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Density of integral points in the Betti moduli of quasi-projective varieties

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Pith's one-line read This paper proves that for $G=\mathrm{SL}_{2,\mathbb{Z}}$ or $G=\mathrm{PGL}_{2,\mathbb{Z}}$, integral points are potentially Zariski-dense in the relative character variety $X_{G,C}(Y)$ of any smooth quasi-projective complex variety with…

desk verdict Strong new theorem on potential density of integral points for rank-2 character varieties; the orbicurve reduction is fine for PGL2, and the paper should clarify the status of the SL2 case. read the letter →

arxiv 2507.00167 v1 pith:EQCYZ2ZH submitted 2025-06-30 math.AG math.GTmath.NT

Simone Coccia , Daniel Litt This is my paper · ORCID
classification math.AGmath.GTmath.NT MSC 14G0514D2014H6020H10
keywords potentialdensityintegralpointscharactervarietiesBettimodulimappingclassgroupSL2localsystemsquasi-projectiveDehntwists
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves Conjecture 1.1.1 in the two lowest-rank cases: for $G=\mathrm{SL}_{2,\mathbb{Z}}$ or $G=\mathrm{PGL}_{2,\mathbb{Z}}$, integral points are potentially Zariski-dense in the relative character variety $X_{G,C}(Y)$ of any smooth quasi-projective complex variety $Y$, for any choice of algebraic-integer boundary traces $C$. This is the first positive-dimensional confirmation of the integrality conjecture for rank-two local systems, which previously was known only for isolated rigid representations. The proof proceeds by reduction to punctured Riemann surfaces: a classification theorem for rank-two local systems on quasi-projective varieties says that a Zariski-dense representation either factors through an orbicurve or is rigid, quasi-unipotent, and integral, so the whole problem collapses to relative character varieties of surface groups. There the authors construct, by gluing integral local systems on pairs of pants, a single integral representation whose orbit under the pure mapping class group is Zariski-dense in the relevant variety. The interest is that these Betti moduli spaces sit on the arithmetic side of non-abelian Hodge theory, where potential density of integral points was open in positive dimension except in special families of cubic surfaces.

What carries the argument

The motor of the proof is the pair consisting of pants-decomposition trace coordinates and $P$-good integral points. A pants decomposition $P=a_1\cup\dots\cup a_{3g-3+n}$ of $\Sigma_{g,n}$ gives a trace map $\mathrm{tr}_P:X_{g,n,k}\to\mathbb{A}^{3g-3+n}$. A point is $P$-good when each coordinate of $\mathrm{tr}_P$ lies outside $E=\{\zeta+\zeta^{-1}:\zeta\text{ a root of unity}\}$ and the fiber is perfect, meaning traces avoid $\pm 2$ and the restriction to each pair of pants is irreducible; then a parametrization identifies the fiber with an algebraic torus $\mathbb{G}_m^{3g-3+n}$ on which Dehn twists act by independent coordinatewise multiplications, so the twist orbit, and hence the full pure mapping class group orbit, is Zariski-dense. The arithmetic half is a gluing toolbox: the sets $M_K$ and $N_K$ of integral matrices with unit off-diagonal entries are used to build integral $\mathrm{SL}_2(\mathcal{O}_L)$-representations on pairs of pants, once-punctured tori, and two-holed tori that are $P$-good, and these are glued along common boundary monodromy to cover every $\Sigma_{g,n}$ with $3g-3+n>0$. For $\mathrm{PGL}_2$ and the closed-surface case, the same dynamics is run on $X_{g,-I}$, the relative variety of representations with monodromy $-I$ around a puncture, where the fiber is a smaller torus $\mathbb{G}_m^{3g-3}$.

What would settle it

Exhibit a smooth quasi-projective variety $Y$ and a connected component of $X_{\mathrm{SL}_2,C}(Y)$ of positive dimension whose generic representation has Zariski-dense image, has some boundary trace outside $E=\{\zeta+\zeta^{-1}\}$, and does not factor through any orbicurve; the proof in Section 7.4 asserts that no such component exists, so one explicit example would falsify the reduction. Alternatively, on the surface side, find a boundary tuple $k$ for which the relative character variety $X_{g,n,k}$ has no Zariski-dense set of $\mathcal{O}_L$-points after any finite extension $L$, contradicting Theorem 5.0.4.

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Extended reading notes

Core claim

The central result is Theorem 1.1.2: for $G=\mathrm{SL}_{2,\mathbb{Z}}$ or $\mathrm{PGL}_{2,\mathbb{Z}}$, a number field $K$, and a tuple $C\in (G/\mathrm{ad}G)(\mathcal{O}_K)^n$ fixing the traces of monodromy around each boundary component of a simple normal crossings compactification, there is a finite extension $L/K$ such that $\mathcal{O}_L$-points are Zariski-dense in $X_{G,C}(Y)$. For a surface $\Sigma_{g,n}$, potential density is obtained by exhibiting an integral representation $\rho$ that is $P$-good for a pants decomposition $P$: the traces of $\rho$ along the pants curves avoid the set $E=\{\zeta+\zeta^{-1}:\zeta\text{ a root of unity}\}$ and the corresponding fiber of the trace map is perfect. A dynamical lemma then shows that the orbit of $\rho$ under the pure mapping class group is Zariski-dense in the whole relative character variety, and since the action preserves integrality, the density of $\mathcal{O}_L$-points follows. For arbitrary $Y$, the same statement is forced by the rank-two classification: any Zariski-dense local system is either rigid and quasi-unipotent, hence integral, or pulled back from an orbicurve, and relative character varieties of orbicurves are disjoint unions of those of surface groups. For $\mathrm{PGL}_2$, the proof first handles the locus of representations with monodromy $-I$ around a puncture, then lifts the resulting density from $\mathrm{PGL}_2$ to $\mathrm{SL}_2$ through a finite morphism.

Load-bearing premise

The proof rests on a cited classification theorem saying that every Zariski-dense rank-two local system on a smooth quasi-projective variety either factors through an orbicurve or is rigid, quasi-unipotent, and already integral; if that classification has a counterexample, the reduction of the whole problem to punctured surfaces fails.

Editorial extensions

If this is right

  • Conjecture 1.1.1 is settled for $G=\mathrm{SL}_{2,\mathbb{Z}}$ and $G=\mathrm{PGL}_{2,\mathbb{Z}}$: every relative character variety of either group, with arbitrary algebraic-integer boundary data on any smooth quasi-projective $Y$, has potentially Zariski-dense integral points.
  • This supplies the first positive-dimensional cases of the rank-two integrality conjecture for local systems, going beyond isolated rigid representations.
  • For surfaces, the construction works over a biquadratic extension of the field generated by the boundary traces, and the cubic $x^2+y^2+z^2-xyz=3$ shows that some field extension is genuinely necessary.
  • Because these relative character varieties are log Calabi-Yau, the theorem confirms the expected potential density of integral points for log Calabi-Yau varieties in these cases.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The gluing construction depends on infinitely many units in the ring of integers; a natural test is whether enlarging a field with finite unit group, such as an imaginary quadratic field, defeats the method, and whether some other source of integral matrices then supplies the same density.
  • If analogous unit-matrix sets exist for $\mathrm{SL}_n$, the same pants-decomposition dynamics would suggest potential density for higher-rank Betti moduli on punctured surfaces, with the main difficulty being an $n$-dimensional analogue of the $P$-good conditions.
  • One could test the orbicurve reduction directly on a concrete orbicurve with nontrivial generic stabilizer, such as a weighted projective line, to see how the disjoint-union structure of its relative character varieties behaves; the paper treats this case only through the classification theorems.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proves potential density of integral points in the relative SL2 and PGL2 character varieties of a smooth quasi-projective complex variety Y, with fixed algebraic-integer trace data along the boundary of a simple normal crossings compactification. The proof combines a detailed surface-group theorem (explicit integral P-good representations with Zariski-dense pure mapping class group orbits, constructed via the matrix families M_K, N_K, L_K and gluing along pants decompositions) with a reduction from arbitrary quasi-projective Y to curves using the Corlette–Simpson and Loray–Pereira–Touzet classification of rank-two local systems. The main theorem (Theorem 1.1.2) would resolve Conjecture 1.1.1 for G = SL2 and PGL2 and would give the first positive-dimensional cases of Simpson's integrality conjecture for rank 2.

Significance. If the result is correct, it is a substantial advance: it settles Campana's conjecture for these SL2/PGL2 relative character varieties and provides the first positive-dimensional evidence for Simpson's integrality conjecture beyond rigid local systems. The surface-group part is a strong technical contribution: the explicit matrix families, the P-good point constructions, the X_{g,-I} technology, and the careful bookkeeping of field extensions are detailed and largely self-contained. The paper also clearly identifies which ingredients come from external classification theorems. However, the quasi-projective reduction in §7.4 is the most fragile part of the manuscript, and two load-bearing gaps there and in Proposition 3.1.10 currently prevent the main theorem from being proven as written.

major comments (3)
  1. [§3.1, Lemma 3.1.8] Lemma 3.1.8 is false as stated for N > 1. The proof asserts that the orbit of any point of G_m^{3g-3+n} under the group generated by the diagonal transformations T_{z_i} is Zariski dense when each t_i lies in A^1(Q) \ E. This is only true if the λ_i are multiplicatively independent; for example, if N = 2 and λ_1 = 2, λ_2 = 4, the orbit is contained in the subtorus x_2 = x_1^2 and is not Zariski dense in G_m^2. The lemma is used in the final step of Proposition 3.1.10 to pass from Zariski density of tr_P(Γ_{g,n} · p) in A^{3g-3+n} to Zariski density of Γ_{g,n} · p in X_{g,n,k,Q}. That step requires the lemma for the full pants decomposition, i.e. for N = 3g-3+n, which can be arbitrarily large. Without an additional multiplicative-independence hypothesis (or a different argument, e.g. citing the known orbit-closure results of Golsefidy–Tamam), the proof of Proposition 3.1.10, and hence of Theorem 5.0.2 and Theorem 5.0.4, is incomplete.
  2. [§7.4, proof of Theorem 1.1.2] The assertion that relative character varieties of orbicurves are disjoint unions of relative character varieties of surface groups is false for SL2 when the orbicurve has a stacky point of order 2. Such a point forces the local monodromy of an SL2 local system to be the central matrix -I. But in a surface relative character variety, fixing the boundary trace to -2 does not force -I: by Proposition 3.1.3, X_{1,1,-2} is the Markoff surface x^2 + y^2 + z^2 - xyz = 0, while the locus with monodromy -I at the puncture is the single point (0,0,0). Thus the SL2 relative character variety of the orbicurve is a proper subvariety, not a union of components, of the natural surface relative character variety. Consequently Theorem 6.0.1 cannot be invoked to conclude potential density in the orbicurve variety. The proof needs to treat order-2 stacky points separately, for example by using the X_{g,-I} technology developed in §6.2–§6.3, which is not done in this step.
  3. [§7.4, final SL2 lifting step] The lifting step from PGL2 back to SL2 does not control the boundary trace conditions. After proving potential density of integral points in the PGL2 relative character variety W, the proof invokes Lemma 7.3.1 to lift each OK'-point of W to a point of the SL2 relative character variety W'. But Lemma 7.3.1 only produces some SL2 lift; it does not guarantee that the lifted representation satisfies the prescribed tuple C' of boundary traces. For example, if C'_i = -2 at a boundary component, the corresponding PGL2 boundary datum is f = 4, which also contains unipotent monodromy. A Zariski-dense set of PGL2 points with f = 4 may avoid the locus that lifts to a representation with boundary trace -2. Therefore the density statement for W does not by itself imply density for W', and the claim that 'all these points lift to W'' needs a separate argument that respects the boundary traces, presumably again involving the X_{g,-I} loci.
minor comments (4)
  1. [§4.1, Lemma 4.1.1] In the proof, the system is said to be solved for 'x, y, z, w ∈ Z', but the context is a number field K and its ring of integers; this should read O_K (or O_L after passing to the quadratic extension).
  2. [§5, proof of Proposition 5.0.1] In the case n = 1, g > 1 and k ∈ E, the text reads 'we pick M ∈ N with tr M /∈ E'; the symbol N should be N_K.
  3. [§7.1, Remark 7.1.1] The remark asserts that relative character varieties are independent of the choice of simple normal crossings compactification, but says only 'we leave verifying the details to the reader'. Since this is used freely in §7, a short proof or a precise reference would improve the exposition.
  4. [§6.2, Definition 6.2.1] The notation γ for a pants curve in the definition of X_{g,-I} conflicts with the earlier use of γ for a loop around a puncture in §3.1; this is a minor notational clash that could confuse readers.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof derives potential density from explicit integral-point constructions and external classification results.

full rationale

The paper's central derivation is not circular. The surface-group case is proven by explicitly constructing integral P-good representations (Section 4 and Proposition 5.0.1) and then invoking Whang's external dynamics theorem (Proposition 3.1.7, cited from [Wha20a]) via Proposition 3.1.10 to conclude that the pure mapping class group orbit is Zariski dense. The integrality of the constructed points is verified directly through explicit matrices in MK, NK, and LK, not assumed from the density statement. The PGL2 case is reduced to SL2 by dominant maps from SL2 relative character varieties, and the X_{g,-I} spaces are handled by their own explicit construction in Section 6.3. The quasi-projective case uses external classification results of Corlette--Simpson [CS08] and Loray--Pereira--Touzet [LPT16] as inputs, not as renamed versions of the theorem being proved. Self-citations appear only in the motivation and outlook paragraphs ([Lit24, Question 5.4.3(2)] and [LL25]) and are not load-bearing for any proof step. The skeptical concern about Section 7.4 (that relative character varieties of orbicurves may not be disjoint unions of surface-group relative character varieties when an order-2 stacky point forces monodromy -I) is a potential mathematical gap in the reduction, not a circularity: the paper does not define the orbicurve statement in terms of the surface statement, nor does it fit parameters and rename them as predictions. Accordingly, no circular step can be exhibited by quotation, and the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted constants. Its central claim rests on a chain of external classification and structural theorems: Corlette-Simpson and LPT for orbicurve factorization, CS08 Theorem 7.3 for integrality of rigid quasi-unipotent components, and Whang's torus parametrization of perfect fibers. The orbicurve-to-surface-group decomposition is asserted without proof. There are no free parameters fitted to data.

assumptions (6)
  • domain assumption Corlette-Simpson [CS08, Theorem 1]: a Zariski-dense rank-2 local system on a quasi-projective variety with quasi-unipotent monodromy at infinity factors through a map to an orbicurve if its connected component in the character variety has positive dimension.
    Used in Section 7.4, Case 2, to reduce Theorem 1.1.2 from arbitrary Y to punctured surfaces.
  • domain assumption Loray-Pereira-Touzet [LPT16, Theorem A]: the same orbicurve factorization holds when the local system is not quasi-unipotent at infinity.
    Used in Section 7.4, Case 2, when C is not quasi-unipotent.
  • domain assumption Corlette-Simpson [CS08, Theorem 7.3]: zero-dimensional components of relative character varieties with quasi-unipotent boundary traces are integral.
    Used in Section 7.4, Case 1, to assert that an isolated representation is defined over a ring of integers. This is a critical point that the text does not verify in detail.
  • domain assumption Whang [Wha20a, Proposition 4.3]: for a perfect fiber X^P_{k,t}, the Dehn twist action is lifted by a torus action on G_m^{3g-3+n}, with a surjective finite morphism to the fiber.
    Quoted as Proposition 3.1.7 and used in Lemma 3.1.8 to prove Zariski density of mapping class group orbits.
  • domain assumption Whang [Wha20a, Lemma 3.3]: an SL2 local system on a pair of pants is irreducible unless the three boundary traces satisfy x^2+y^2+z^2-xyz=4.
    Used in Corollary 4.1.2 and in Definition 3.1.5 to identify perfect fibers.
  • domain assumption Relative character varieties of orbicurves decompose as disjoint unions of relative character varieties of punctured surface groups.
    Stated without proof in Section 7.4 ('Here we use that...'). This is load-bearing for the reduction from orbicurves to Sigma_{g,n}.

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Pith. "Pith review of Density of integral points in the Betti moduli of quasi-projective varieties." pith.science (2026). https://pith.science/paper/EQCYZ2ZH

@misc{pith2026250700167,
  author       = {Pith},
  title        = {Pith review of: Density of integral points in the Betti moduli of quasi-projective varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EQCYZ2ZH}},
  note         = {Machine review of arXiv:2507.00167}
}
abstract

Let $Y$ be a smooth quasi-projective complex variety equipped with a simple normal crossings compactification. We show that integral points are potentially dense in the (relative) character varieties parametrizing $SL_2$-local systems on $Y$ with fixed algebraic integer traces along the boundary components. The proof proceeds by using work of Corlette-Simpson to reduce to the case of Riemann surfaces, where we produce an integral point with Zariski-dense orbit under the mapping class group.

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