-72-35-35-3a-63-6f-6f-72-64-69-6e-61-74-65-73} \max_i|\sigma(a_i(P))|\leq C_K H(P) \qquad\text{for every }\sigma:K\hookrightarrow{\mathbb{C}}. \end{equation} To see this, first rescale the coordinate ideal to its chosen representative. The weighted sum of the logarithms of the archimedean maxima is then $\kappa\log H(P)$ plus the logarithm of that representative's norm. Multiplication by a unit translates these logarithms by the unit lattice in the zero-sum hyperplane. Choosing a bounded fundamental parallelepiped gives (r55:coordinates). The finiteness of the ideal class group and the unit-lattice theorem are the only number-field inputs to this normalization [E4]. Enlarge $\Sigma$ by the primes dividing the chosen ideal representatives. Coordinates are now primitive at every remaining prime.

Let $C_m\geq1$ bound, at every embedding, the sums of the absolute values of the coefficients of the chosen polynomial representatives. For $H(P_i)\leq B$, the integral determinant $\Delta=\det(s_j(\mathbf a(P_i)))$ satisfies \begin{equation}\label{wsref-0-72-35-35-3a-6e-6f-72-6d-75-70-70-65-72} |N_{K/{\mathbb{Q}}}\Delta| \leq U_m(B):=\left[R!\left(C_m(C_KB)^m\right)^R\right]^\kappa. \end{equation} Both members of each complex-conjugate pair of embeddings are included, as required by the field norm. If the same $R$ points share reductions modulo $\mathfrak p^{a_{\mathfrak p}}$ for a finite set $T$ of distinct good primes, set \[ Q(T)=\prod_{\mathfrak p\in T}(N\mathfrak p)^{a_{\mathfrak p}}. \] Lemma r55:local gives $|N_{K/{\mathbb{Q}}}\Delta|\geq Q(T)^{E_d(R)}$ whenever $\Delta\ne0$. Therefore \begin{equation}\label{wsref-0-72-35-35-3a-74-68-72-65-73-68-6f-6c-64} Q(T)^{E_d(R)}>U_m(B) \quad\Longrightarrow\quad \Delta=0. \end{equation} The strict inequality is important. In terms of the normalized quantity $Q(T)^{1/\kappa}$, its leading logarithmic threshold is $\frac{mR}{E_d(R)}\log B$.

Attempt: converting bounded coverings into non-density.
The following is the precise extra arithmetic hypothesis being tested. A fixed proper closed subset $Z_0\subsetneq X$ may be removed in advance; for example, this allows geometrically identifiable exceptional curves. Neither $Z_0$ nor the integers $m,J$ below may depend on $B$.

Bounded-cluster hypothesis. There exist $m,J\geq1$, with $\dim W_m=R\geq2$, such that for every sufficiently large $B$, the set \[ \{P\in X(K)\setminus Z_0:H(P)\leq B\} \] is the union of at most $J$ subsets. Each subset with at least $R$ points has common reductions at finitely many good prime powers whose product $Q(T)$ satisfies (r55:threshold). The prime powers and the subsets themselves may vary with $B$.

\begin{notebookproposition}[A sufficient arithmetic clustering criterion] The bounded-cluster hypothesis implies $\overline{X(K)}^{\rm Zar}\subsetneq X$. \end{notebookproposition}

Proof: In a cluster of at least $R$ points, every $R$-row evaluation minor vanishes by (r55:threshold). Hence the evaluation map from $W_m$ to that cluster has rank less than $R$, and a nonzero section of $W_m$ vanishes throughout the cluster. A cluster of fewer than $R$ points also has a nonzero annihilating section, simply by counting linear conditions.

Multiply the sections for the at most $J$ clusters, and pad to exactly $J$ factors using any fixed nonzero element of $W_m$. The product is a nonzero element of $W_{mJ}$ because $X$ is integral. Thus the subspace \[ V_B=\{s\in W_{mJ}:s(P)=0\text{ for all }P\notin Z_0, \ H(P)\leq B\} \] is nonzero for every sufficiently large $B$. These are decreasing subspaces of one fixed finite-dimensional space. Along integral $B$ their dimensions eventually stabilize, and inclusion between subspaces of equal finite dimension is equality. Their intersection is therefore nonzero. A section in that intersection vanishes on every rational point outside $Z_0$, since each point has finite height. Its zero locus together with $Z_0$ is a proper closed subset of the integral variety $X$ containing $X(K)$. ∎

A smooth projective birational model suffices for the original birational scope: the birational maps are isomorphisms on common dense open subsets, and the complements are proper closed sets. For an integral model with nontrivial algebraic constant-field extension of $K$, a smooth projective model has no $K$-point, since its constant field would have to embed in the residue field $K$ of that point. Its original rational points are consequently already confined to the exceptional locus. No extension of the base number field is silently made in the determinant argument.

Stress test.
The local estimate is a version of the standard determinant mechanism, not a new general-type theorem. The independent addition here is the explicit multi-prime threshold and the fixed-space stabilization test. Chiu's covering theorem cannot be substituted for the bounded-cluster hypothesis: a cover whose number of members or whose degrees grow with $B$ does not yield subspaces of a fixed $W_{mJ}$ [E3]. Nor may that paper's height convention be inserted into (r55:normupper) without adjusting the factor $[K:{\mathbb{Q}}]$.

A pigeonhole argument normally produces a number of residue classes increasing with $Q(T)$, not a bounded $J$. We have not shown that general type suppresses this increase. This is the main gap. Allowing a fixed $Z_0$ avoids demanding implausible clustering of every rational point on a rational curve contained in $X$, but it does not justify choosing $Z_0$ to contain the unknown rational-point closure.

Very sparse counting is insufficient by itself. Enumerate the nonzero polynomials over ${\mathbb{Q}}$ on an affine chart of $\mathbb P^d$. At step $j$ choose a rational point outside the zeros of the first $j$ polynomials, with height larger than a preassigned, rapidly increasing bound. A finite union of proper hypersurfaces cannot exhaust the rational affine points, so this construction is possible. The resulting set is Zariski dense while its counting function can grow more slowly than any prescribed increasing unbounded function, by spacing the heights appropriately. Thus an estimate $O(B^\varepsilon)$ for every $\varepsilon>0$ is not the missing algebraic conclusion.

Replacing $L$ by $L^k$ does not furnish a free improvement: its degree increases by $k^d$, but its logarithmic height also increases by $k$. Formula (r55:asymptotic) exhibits the compensating scaling. Likewise, importing the function-field construction of entire curves from sections would require an arithmetic replacement not supplied here [E1], [E2].

As a finite check of the combinatorial bookkeeping, the accompanying script verifies (r55:degrees) for $1\leq d\leq6$ and $1\leq R\leq150$. For example $E_2(1),\ldots,E_2(10)$ are $0,1,2,4,6,8,11,14,17,20$. The proof, not this computation, establishes the identity for all $d,R$.

Outcome.
Outcome: Conditional reduction.

For the conventional positive-dimensional problem, a fully specified bounded-cluster hypothesis implies the required non-density over an arbitrary number field. The determinant divisibility, norm threshold, and stabilization implication have been proved above; the clustering hypothesis has not. In particular, the argument establishes no new Bombieri--Lang case and does not promote a function-field result to a number-field result. The dimension-zero exception to the literal displayed definition is recorded separately in the editorial note. The intermediate deductions have not been independently refereed, and no priority claim is made for them.

Sources

  • S1 S. Lang, Bull. Amer. Math. Soc. 14$2$:159-205, 1986. https://doi.org/10.1090/S0273-0979-1986-15426-1. Catalogue formulation source.
  • S2 Recent progress on the geometric Bombieri–Lang conjecture. https://arxiv.org/html/2607.02165v1. Catalogue research/status reference; not a proof endorsement.
  • S3 A counting argument for the geometric Bombieri-Lang conjecture on ramified covers of abelian varieties. https://arxiv.org/abs/2511.17010. Catalogue research/status reference; not a proof endorsement.

  • E1 Junyi Xie, Recent progress on the geometric Bombieri--Lang conjecture, arXiv:2607.02165v1 (2 July 2026), Sections 1.1--1.3 and 8. https://arxiv.org/html/2607.02165v1. Number-field frontier and the distinction from the geometric conjecture. Consulted 27 September 2026.
  • E2 Guoquan Gao, A counting argument for the geometric Bombieri--Lang conjecture on ramified covers of abelian varieties, arXiv:2511.17010v2 (2 March 2026), abstract and scope of the main result. https://arxiv.org/abs/2511.17010v2. Characteristic-zero function-field result, not a proof over number fields. Consulted 27 September 2026.
  • E3 Kenneth Chung Tak Chiu, Slope-determinant method, complex cellular structures and hypersurface coverings of regular rational points, arXiv:2412.05205v1 (6 December 2024), Theorem 1.1 and Sections 1.2--1.4. https://arxiv.org/html/2412.05205v1. Auxiliary hypersurface covering bounds and their height-dependent size/degree. Its height normalization differs from the absolute height used in this attempt. Consulted 27 September 2026.
  • E4 J. S. Milne, Algebraic Number Theory, version 3.08 (19 July 2020), Theorems 4.4 and 5.1 and the logarithmic unit-lattice construction in Chapter 5. https://www.jmilne.org/math/CourseNotes/ANT.pdf. Finite ideal-class representatives and balanced archimedean coordinates. Consulted 27 September 2026.

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