On Boston's Unramified Conjecture for GL2 and McLeman's (3,3)-Conjecture
Abstract
Let p be an odd prime number. Based on recent work of Zhang, we prove that for a finite set S of primes of Q containing ∞ but not p, any continuous odd representation GQ,S → GL2(A) over a complete Noetherian local ring A with finite residue field of characteristic p has finite image, where GQ,S denotes the Galois group of the maximal extension of Q unramified outside S. This proves the two-dimensional odd case of Boston's strengthening of the unramified Fontaine--Mazur conjecture over Q. Furthermore, we show that for an imaginary quadratic field K, any continuous conjugate self-dual two-dimensional p-adic representation of its pro-p Galois group GK,S(p) has finite image, provided the primes in S satisfy a modest condition. As an application, we resolve the sufficiency direction of McLeman's (3,3)-conjecture on p-class field towers for p>3. Namely, we prove that if K is an imaginary quadratic field with p-class rank two and the Galois group GK,∅(p) of the maximal unramified p-extension of K has Zassenhaus type (3,3), then the p-class field tower of K is finite. For p=3, the results of Ahlqvist and Pink give the same finiteness conclusion in ten of the thirteen possible cases for the fourth Zassenhaus quotient.
Related
- The unramified Grothendieck--Serre conjecture
- Arithmetic jet spaces and the unramified Zilber--Pink conjecture for subvarieties of abelian varieties
- Maximal unramified 3-extensions of imaginary quadratic fields and \textrm SL\sb 2(\mathbb Z\sb 3)
- Unramified Geometric Iwasawa theory of function fields
- On finiteness properties of the unramified Iwasawa module of a ℤp-extension with restricted ramification
BibTeX
@misc{luo2026bostons,
title = {{On Boston's Unramified Conjecture for $\GL_2$ and McLeman's $(3,3)$-Conjecture}},
author = {Yufan Luo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37252},
eprint = {2609.37252},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37252},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - On Boston's Unramified Conjecture for GL_2 and McLeman's (3,3)-Conjecture
AU - Yufan Luo
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37252
UR - https://arxiv.org/abs/2609.37252
AB - Let p be an odd prime number. Based on recent work of Zhang, we prove that for a finite set S of primes of Q containing ∞ but not p, any continuous odd representation G_Q,S → GL_2(A) over a complete Noetherian local ring A with finite residue field of characteristic p has finite image, where G_Q,S denotes the Galois group of the maximal extension of Q unramified outside S. This proves the two-dimensional odd case of Boston's strengthening of the unramified Fontaine--Mazur conjecture over Q. Furthermore, we show that for an imaginary quadratic field K, any continuous conjugate self-dual two-dimensional p-adic representation of its pro-p Galois group G_K,S(p) has finite image, provided the primes in S satisfy a modest condition. As an application, we resolve the sufficiency direction of McLeman's (3,3)-conjecture on p-class field towers for p>3. Namely, we prove that if K is an imaginary quadratic field with p-class rank two and the Galois group G_K,∅(p) of the maximal unramified p-extension of K has Zassenhaus type (3,3), then the p-class field tower of K is finite. For p=3, the results of Ahlqvist and Pink give the same finiteness conclusion in ten of the thirteen possible cases for the fourth Zassenhaus quotient.
ER -