bibkey: fretwellroberts2026eisenstein
authors: Dan Fretwell and Jenny Roberts
year: 2026
title: Hilbert modular Eisenstein congruences of local origin
doi: 10.1016/j.jnt.2025.09.009
claim: The stated congruence-existence results concern parallel holomorphic weight at least three; they do not by themselves produce the weight-zero golden Maass tower.
strata_touched: []
license: citation-only
triage: anchor

Eisenstein congruences and the weight boundary of the WSS application

Primary record

Dan Fretwell and Jenny Roberts, Hilbert modular Eisenstein congruences of local origin, Journal of Number Theory 280 (March 2026), 861-896, DOI 10.1016/j.jnt.2025.09.009. The corresponding arXiv record is 2411.06987v2, revised March 3, 2026.

https://doi.org/10.1016/j.jnt.2025.09.009 https://arxiv.org/abs/2411.06987

The primary publisher and arXiv abstracts state existence of Eisenstein congruences over arbitrary totally real fields in parallel weight k>=3, between eigenforms at level mp and Eisenstein series at level m, with p a prime ideal not dividing m. The moduli come from Hecke L-values and Euler factors; the work also treats a newform question and allows nontrivial character. This note uses those abstract-level scope statements and the publication metadata. It does not assert that all technical hypotheses or proofs of the full paper have been checked here, and cites no uninspected theorem number as an input.

Precise transfer boundary

Problems/wall-sun-sun-golden-unit-lift.md, CTG.4-CTG.7, constructs an Eisenstein-type coefficient system b(n)=sum_(d|n)chi_5(d) and computes its exact congruence module against the COMPLETE finite-order golden dihedral tower. The family consists of weight-zero Maass forms with parameter zero and even Artin representations. Neither parallel holomorphic weight at least three nor the other hypotheses of a holomorphic congruence-existence theorem follow from that construction.

Accordingly, CTG proves its module and explicit trace-combination statements by the polynomial relation Q_k(2)=p^k and the actual integral coefficient lattice. It does not apply this paper to establish existence of the golden character. A nonzero abstract congruence ring alone is not a new WSS prime.

This reference supplies a relevant frontier comparison for how independent L-value or local-factor information can support a congruence existence theorem in its proper category. Any transfer to the fixed-golden Maass problem requires a separate theorem retaining its weight, parity, conductor and field conditions. No such transfer theorem is claimed in this note.