For every global function field K of characteristic p, every integer n≥ 1, and every effective divisor D on K, there are only finitely many isomorphism classes of continuous, semisimple, geometric representations rho:G_K→GL_n(overlinemathbf F_p) whose conductor satisfies mathfrak f(rho)≤ D.

Moon–Taguchi bounded-conductor finiteness conjecture

Khare and Moon proposed the number-field conjecture; Moon and Taguchi formulated its function-field analogue. It asks for finiteness of geometric representations with conductor bounded by a fixed divisor.

Known results

Moon and Taguchi: solvable-image representations.
Böckle and Khare: large image, $p\nmid n$ , and everywhere tame ramification.
Further cases with $p\nmid n$ were implicit in earlier work.
A 2026 paper claims the conjecture when $p$ is odd or $n=2$ .

September 2026 equal-characteristic refutation

On September 10, 2026, Luo and Xu reported that bounded-conductor finiteness fails in equal characteristic without lifting, while identifying a positive lifting regime. Combined with cross-characteristic results, this gives a claimed complete resolution in dimension $2$ across characteristics; higher dimensions remain unresolved in the unrestricted characteristic- $2$ case.

Current status (as of September 2026): The two-dimensional case is claimed settled, but the equal-characteristic refutation and the stated higher-dimensional boundaries remain unverified.

Moon–Taguchi bounded-conductor finiteness conjecture

Primary source

Additional references

Progress summary

Solutions 0