Potential integral density on curve character varieties

The explanation

Take two grids of whole numbers, A = [[1, 1], [1, 2]] and B = [[1, −1], [−1, 2]]. Each has determinant 1. Each has trace 3, where the trace is the sum of the two diagonal entries. Their product AB also has trace 3. Now multiply A, then B, then the inverse of A, then the inverse of B. The result is [[−1, 0], [−6, −1]]. Its trace is −2. Its only eigenvalue is −1, and no change of coordinates makes it diagonal.

These matrices describe a torus, the surface of a ring doughnut, with one point removed. Walk once around the ring and apply A. Walk once through the hole and apply B. Walk around the missing point and you get that last product. Such a rule, a matrix for every loop that respects how loops combine, is called a representation. Changing coordinates gives the same representation in disguise. Here three traces, x = 3, y = 3 and z = 3, pin it down. They obey one equation with t, the loop's trace: x² + y² + z² − xyz − 2 = t. Indeed 27 − 27 − 2 = −2.

Fix t = −2 and the equation becomes x² + y² + z² = xyz. Whole-number solutions keep coming. Replace (x, y, z) by (y, z, yz − x). From (3, 3, 3) you reach (3, 3, 6), then (3, 6, 15), then (6, 15, 87). Each still fits: 36 + 225 + 7569 = 7830 = 6 × 15 × 87. The real question is where such whole-number points go. Do they spread over the whole solution space, or stay trapped on a thinner piece? Being trapped means extra polynomial equations hold at all of them. Not trapped is called Zariski dense.

The paper asks this for every surface of this type, in every matrix size. Mathematicians call such a surface a smooth complex curve. It may have any number of handles and any number of puncturesSource 1. The matrices are r by r with determinant 1. The space of all representations, up to coordinates, is the character variety. At each puncture one may also fix the type of matrix allowed, its conjugacy class. The paper requires these classes to be quasi-unipotent, meaning every eigenvalue is a root of unity. The −1 in our example qualifies.

Whole numbers alone are not enough. Take the family x² + y² + z² − xyz = k. Ghosh and Sarnak showed integral points are dense for almost all suitable k. They also found infinitely many suitable k with no integer points at all. So the question allows one enlargement of the whole numbers. Pick a number field, a finite extension of the fractions, and use all its algebraic integers. Daniel Litt asked whether integral points then become dense, and whether one number field suffices. Simpson, Esnault and Groechenig, and de Jong and Esnault had studied when such matrices can be made integral. De Jong and Esnault worked one prime at a time, which is a different question. Coccia and Litt proved density for 2-by-2 matrices.

This paper claims to prove the answer is yes for every such curve and every matrix sizeSource 2. One number field works for the whole space at once, including every piece of itSource 3. It also handles boundary matrices that cannot be made diagonal, like our −2 loop. For a curve with punctures and no boundary conditions this was already easy, using whole numbers aloneSource 4. The hard cases are fixed boundary types and closed surfacesSource 5. There one equation ties all the matrices together. For those cases the paper even names the field: add √2, certain roots of unity, and r-th roots of unitsSource 6.

The obstacle is that large matrices with fixed boundary types are tangled. Writing complex solutions by formulas does not produce integral onesSource 7. The paper splits each boundary matrix into simpler blocks along nested subspacesSource 8. It then lowers the matrix size one step at a time. The lost pieces come back as free whole-number choices. For closed surfaces, repeated twists of the surface spread the integral points, much like the move in our exampleSource 9. The exact boundary types are imposed only at the very end. Doing it earlier fails, because simplifying a representation can shrink its blocksSource 10.

The notion of integral here is specific. A point counts if its coordinates in the whole character variety are algebraic integers. The boundary type is required only over the complex numbers. Reduced modulo a prime, the boundary matrix may fall into a simpler type. The result covers curves only; higher-dimensional spaces remain openSource 11. For 2-by-2 matrices the conclusion already follows from Coccia and Litt, who allow more boundary traces thereSource 12,Source 13.

How sure is this? A computer proof checker called Lean has checked one supporting step. It shows that pieces built in the paper's surface construction fit together along every seam. The density theorem itself is outside that check, as are the boundary and every-piece conclusions. No specialists have published a review yet.

What it doesn't show. This does not cover spaces of higher dimension than curves, and it does not count the integral points.