For every odd prime p and every imaginary quadratic field K whose p-class rank is 2, if the Galois group G_K,varnothing(p) of the maximal unramified pro-p extension of K has Zassenhaus type (3,3), then the p-class field tower of K is finite.

Boston’s unramified Fontaine–Mazur conjecture; McLeman’s (3,3)-conjecture

The problem concerns Boston’s unramified Fontaine–Mazur conjecture and McLeman’s conjecture for the $(3,3)$ case, linking Galois representations with finiteness of unramified class-field towers.

Known results

A 2024 paper proves Boston’s conjecture when the representation image contains an open solvable subgroup and derives related deformation-ring consequences, conditional on the conjecture.

September 2026 partial advance

Yufan Luo’s preprint reports finite image for the specified two-dimensional odd Boston case and finiteness of the relevant class-field tower in the $(3,3)$ case. A separate 2026 study shows that the relevant Galois groups are Schur $\sigma$ -groups and presents supporting evidence for McLeman’s “if” direction. These are specified-case advances, not complete proofs.

Current status (as of September 2026): Specified cases and directions are reported as proved, but the two conjectures remain open in general and the recent claims are unverified.

Boston’s unramified Fontaine–Mazur conjecture; McLeman’s (3,3)-conjecture

Primary source

Additional references

Progress summary

Solutions 0