Integral points on character varieties of curves

1 Introduction

Let X be a smooth connected complex algebraic curve and let r\geq1. Its \SL_r-character variety parametrizes semisimple representations of \pi_1(X) up to isomorphism. Although X need not be defined over a number field, this character variety has a natural integral model: a finite presentation of its fundamental group gives a representation scheme over \Z, and we take the spectrum of its ring of conjugation invariants. Write this affine scheme as M_r(X).

An arithmetic question is whether its integral points become Zariski dense after extending the ground field once. Here integrality always refers to the full ring of integers. We prove this for curves, retaining arbitrary quasi-unipotent conjugacy classes at the punctures.

Let \overline X be the smooth projective completion of X, and write \overline X\setminus X=\{x_1,\ldots,x_s\}. For each i, fix a positive meridian \gamma_i about x_i. A matrix is quasi-unipotent if all its eigenvalues are roots of unity. Choose quasi-unipotent conjugacy classes C_i\subset\SL_r(\C), and let Y be the reduced locus in M_r(X)_\C whose semisimple representations have \rho(\gamma_i)\in C_i. This condition allows nonsemisimple boundary matrices: semisimplicity of a representation is a condition on the whole group action. The locus Y is locally closed, as explained in Section 2.

For a number field L\subset\C, write \cO_L for its full ring of integers. We use ambient integrality on Y: a point extends to M_r(X) over \cO_L, and its complex semisimple representation has the specified boundary classes. It need not avoid smaller Jordan classes after reduction at every finite prime. Requiring a section of a strict locally closed integral model is a different condition.

The extension in (1) is finite because the unit group of \cO_F is finitely generated. It is fixed for the entire locus, including all its components. No restriction on the genus or on the Jordan forms is imposed.

For a nonprojective curve without boundary conditions, density already holds over \Z: the fundamental group is free and \SL_r(\Z) is Zariski dense in \SL_r(\C). The substantive cases here are prescribed boundary monodromy and projective curves, where the surface relation couples the matrices. We construct actual integral representations and then pass to their characters; the exact boundary conditions are selected only after this passage.

1.1 History and relation to previous work

The arithmetic study of local systems includes Simpson’s conjecture that rigid irreducible representations of smooth projective fundamental groups are defined over rings of integers [20, p. 9]. Esnault and Groechenig prove integrality for irreducible cohomologically rigid local systems with finite determinant and quasi-unipotent boundary monodromy on smooth quasiprojective varieties [4, Theorem 1.1]. Here an integral local system may have a finite projective lattice over a number ring; freeness is not part of that assertion. Without a rigidity assumption, de Jong and Esnault prove that the existence of an irreducible complex local system with torsion determinant and quasi-unipotent boundary monodromy implies the existence of an absolutely irreducible integral \ell-adic local system for every prime \ell, preserving the determinant and the semisimplified boundary classes [3, Theorem 1.1]. This prime-by-prime existence is distinct from density over the full ring of integers of one number field.

Litt asks for potential integral density on character varieties of smooth projective varieties and also formulates a boundary variant [13]; his survey explicitly asks whether one number field suffices [14, Question 5.4.3(2)]. Coccia and Litt formulate a related conjecture for Chevalley groups with fixed boundary data in the adjoint quotient [2, Conjecture 1.1.1]. Theorem 1.1 answers the determinant-one form of Litt’s question affirmatively for curves, in every rank, and proves the \SL_r curve case of the Coccia–Litt conjecture for quasi-unipotent boundary data. The higher-dimensional, higher-rank question remains outside its scope.

Coccia and Litt prove full-ring potential integral density for \SL_2 and \mathrm{PGL}_2 on smooth varieties admitting a smooth projective compactification with simple normal crossings boundary [2, Theorem 1.1.2]. Their surface theorem allows arbitrary algebraic-integer boundary traces and gives an extension of degree at most four over the field generated by those traces [2, Theorem 5.0.4 and Remark 5.0.5]. Their ordinary and twisted surface theorems also imply the ambient-integral exact-class statement in rank two, as explained in Remark 8.1. Their use of integral representations and mapping-class-group dynamics is an arithmetic predecessor of the closed-surface argument below. Theorem 1.1 treats every rank with exact quasi-unipotent boundary data; in rank two, Coccia and Litt allow a broader range of boundary traces.

Earlier arithmetic work includes Whang’s descent framework for rank-two integral characters with fixed integer boundary traces [22]. For the Markoff cubic family x^2+y^2+z^2-xyz=k, Ghosh and Sarnak prove integral density for almost all locally admissible integer parameters, and also exhibit infinitely many admissible parameters with no integral points [7, Theorem 1.2]. These results concern ordinary integral points and help distinguish that question from density after one finite extension of the ground field.

On the geometric side, Mellit uses flags and braid operations to stratify a vector bundle over a character variety into torus and affine-space pieces, for generic semisimple data with a regular semisimple puncture [15, Theorem 1.2]. Su constructs an open dense torus for nonempty character varieties with generic semisimple boundary data and a regular semisimple puncture [21, Theorem 0.5]. These results concern complex geometric decompositions. The present argument must also preserve integral splittings and identified graded modules while treating arbitrary Jordan types and nongeneric components. It therefore carries out the local operations over the full ring of integers, rather than obtaining integral points merely from a complex coordinate parametrization.

1.2 The proof

We first work with \GL_n rather than \SL_n. This makes direct sums, subspaces and quotients available without determinant restrictions. A boundary matrix with a prescribed Jordan type admits an invariant flag on whose successive quotients it acts by scalar roots of unity. We attach disks carrying these scalar local systems to the boundary. The resulting object consists of oriented surfaces of various ranks, joined along boundary pieces by flags and their graded identifications. Section 4 defines exactly which joins are allowed.

The main construction proves integral density for all such diagrams by induction on the largest rank. A surface of maximal rank with boundary is cut into disks. Two flags at the ends of a cut are compared by refining them to their common graded pieces and interchanging adjacent pieces. On each disk, a proper subspace chosen from one boundary flag is preserved by all puncture monodromies, since these are scalar. Passing to that subspace and its quotient lowers the rank. The lost extension data are recovered by successive choices of linear maps. Choosing the subspace as a prefix of the starting flag makes the lifted flag unique on that interval, so returning around the boundary introduces no further equation. The reduction includes the complementary swaps needed for nonregular boundary data (Section 5).

Closed surfaces enter at the same induction step. Cutting along one nonseparating circle and fixing a convenient spectrum brings them into the preceding case. Powers of a Dehn twist then enlarge the closure of the integral points. At one explicit smooth representation, the resulting torus motions vary the spectrum in every direction. Irreducibility of the twisted surface equation finishes the argument. We give a finite-field proof of irreducibility, following the character-counting method of Liebeck and Shalev [12]. Sections 6 and 7 carry out this closed-surface step; the latter then assembles the rank induction.

Finally, all handle determinants of our integral representations are units in \cO_F. Adjoining their rth roots uniformly by (1) permits determinant-one twisting. Exact Jordan conditions are then selected on the character quotient. This order is essential: taking the semisimplification of a representation can reduce its boundary Jordan blocks. Section 8 completes this descent.

The reusable ingredient is the diagram density theorem. It transfers integral density through rank reduction using free graded modules and linear extension parameters.

Three standard inputs enter at distinct points. Finite generation of integral invariants gives the affine model in Section 2. Dirichlet’s unit theorem gives the single field in Section 3. The Lang–Weil estimate converts the finite-group count proved in Section 6 into geometric irreducibility. The flag reconstruction, character estimates, twist directions and exact-Jordan descent are proved below.

2 The integral model and exact boundary conditions

We specify the integral model once and record how to select exact boundary classes on its complex quotient. In particular, we distinguish semisimplification of a representation from diagonalization of its individual matrices.

2.1 Representation schemes

For a finitely presented group \Gamma and an integer r\geq1, let \mathscr R_r(\Gamma) be the affine \Z-scheme representing

Concretely, choose a matrix for each generator and impose the matrix relations of a finite presentation. Simultaneous conjugation defines an action of \SL_{r,\Z}. With A=\Z[\mathscr R_r(\Gamma)], put

This definition is independent of the chosen presentation, by the representing property. The invariant ring is finitely generated over \Z by finite generation for Chevalley-group invariants [5, Theorem 3]. Here the base \Z is Noetherian, A is a finite-type commutative algebra, and \SL_{r,\Z} is a split Chevalley group with its rational conjugation action. This input does not require A to be reduced or flat over \Z.

Flat base change commutes with these invariants: they are the kernel of the difference between the coaction and the map a\mapsto a\otimes1, and tensoring with a flat algebra preserves that kernel. In particular, the complex fiber of (2) is the usual affine character quotient. Its complex points correspond to the unique closed orbits in the fibers of the quotient [18, pp. 184–185]. The representation scheme is closed in the space of generator tuples: inverses in \SL_r are polynomial adjugates, so the relators are polynomial equations. Kraft’s matrix-tuple closed-orbit criterion, recalled in [1, Section 3.4], therefore identifies these orbits with semisimple representations. Conjugation by \GL_r(\C) gives the same orbits as conjugation by \SL_r(\C), since any invertible matrix can be multiplied by a scalar to have determinant one.

For the curve X, we use \Gamma=\pi_1(X) and write M_r(X)=M_r(\Gamma). Every representation \Gamma\to\SL_r(\cO_L) gives a point of M_r(X)(\cO_L) by evaluating invariant functions. Thus it suffices to construct actual integral matrices. No assertion about generation of invariants in positive characteristic is needed for this implication.

2.2 Rank conditions on the quotient

Write \mathscr R=\mathscr R_r(\Gamma)_\C and let q:\mathscr R\to M_r(\Gamma)_\C be the quotient map. We use reduced closed subsets when imposing additional conditions. An invariant closed subset D\subset\mathscr R has closed image under q. Moreover,

Indeed the closure of every orbit in the fiber contains that closed orbit, and a closed invariant subset contains the closure of each of its orbits. These are the standard closed-orbit properties of an affine quotient by a reductive group in characteristic zero [18, pp. 184–185].

Fix the characteristic polynomial of each boundary matrix and let \lambda range over its eigenvalues. A Jordan form is determined by these eigenvalues and the numbers

Here the expression on the right means the rank for any matrix in the chosen class C_i. Let D be the invariant closed subset defined by the fixed characteristic polynomials and the inequalities

Let D_{\mathrm{sm}}\subset D be the finite union of the closed subsets on which at least one of these inequalities is strict; indices with d_{i,\lambda,k}=0 contribute an empty subset. The matrices have exactly the prescribed classes precisely on D\setminus D_{\mathrm{sm}}.

The lemma permits us to construct integral representations in closed rank bounds and impose the exact conditions after passing to characters. Density will then meet every open part of the desired locus, including those in individual irreducible components.

3 Arithmetic parameters and integral splittings

Our constructions will use matrices, split flags, and gluing maps over one ring of integers. We first record the parameter sets that are dense over that ring and the finite extension needed to remove determinants.

Fix the rank bound r\geq 1 and a number field K\subset\C containing the prescribed boundary eigenvalues. Choose distinct primes p,q>r, fix a primitive pq-th root of unity \zeta, and put

For a projective curve we take K=\mathbb Q. Throughout the proof, “integral” means over the full ring R, or over the full ring of integers of a specified finite extension. In particular, an invertible integral matrix belongs to \GL_d(R): its determinant is a unit.

We will always supply free graded modules as part of integral flag data. This requirement permits the following elementary splitting argument over R, without any assumption on its ideal class group.

In particular, every matrix changing between two such integral splittings is an R-isomorphism; its inverse is integral. This argument concerns flags whose graded modules are explicitly free. We will establish that property for each flag used below.

The unit differences in Lemma 3.3 allow spectral projectors to be formed over R. The freeness of their images, when needed, will follow from the free graded lines of the accompanying flag.

The field L depends only on the fixed initial data. Later we will take roots of determinants of matrices over R; Lemma 3.4 handles all those roots simultaneously, regardless of the representation or irreducible component.

4 Filtered local systems on surfaces

The proof uses auxiliary surfaces whose boundary local systems are related by filtrations. Cutting these surfaces and refining their filtrations will reduce the rank. We first define the auxiliary objects and the linear algebra needed to reconstruct them after a cut.

Use the field F and its full ring of integers R from Section 3. All prescribed scalars below are roots of unity in F. A local system over R is required to have finite free fibers. Its parallel transports are thus isomorphisms over the full ring R.

4.1 Surfaces, seams, and their solutions

A surface diagram has finitely many compact oriented surfaces, called its facets. Each facet has an assigned rank between 0 and r and finitely many interior punctures. A scalar monodromy is prescribed at each puncture, using the positive orientation around that puncture.

Every boundary circle is either left unmarked or divided into intervals by finitely many marked points. These boundary pieces are grouped into seams. A seam has a distinguished parent side and an ordered list of child sides, written

All its sides are parametrized by the same interval or circle. The boundary orientation of each child is opposite to that of the parent. Different side occurrences may belong to the same facet. An unmarked circle seam uses entire circles and has no permutation of its sides.

At an interval endpoint, the boundary of each facet pairs its two incident side germs. We permit precisely the following vertices.

1. 1.

   A parent list continues with the same ordered children, or with two adjacent children interchanged. Each child continues to itself.

2. 2.

   A child W of one parent list is replaced by an ordered list W'_1,\ldots,W'_s, and there is a third seam W:(W'_1,\ldots,W'_s). The parent germs continue to each other; the two occurrences of W are paired; the two occurrences of each W'_j are paired; and unchanged children continue to themselves. The reverse operation is also permitted.

The specification concerns distinct occurrences of germs, even when some belong to one facet. No embedding of the diagram in a surrounding space is required.

A solution assigns to each punctured facet a local system of its rank, with the prescribed scalar monodromies. At a seam it assigns a parallel filtration of the parent and parallel identified quotients

The conditions at vertices are as follows. At a continuation, all data continue. At a split or merge, the finer parent filtration is obtained by refining the W-quotient with its given filtration, and the identified quotients agree. At an interchange of A,B, the coarsened filtration with quotient A\oplus B is unchanged. Within that quotient the two intermediate subspaces are complementary, and the maps to the identified quotients are the natural projections. Thus the interchange includes a specified direct-sum decomposition, not just a pair of transverse flags.

Choose one frame on each connected facet. Holonomies along a finite set of paths, the seam flags, and their identified quotients describe the solutions by finitely many algebraic equations and open conditions. They form a complex variety, denoted \mathcal S(D) for a diagram D. Changing the auxiliary paths gives the same data through composition and inversion of transport matrices.

An integral solution uses free R-local systems and filtrations whose identified graded modules are the specified free child fibers. Complementarity at an interchange means a direct sum over R. In particular, each filtration splits on a fiber: successive free quotients are projective. This definition does not assert that every submodule of a free R-module is free.

We prove Theorem 4.1 by induction on the largest rank, normalizing identity seams before separating closed facets. The nonclosed step is proved in Section 5. The rank-one and genus-zero closed cases are elementary, and the genus-one case is Lemma 6.2. At each remaining rank, Lemma 7.1 gives the closed-surface step; it uses the normalized nonclosed step at that same rank and the completed induction at smaller ranks.

4.2 Transferring density through choices

We apply this lemma with B_\tau the solutions of a simpler diagram and T_\tau the choices for reconstruction. The index \tau records ranks of finitely many intersections and projection images. Its finiteness follows from the fixed rank bound. The forward reduction need not be a morphism on the entire original solution variety; it is enough that every original solution has a reconstruction in one of the finitely many spaces T_\tau.

The choice spaces used below are locally affine bundles, or products of these with general linear groups. Their projections are therefore open. Over an integral solution, the stated free splittings identify their parameters with matrices over R and with invertible matrices over R. Both sets are Zariski dense in the corresponding complex spaces by Lemma 3.1. We may include arbitrary frame changes among the choices; coverage up to isomorphism then suffices for framed coverage.

4.3 Comparing two flags

The following lemma includes the compatibility of the identified quotients. That compatibility is needed when a cut surface is sewn back together. Flag decompositions and braid operations also underlie the geometric stratifications of Mellit [15, Theorem 1.2] and Su [21, Theorem 0.5]. Here we formulate reconstruction over one ring of integers with specified free graded modules, so that every change of splitting has integral matrix parameters.

4.4 Lifting a filtered exact sequence

Replacing a local system by a subspace U and quotient Q loses both the position of a boundary flag in U\oplus Q and the identifications of its quotients with the child systems. The next lemma recovers both. Its second part concerns a further loss of information at a vertex: a coarse quotient is identified with a space W carrying an internal flag or decomposition, whereas an adjoining interval retains only the resulting finer flag and its fine quotient identifications.

For example, let U,Q be lines and take W=U\oplus Q with its standard exact sequence. An identification of V with W inducing the identity on U,Q can vary by any element of \Hom(Q,U). If we retain only the flag 0\subset U\subset V and its two identified quotients, all those choices give the same data. The coarse stabilizer is zero and the refined-interval stabilizer is all of \Hom(Q,U). The reversal comes from forgetting the full coarse identification, not from imposing fewer subspaces.

At a vertex there will be two adjoining intervals. We use the surjection to either one separately: an incoming lift extends to a vertex lift, which then determines an outgoing lift. We do not prescribe two interval lifts independently or assert that every such pair comes from a vertex lift.

5 Reducing the rank of a surface diagram

We prove the nonclosed step in the diagram-density induction. Call a diagram normalized if all its facets have positive rank and every seam has at least two children; call these seams proper. The first subsection explains how to reach this form. This operation may create closed facets, so it must precede their separation from the facets with boundary.

We prove the proposition by constructing finitely many lower-rank diagrams from which every original solution can be recovered. Over each lower solution variety, reconstruction consists of successive locally affine bundles and changes of frames. Every integral lower solution admits lifts, with dense integral choices in each fiber. Lemma 4.2 will then give the conclusion.

There are two geometric stages. First, removing bands cuts the rank-n facets to disks; the two-flag lemma recovers each band through affine matching choices. Next, a proper subspace and its quotient replace each disk. The vertex constructions ensure that arbitrary lower data admit the successive lifts needed to reconstruct the disk, including closure around its boundary.

5.1 Removing identity seams

Delete every zero-rank facet and every zero child. Interchanges with a zero child become continuations. A seam with one positive child is an identity seam; its specified isomorphism sews its two sides together. Opposite boundary orientations give an oriented surface. At a split vertex with only one positive subblock, this sewing turns the split into continuation. If the original parent list has just one positive block, sewing identifies its refined list with the additional seam. If all three lists have one positive block, three corner sectors sew to a disk, and the refinement compatibility is precisely the composition condition for the local system to extend over its center. Identity circle seams are sewn in the same way.

These operations use occurrences of boundary germs and therefore also apply to seams incident several times to the same facet. Solutions of the sewn diagram restrict to solutions of the old one; allowing frame changes gives all old solutions. The construction is valid over R. We can consequently work with normalized diagrams. A facet of maximal rank n can then only be a parent, and every child on its boundary has rank less than n.

The closed rank-n facets of a normalized diagram are independent factors of its solution variety. Set them aside when applying Proposition 5.1; Sections 6 and 7 will supply their density. We now prove the proposition for the remaining diagram, using only the smaller-rank induction hypothesis. If it has no rank-n facets, that hypothesis already applies.

5.2 Cutting the maximal-rank facets to disks

In each nonclosed rank-n facet choose disjoint properly embedded arcs which cut the underlying surface to disks. Their endpoints are distinct interior points of seam intervals; an unmarked circle seam may first be marked. The arcs avoid all punctures, which do not affect this topological choice. Remove narrow bands around the arcs.

Consider one band. At its two short ends, the old seams give flags of types F_\bullet and H_\bullet. On each long side put the two-flag sequence of Lemma 4.3, with the same array of block ranks on the two sides. Across the band put a lower-rank strip for each block of the running list. Its two long sides are children of the two parent sides. At a short end it extends the corresponding old child facet. To split a block on both long sides, put a transverse seam across its strip; its endpoints give the two split vertices. Merges reverse this construction. At an interchange put an interchange vertex on both parents and continue each labeled strip unchanged. Strips may cross one another in this description: the diagram is specified by its side incidences, not by a planar embedding. Figure 1 shows one strip occurrence and the two-by-two example of the common gallery; its blocks W_{ij} have ranks m_{ij}.

All strips inherit the band orientation. Their long-side boundary orientations are opposite to those of the corresponding parents, and the short-end attachments sew opposite orientations. Thus the result is again an oriented surface diagram. Its only rank-n facets are disks, possibly with punctures. Every new strip has rank less than n, since it refines a child of a proper seam. Zero blocks can be deleted.

We verify reconstruction before applying density. A solution of the cut diagram gives two pairs of flags, one on each side of the band. The old child transports prescribe isomorphisms on their F-graded and H-graded modules. Transverse split and merge seams make those isomorphisms preserve the refinements. Transports on the fully refined strips and the natural maps at interchanges make them agree on the double-graded modules. Lemma 4.3 therefore supplies a full isomorphism respecting the entire gallery, not just its endpoint flags. Use it to sew back the rank-n band. The old seam flags extend across its short ends. Different bands give independent matching choices.

Conversely, transport the two endpoint flags of an old solution into a band. Their intersection ranks specify one of finitely many arrays; a simultaneous splitting gives its two-flag sequences and all the strips, matched by the old band transport. This produces a cut solution reconstructing the old one. The matching choices form locally affine bundles by Lemma 4.3, and over integral cut solutions they have dense integral parameters. Thus Lemma 4.2 reduces the problem to diagrams whose maximal-rank facets are disks. The added topology of the lower-rank facets is retained and causes no restriction in the induction hypothesis.

5.3 Replacing a disk by a subspace and a quotient

Let V be one rank-n disk. At one starting boundary interval, choose a nonzero proper prefix of its seam flag, ending at index k. Such a prefix exists because the seam is proper. In an old solution this subspace extends to a parallel subsystem U throughout the punctured disk, because every puncture monodromy is scalar. Put Q=V/U. Both ranks are less than n, and both systems have the same scalar puncture monodromies as V.

The prefix will make the reconstruction close around the boundary. At this interval the induced flags satisfy Q_i=0 for i\leq k and U_i=U for i\geq k. Once these flags and their identified quotients are fixed, they have a unique lift in U\oplus Q: before k it is the given flag in U, and after k it is the inverse image of the flag in Q. Equivalently, the stabilizer in Lemma 4.4 is all of H=\Hom(Q,U). These zero-rank conditions will be part of the chosen lower diagram, so the unique starting lift exists for every lower solution, not only for one extracted from an old solution.

On an interval with old seam list W_1,\ldots,W_m, the old flag induces flags on U,Q and exact sequences

Choose the ranks of all these modules, and of the modules at the finitely many vertices below. There are finitely many possibilities; we use the possibilities arising from old complex solutions. The actual construction of the lower diagram depends only on these ranks, not on the old solution. In particular, at the starting interval the Q-graded pieces have rank zero for i\leq k, and the U-graded pieces have rank zero for i>k. Intersections and projection images are extracted only from complex solutions. Integral reconstruction will instead use the specified free child modules; it does not require intersections of arbitrary R-lattices to be free.

Replace V by two disks U,Q, with the same orientation and scalar punctures. On a regular interval replace its seam by

Here u_i is an actual strip joining the U side to the W_i side, as child on both, and q_i similarly joins Q to W_i. Its transport gives the maps in (4). Along an unmarked seam circle these may be cyclic strips. The old parent and child have opposite boundary orientations, so the strips sew with the required orientations.

We describe the two vertex replacements. In these descriptions, time increases along the positive boundary orientation of V, and along those of U,Q. It therefore runs opposite to the old child boundary orientation. Unaffected strips and lists continue.

Each replacement has a common reconstruction requirement. At a split, the coarse quotient is W with its internal flag; at an interchange, it is A\oplus B with its given decomposition. The lower data must supply an exact sequence with this middle term whose restrictions are the exact data on both neighboring intervals. Lemma 4.4 will then extend an incoming interval lift to the vertex, and that choice will determine the outgoing lift.

A split or merge

Suppose an old block W splits into W'_1,\ldots,W'_s. Before the event its list is W:(w,q). Split the w,q connections into w_1,\ldots,w_s and q_1,\ldots,q_s by transverse seams, refining the lists at both ends of each connection. On W, use the two-flag sequence to interleave them as

Merge each pair in this list to W'_j, introducing the side of W'_j through the merge vertex and its seam W'_j:(w_j,q_j). The resulting list on W continues onto the old additional seam W:(W'_1,\ldots,W'_s).

At this merge vertex the W'_j corner joins its occurrence as a child of W to its occurrence as parent of w_j,q_j. The strip boundaries formerly at W continue at W'_j, and their other ends remain at U,Q. This describes all germ pairings. The W lists run opposite to its boundary orientation toward the old side seam; the W'_j sides run into that seam as children and out along their new strips as parents. Hence the orientations agree. Reverse this construction for an old merge.

Lemma 4.3 says exactly what the construction records: the flag of W with two pieces w,q is compared with its flag having pieces W'_j. Their intersections induce the refined flags on U,Q, and the matched double-graded maps give the exact sequences for the W'_j. Every old complex solution admits this replacement. Conversely, any lower solution supplies these compatible refined exact sequences, including all their identified quotients.

Here the seam W:(w,q) supplies the coarse exact sequence, and the retained seam W:(W'_1,\ldots,W'_s) supplies its internal flag. The two-flag gallery makes their induced fine quotient maps agree. Thus every lower solution supplies the coarse exact sequence with middle term W and its fixed internal flag, not only the sequences on the fine quotients W'_j. The restriction of a vertex lift to the coarse interval is the identity. On the fine interval we forget the full identification with W. For the exact sequences 0\to w_j\to W'_j\to q_j\to0, free compatible splittings give

The restriction to that interval is therefore an affine surjection with free integral parameters. The same two restrictions handle the reverse merge.

An interchange and the negative scalar

Now suppose the old event interchanges A,B. Their coarsened quotient of V is identified with A\oplus B. In an old solution write U_*\subset A\oplus B and Q_*=(A\oplus B)/U_* for the induced subquotients. Put

The modules \widetilde a/a and \widetilde b/b have the same rank, denoted h. Put x=A/\widetilde a and y=B/\widetilde b. The required before and after lists are

These symbols name ranks and strip occurrences in the new diagram; the intersection description explains how every old solution determines such data.

Split the incoming \widetilde b connection into b,h, and the incoming A/a connection into h,x. The four lists become

Interchange a,b on U and x,y on Q, leaving those connections unchanged. Replace the two h connections

using one punctured disk of rank h, with scalar monodromy -I. Its boundary contacts, all as child, occur in the cyclic order U,B,Q,A. Along U it runs from positive to negative time, then crosses the incoming (U,B) strip end; along B it runs from negative to positive time, then crosses the outgoing (B,Q) end; along Q it runs from positive to negative time, then crosses the incoming (Q,A) end; along A it runs from negative to positive time, then crosses the outgoing (A,U) end. This specifies the oriented sewing to all four strip ends. Finally merge a,h to \widetilde a and h,y to B/b. If h=0, delete this disk.

The lower solution provides maps p_A:U_*\to A, p_B:U_*\to B, i_A:A\to Q_* and i_B:B\to Q_*, with the images and kernels specified in the table. Write

These are the rank-h subquotients at the four contacts. The maps p_A,p_B induce isomorphisms from h_U to h_A,h_B, and i_A,i_B induce isomorphisms from h_A,h_B to h_Q. Figure 2 displays the two resulting paths from h_U to h_Q.

The two composites i_Ap_A,i_Bp_B factor through h_U to h_Q. They are opposite, because they are the two transports around the punctured h-disk. Hence the following sequence is a complex:

It is exact over R as well as over \C. The kernels of p_A,p_B are respectively b,a, and their intersection is zero by the interchange on U. The images of i_A,i_B span Q_* because their x,y quotients are complementary. If i_A(z)+i_B(t)=0, projection modulo \operatorname{im}i_A gives t\in\widetilde b. Lift t under p_B and subtract the image of that lift; the remaining vector is in \ker i_A=a, which is again in the image of U_*. The needed lifts exist over R because the named quotients are free.

For clarity, in split coordinates the maps are

Thus the scalar -I supplies exactly the sign needed in (5); no division by 2 is involved. Conversely, any old exact sequence with the induced flags gives the refinements and maps above, and its two induced h-maps have this opposite sign. They therefore extend over the punctured disk.

The exact sequence with middle term A\oplus B is the coarse vertex data required for reconstruction. In the displayed free coordinates, forgetting its full identification permits the additional stabilizer block \Hom_R(y,a) on the incoming interval and \Hom_R(x,b) on the outgoing interval. All unchanged blocks cancel from these quotients. Each restriction is considered separately: split its refined flags on U,Q and group the free summands to obtain the coarse flags. Both restrictions are affine surjections with free integral parameters. No simultaneous choice of both interval lifts is required.

All replacements occur in separate side intervals and corner neighborhoods. They preserve the other seams of every lower-rank facet, including repeated occurrences of the same facet. Parallel maps on the strips extend across the specified cross sections, so the construction also applies to cyclic boundary lists.

5.4 Reconstruction and closure around a disk

We now reconstruct a disk from an arbitrary solution of its lower diagram. Put V=U\oplus Q. All puncture monodromies are the prescribed scalars, so this is a local system with the required monodromy. Moreover H=\Hom(Q,U) is constant on the punctured disk: the identical scalar monodromies of U,Q act trivially on it.

The starting interval has the unique lift already established. Continue it by parallel transport to the first vertex. The relevant vertex construction gives an affine space of extensions of this incoming lift. Choose one and restrict it to the outgoing interval, then repeat around the boundary. Every step is surjective with affine fiber, and over integral lower data that fiber has free integral parameters. On return, the starting interval still has just one lift, so the propagated lift agrees with it automatically. This includes all identified quotients: they belong to the lift spaces and are not additional data to be matched afterward. With no vertices, the same unique lift is parallel around the unmarked seam circle. Thus closure imposes no further equation.

For each fixed rank pattern these successive choices form a finite tower of locally affine bundles over the lower solution variety. Indeed the flags have constant ranks, so they can be split locally, and the preceding matrix-block descriptions give each event fiber. Every integral lower solution admits integral choices, dense in those fibers by Lemma 3.1. Conversely, every old complex solution is recovered from its induced lower data and its original lifts, up to the allowed frame changes. Its disk local system is isomorphic to U\oplus Q because all puncture generators act by the prescribed scalars.

Perform the construction on every rank-n disk. No new rank-n facet has been introduced. After normalizing zero blocks and identity seams, the induction hypothesis gives density on each resulting lower diagram. Lemma 4.2 gives density on the disk diagrams and then on the original nonclosed diagram. This proves Proposition 5.1.

Closed maximal-rank facets remain. The next two sections treat their scalar-monodromy equations; after the closed-density lemma we assemble the full induction proving Theorem 4.1.

6 Closed surfaces with scalar monodromy

Closing a surface with prescribed scalar monodromies leads to a product of commutators. We first settle the case of genus one, and then prove the irreducibility needed for the higher-genus argument. For n\geq1, g\geq0, and c\in\C^\times, write

As throughout the paper, this is a reduced variety. Taking determinants shows that it is empty unless c^n=1. For genus zero it is a point if c=1, and is empty otherwise. For rank one and c=1, it is the torus (\C^\times)^{2g}.

6.1 Genus one

The density of simultaneously diagonalizable commuting pairs goes back to Motzkin and Taussky [17]; see also Gerstenhaber [6] and Guralnick [9]. We use the regular-centralizer proof described by Guralnick and Sethuraman [10, Section 3, Proposition 6], then record its integral consequence over the full ring in use here.

In our application R=\cO_F, and the unit 1+\sqrt2 has infinite order. Thus these lemmas, as well as the rank-one torus case, apply over the fixed ring already chosen.

6.2 Irreducibility in higher genus

The proof below follows the character-count and Lang–Weil method of Liebeck–Shalev [12, Section 7, Lemma 7.1 and the proof of Theorem 1.8]. We give the elementary estimates for \SL_n, and use a lower bound for the dimension of every component to obtain irreducibility of the entire fiber.

Proposition 6.3 is a geometric statement. The roots used in its last paragraph are taken in \C; the arithmetic passage to determinant one over a single number field will be treated separately.

7 From one spectrum to a closed surface

The remaining step in the diagram induction concerns a closed facet. Cutting a nonseparating circle reduces it to a facet with boundary, but prescribes the spectrum of the gluing monodromy. We first obtain density with this spectrum fixed. A Dehn twist then supplies the missing directions.

Multiplication by a centralizer element is the algebraic form of the generalized twist motions described by Goldman [8, Section 4.6]. We use the matrix relation directly, then verify the needed torus closure and tangent directions at an explicit representation.

7.1 Completion of the rank induction

8 Boundary classes and the character quotient

We now apply Theorem 4.1 to the surface underlying the curve. Flags encode the boundary classes by closed rank bounds. We then make the handle determinants one and select the exact classes on the character quotient. This order is important: semisimplifying a representation can decrease the Jordan blocks of a boundary matrix.

8.1 Flags for arbitrary Jordan classes

Let \overline X be the smooth compactification of the curve, of genus g, and replace each missing point by a boundary circle. For a prescribed boundary class, write \nu_\lambda for its Jordan partition at the eigenvalue \lambda, and write

for the column lengths of this partition. Its generalized \lambda-eigenspace has dimension n_\lambda=|\nu_\lambda|. Order the eigenvalues once and for all. On the boundary put a flag whose graded pieces, in that order, have dimensions c_{\lambda,1},c_{\lambda,2},\ldots and scalar monodromy \lambda. Realize each piece by a punctured disk cap, choosing the scalar according to the meridian orientation.

Every matrix in the prescribed class admits this flag: on its generalized \lambda-eigenspace use the successive kernels of powers of A-\lambda I. Conversely, any matrix with the prescribed flag has the required characteristic polynomial and satisfies

where missing columns are zero. To see this, restrict the flag to the generalized \lambda-eigenspace. The nilpotent part lowers its flag by one step, so the first k steps lie in its kth kernel. On the other generalized eigenspaces, A-\lambda I is invertible. These observations give (10). Restriction to the generalized eigenspaces is valid because their polynomial projectors preserve every invariant flag. This argument takes place over \C and requires no unit condition on differences between the prescribed boundary eigenvalues.

Use the compact surface as a rank-r parent facet and attach the caps just described. In the projective case no caps are needed. Let Z be the reduced Zariski closure, in the framed \GL_r representation variety, of the main-surface representations obtained from this diagram. Theorem 4.1 gives a dense set of actual \GL_r(R) matrix points in Z. The preceding flag argument shows that Z contains every representation with the prescribed exact classes and lies within the closed conditions consisting of their characteristic polynomials and (10). In particular all its peripheral determinants are one.

The set Z is invariant under simultaneous conjugation. It is also invariant under twisting by characters of \pi_1(\overline X): such a twist independently multiplies each of the 2g handle matrices by a scalar and leaves all peripheral and cap data unchanged. These two invariances also hold for its closure.

8.2 A single field makes all determinants one

Put S=Z\cap\Hom(\pi_1(X),\SL_r), taking reduced varieties, and let T=(\mathbb G_m)^{2g}. Write \delta:Z\to T for the handle determinants. Scalar twisting defines

It is the base change of the power map [r]:T\to T. More explicitly,

with inverse (z,t)\mapsto(t^{-1}\cdot z,t). Twist invariance of Z ensures that this inverse takes values in S\times T. Consequently f is finite, étale and surjective, and in particular open.

Let L/F be the fixed finite extension of Lemma 3.4, containing all rth roots of all units of R. If z is one of the integral matrix points dense in Z, every coordinate of \delta(z) is a unit of R. Every preimage of z under f therefore has t\in (\cO_L^\times)^{2g} and s=t^{-1}\cdot z an actual \SL_r(\cO_L) representation. Since f is open, the full preimage of this dense set is dense in S\times T. Projection to S shows that actual \SL_r(\cO_L) representations are dense in S. The field L is chosen before the points and is the same for all components. For g=0, the torus T is trivial and this argument is the identity map.

8.3 Selecting the exact classes after semisimplification

Let \mathcal R=\Hom(\pi_1(X),\SL_r) over \C and let q:\mathcal R\to M_r(X)_\C be its affine character quotient. Let C\subset\mathcal R be the invariant closed locus defined by the prescribed characteristic polynomials and all the bounds (10). Only 1\le k\le r is needed. Let C_{\mathrm{bad}}\subset C be the union of the loci where at least one rank is strictly smaller than the rank in the prescribed class. This is a finite union of invariant closed subsets, obtained by lowering one of the positive rank bounds by one.

Lemma 2.1 identifies the exact character locus as

In particular it is open in q(C), including when semisimplification can shrink some boundary blocks.

We have S\subset C, and S contains every representation with the exact prescribed boundary classes. Hence q(S) is a closed subset of q(C) containing Y. The dense integral representations in S give a dense subset A\subset q(S) of ambient integral character points: evaluating integral conjugation invariants on their entries gives points of the specified model over \cO_L. No integral matrix realization of their semisimplifications is needed. By (12), Y is open in q(S), so A\cap Y is dense in Y.

This proves Theorem 1.1 for exact classes. It proves density in the entire reduced locus, hence in every irreducible component, without an irreducibility assumption on Z, S or Y. It also retains the genus-zero and nonregular boundary cases. The selection concerns only the generic-fiber character; it imposes no avoidance of q(C_{\mathrm{bad}}) at finite primes.

Finally, fixed characteristic polynomials allow only finitely many Jordan partitions. Their exact strata cover the fixed-polynomial locus. The same F and L work for all of them, because these fields depend on the rank and the boundary eigenvalues, not on the partitions. Taking the finite union of the dense sets proves the fixed-polynomial assertion as well. Empty strata are harmless.

For completeness, if X is nonprojective and no boundary conditions are imposed, its fundamental group is free of rank 2g+s-1. The group \SL_r(\Z) is Zariski dense in \SL_r(\C): its closure contains every elementary one-parameter unipotent subgroup, and these generate \SL_r(\C). Thus the integral matrix tuples are dense in the free-group representation variety, and their characters are dense in its quotient, already over \Z. The cases of a free group of rank zero and of r=1 give a point. Together with the projective case proved above, this establishes the unrestricted assertion of Theorem 1.1.