Integral points throughout character varieties of curves

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Family 027 · Number theory
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-10

1. The question

Imagine carrying a vector around loops on a surface. After returning to its starting point, the vector may have changed by a matrix. A compatible collection of these matrices is called a representation of the surface’s fundamental group. Such representations vary in families, described by algebraic spaces called character varieties.

These spaces are usually studied over the complex numbers. The arithmetic question is whether they contain many points whose coordinates are algebraic integers. More precisely, can one choose a single number field so that integral points over its ring of integers are dense in the algebraic sense?

The manuscript claims an affirmative answer for the determinant-one character varieties of all smooth complex algebraic curves, in every rank. It also allows prescribed behavior around missing points of the curve, including exact nondiagonalizable matrix types. The word “single” matters: the number field is fixed for the whole space, rather than enlarged separately for every point.

2. Background and essential definitions

From loops to matrices

The fundamental group $\pi_1(X)$ records loops in a connected space $X$, with multiplication given by following one loop after another. A rank-$r$ representation is a homomorphism

$$\rho:\pi_1(X)\longrightarrow\operatorname{SL}_r(\mathbb C),$$

where $\operatorname{SL}_r$ is the group of $r$-by-$r$ matrices with determinant one. Changing the basis of the vector space conjugates all matrices simultaneously, so it should not change the represented object.

For a curve of genus $g$ with $s$ punctures, generators can be chosen with relation

$$\prod_{j=1}^{g}[A_j,B_j]\prod_{i=1}^{s}M_i=I, \qquad [A,B]=ABA^{-1}B^{-1}.$$

The matrices $M_i$ describe travel around the punctures and are called boundary monodromies. The relation says that these journeys fit together globally.

A representation is semisimple if it is a direct sum of irreducible representations. This condition concerns all the matrices together. It does not mean that each individual matrix is diagonalizable: an irreducible group action can contain nontrivial unipotent matrices.

The character variety $M_r(X)$ parametrizes semisimple representations up to isomorphism. Algebraically, one first writes the matrix relations as polynomial equations, then takes the ring of polynomial functions invariant under simultaneous conjugation. This produces a natural affine model over $\mathbb Z$, even when the original complex curve itself is not defined over a number field.

Integral and dense have specific meanings

A number field $L$ is a finite extension of $\mathbb Q$. Its ring of integers $\mathcal O_L$ consists of elements satisfying monic polynomial equations with integer coefficients. For example, $\sqrt2$ is integral, whereas $1/2$ is not.

An integral point of the character variety means a point defined over this full ring $\mathcal O_L$. One is not allowed to invert an arbitrary finite list of primes. A set is Zariski dense if no additional polynomial equations cut out a proper closed subset containing it. This is algebraic density, not closeness in ordinary Euclidean distance.

Finally, a matrix is quasi-unipotent if all its eigenvalues are roots of unity. Prescribing its conjugacy class specifies both those eigenvalues and its Jordan block sizes. The matrices $I$ and $\begin{pmatrix}1&1\\0&1\end{pmatrix}$ have the same characteristic polynomial in rank two but belong to different conjugacy classes. Exact boundary conditions distinguish them.

3. Development history and previous partial results

The arithmetic of local systems developed partly through questions about rigidity. A local system is another way of describing a representation: it assigns vector spaces locally, with compatible transport along paths. A rigid local system has no appropriate nontrivial deformations. Simpson conjectured integrality for rigid irreducible local systems in the projective setting.

Esnault and Groechenig proved integrality for cohomologically rigid irreducible local systems with finite determinant and quasi-unipotent boundary monodromy. Cohomological rigidity is a precise condition on infinitesimal deformations. This is a theorem about individual rigid objects, not density in large deformable families.

De Jong and Esnault established a different existence result without rigidity. Under the relevant determinant and boundary assumptions, the existence of a complex irreducible local system implies existence of integral irreducible systems for each prime $\ell$, in the $\ell$-adic setting. Such prime-by-prime existence is distinct from obtaining a Zariski-dense collection over the full ring of integers of one number field.

Litt formulated questions about potential integral density, explicitly asking whether a single field suffices. Coccia and Litt subsequently proved full-ring potential integral density in rank two for $\operatorname{SL}_2$ and $\operatorname{PGL}_2$ on a broad class of smooth varieties. Their surface results permit arbitrary algebraic-integer boundary traces and an extension of degree at most four over the trace field. The manuscript explains that their results also imply the ambient-integral exact-class conclusion in rank two.

The new claim extends the curve theorem to every rank, including nongeneric and nonsemisimple quasi-unipotent boundary data. It does not subsume every feature of the earlier rank-two theorem: arbitrary algebraic-integer traces are more general than the quasi-unipotent boundary condition used here.

Geometric predecessors also matter. Mellit and Su obtained decompositions of character varieties with suitable generic semisimple boundary conditions into tractable pieces. Such complex geometry does not by itself guarantee full-ring integral points: parametrizations may require denominators or may fail at special Jordan types. The new argument works with integral gluing data throughout its rank reduction.

4. What the new manuscripts claim

Let $X$ be any smooth connected complex algebraic curve, and let $r\ge1$. Prescribe quasi-unipotent conjugacy classes $C_1,\ldots,C_s$ at its punctures. Let $Y$ be the locus in the complex character variety whose semisimple representations have precisely those boundary classes.

Choose a number field $K\subset\mathbb C$ containing the boundary eigenvalues. The main theorem claims that some finite extension $L/K$ satisfies

$$M_r(X)(\mathcal O_L)\cap Y(\mathbb C) \quad\text{is Zariski dense in }Y.$$

The conclusion holds on every irreducible component, with the same field $L$ for all components. There is no restriction on genus or Jordan block shape. Empty loci contribute no existence assertion.

The theorem also allows prescribing only boundary characteristic polynomials, provided their roots are roots of unity. Without any boundary restrictions, it asserts potential integral density for the entire character variety. For nonprojective curves with no boundary conditions, density already holds over $\mathbb Z$, because their fundamental groups are free and integral determinant-one matrices are Zariski dense.

For the prescribed-boundary and projective cases, the manuscript gives a possible field explicitly in terms of an intermediate extension:

$$F=K(\sqrt2,\zeta_{pq}),\qquad L=F\bigl(\zeta_r,\{u^{1/r}:u\in\mathcal O_F^\times\}\bigr),$$

where $p,q>r$ are distinct primes, $\zeta_m$ denotes a primitive $m$th root of unity, and $\mathcal O_F^\times$ is the group of units. Although the notation involves all units, this extension is finite because the unit group is finitely generated. In the projective case one starts with $K=\mathbb Q$.

Where exactness is imposed

Integrality is measured in the ambient character variety. Exact Jordan conditions are imposed on the corresponding complex semisimple representation. The theorem does not require the same Jordan type to survive reduction modulo every prime ideal of $\mathcal O_L$. Such a requirement would define a different arithmetic problem.

5. Why the advance matters

Importance: 4/5, provisionally. The claimed theorem settles an all-rank curve case of a broad arithmetic question about moduli spaces. Its strength lies in combining a fixed number field, the full ring of integers, every component and exact boundary data.

The statement connects two seemingly different kinds of structure. Character varieties encode the topology of surfaces through matrix equations. Integral points impose arithmetic restrictions at all finite primes simultaneously. Density says these restrictions still leave enough points to escape every proper algebraic subvariety of each component.

The scope also explains the rating’s limits. This is not a solution for character varieties of arbitrary higher-dimensional varieties in every rank, nor for every reductive group or arbitrary boundary eigenvalues. The reusable contribution is a mechanism for carrying integral density through rank reduction and gluing.

6. The main idea

Replace a difficult representation by smaller pieces

The construction initially uses general invertible matrices, $\operatorname{GL}_r$, because subspaces, quotients and direct sums are easier to handle without fixing determinants. A matrix with a specified Jordan type preserves a flag: a chain of nested subspaces. On suitable successive quotients it acts by scalar roots of unity.

The paper turns a surface with such boundary data into a diagram of surfaces connected by flags and identifications of their quotient spaces. Attaching disks carrying scalar local systems turns boundary behavior into gluing instructions. The argument then inducts on the largest rank present in the diagram.

A maximal-rank surface with boundary is cut into disks. On a disk, a selected proper subspace and its quotient have smaller ranks. The missing extension data are recovered by choices of linear maps. Comparing and rearranging flags lets the pieces fit together, including cases with repeated eigenvalues or nonregular Jordan forms.

The arithmetic difficulty is that a convenient complex splitting need not exist integrally. The manuscript carries free graded modules and integral splittings as part of the data. This allows reconstruction over the full number ring, with no unannounced inversion of primes.

Deal with closed surfaces

Closed surfaces have no boundary from which to start the same reduction. The paper cuts along a nonseparating circle and first fixes a convenient spectrum for its monodromy. A Dehn twist is the operation of cutting along a circle, rotating one side and regluing; it acts on surface representations by explicit transformations of their matrices.

Powers of these twists preserve the integral construction while enlarging its algebraic closure. At a suitable smooth representation, the resulting torus motions vary the spectrum in all needed directions. An irreducibility statement for the relevant twisted surface equation then extends density across the closed-surface case. The paper obtains that statement through finite-group character counts and the Lang–Weil link between finite-field point counts and algebraic geometry.

Restore determinants and select the exact classes

The constructed handle determinants are units in $\mathcal O_F$. Adjoining their $r$th roots in the single field $L$ permits scalar corrections that make every determinant one. Finite generation of the unit group makes this correction uniform across the construction.

Finally the argument passes from representations to their characters. This order matters because semisimplifying a representation can shrink Jordan blocks of its boundary matrices. Exact classes are therefore selected on the quotient by excluding the closed loci where a required matrix rank drops. Density in the appropriate closed locus then gives density in the desired open part, component by component.

7. Scope, limitations, and open questions

Potential integral density allows a finite extension of the ground field. It is not an assertion of density over $\mathbb Z$ in every case, and it does not provide a count of points of bounded height.

The fixed boundary eigenvalues must be roots of unity. The all-rank theorem does not cover arbitrary integral eigenvalues merely because earlier rank-two results permit more general traces. Likewise, exact complex Jordan forms should not be confused with preservation of those forms in all finite-characteristic reductions.

Zariski density need not imply Euclidean density or an effective procedure for finding a nearby integral point. The field construction supplies a common arithmetic setting, rather than a sharp bound for the smallest field or the heights of its points.

The higher-dimensional, higher-rank density question remains outside this manuscript. Extending the integral rank-reduction method beyond surface fundamental groups would require additional structure.

8. Sources and further reading

M1 gives the integral model, the exact-class convention and the common field extension. For the preceding rank-two theorem, see Coccia and Litt, Density of integral points in the Betti moduli of quasi-projective varieties. Litt’s Motives, mapping class groups, and monodromy places the density question in a broader setting.

The primary references for the earlier rigidity and existence results are Esnault–Groechenig, Cohomologically rigid local systems and integrality (2018), and de Jong–Esnault, Integrality of the Betti moduli space (2024), both listed in M1. Their conclusions use different hypotheses and notions of arithmetic existence from the density theorem explained here.

Source manuscripts

• M1. OpenAI, Integral points on character varieties of curves (September 25, 2026).

Sources consulted on 10 October 2026 at repository revision fd4aeeb. Download the manuscript citations.