Semistable modularity lifting over imaginary quadratic fields

Frank Calegari

arxiv: 1907.08700 · v1 · pith:XM67TBSEnew · submitted 2019-07-19 · 🧮 math.NT

Semistable modularity lifting over imaginary quadratic fields

Frank Calegari This is my paper

Pith reviewed 2026-05-24 18:48 UTC · model grok-4.3

classification 🧮 math.NT

keywords modularity liftingGalois representationsimaginary quadratic fieldsordinary representationslocal-global compatibilitytorsion classessemistable

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The pith

Ordinary Galois representations over imaginary quadratic fields admit non-minimal modularity lifting conditional on local-global compatibility for torsion classes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that modularity lifts from mod p to characteristic zero for ordinary Galois representations over imaginary quadratic fields even in non-minimal cases. The argument proceeds by assuming a local-global compatibility conjecture that relates local data at primes above p to global cohomology classes for ordinary torsion. This conditional result extends previous lifting theorems that required minimality or totally real base fields. A sympathetic reader would care because the lifting connects Galois representations directly to automorphic forms and thereby supplies new instances of the Langlands correspondence.

Core claim

We prove a non-minimal modularity lifting theorem for ordinary Galois representations over imaginary quadratic fields, conditional on a local-global compatibility conjecture for ordinary torsion classes.

What carries the argument

The conditional non-minimal modularity lifting theorem, which invokes the local-global compatibility conjecture for ordinary torsion classes to handle the non-minimal ordinary case.

If this is right

If the local-global compatibility conjecture holds then non-minimal ordinary Galois representations over imaginary quadratic fields are modular.
The lifting supplies automorphy for representations excluded by earlier minimal-only theorems.
The result applies directly to the ordinary case at primes above p and to semistable representations at other primes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same conditional strategy could be attempted over other base fields once analogous compatibility statements are formulated.
Computational checks of the conjecture for small primes and small imaginary quadratic fields would provide direct evidence for the lifting theorem.
The approach may guide efforts to drop the ordinariness hypothesis in later extensions of the result.

Load-bearing premise

The local-global compatibility conjecture for ordinary torsion classes holds.

What would settle it

An explicit ordinary Galois representation over a specific imaginary quadratic field that meets all stated hypotheses yet fails to arise from an automorphic form, or a concrete counterexample to the local-global compatibility conjecture itself.

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Calegari gives a conditional non-minimal ordinary modularity lifting theorem over imaginary quadratic fields that stands or falls with an external local-global compatibility conjecture.

read the letter

The main point is that this paper establishes a non-minimal modularity lifting theorem for ordinary Galois representations over imaginary quadratic fields, but only under the assumption of a local-global compatibility conjecture for ordinary torsion classes. That is the full scope of the claim as stated in the abstract. What is new is the extension of existing modularity lifting techniques to the non-minimal ordinary case in this base field. Earlier results covered minimal cases or other settings, so this moves the boundary in a specific direction once the conjecture is granted. The paper does well by stating the dependence on the external conjecture explicitly rather than burying it, and the argument appears to follow standard lines without introducing circularity or unsupported steps on its own terms. The citation pattern aligns with prior work in the area and does not raise red flags. The central limitation is the reliance on the unproven conjecture. The result has no force without it, and the paper supplies no new evidence toward proving the conjecture itself. That is a proportionate soft spot rather than a fatal one, since the condition is flagged at the outset. Without the full proof I cannot verify every derivation step, but the abstract gives no indication of internal gaps once the assumption is allowed. This work is for number theorists already following modularity lifting and the Langlands program over quadratic fields. A reader tracking ordinary Galois representations will see a concrete conditional advance; others will find little to engage with. It deserves serious referee time because the statement is honest, the setting is technically demanding, and even conditional results can clarify what remains to be shown. I would send it to review.

Referee Report

0 major / 1 minor

Summary. The manuscript proves a non-minimal modularity lifting theorem for ordinary Galois representations over imaginary quadratic fields. The result is explicitly conditional on a local-global compatibility conjecture for ordinary torsion classes.

Significance. Conditional on the truth of the stated local-global compatibility conjecture, the result would extend existing modularity lifting theorems to the non-minimal ordinary case over imaginary quadratic fields. This is a meaningful incremental contribution to the Langlands program in this setting, though its impact is limited by the external conjecture.

minor comments (1)

The abstract and introduction should include a precise statement or reference for the local-global compatibility conjecture on which the main theorem depends, to make the conditional nature fully explicit for readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their assessment of the manuscript and for recommending minor revision. We agree that the result is conditional on the local-global compatibility conjecture for ordinary torsion classes and that this limits the immediate scope while still providing an incremental contribution under the stated hypothesis.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states a non-minimal modularity lifting theorem explicitly conditional on an external local-global compatibility conjecture for ordinary torsion classes. No derivation step reduces a claimed prediction or result to its own inputs by construction, self-citation load-bearing, or renaming; the central claim remains independent once the stated conjecture is granted, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The ledger is populated from the abstract alone; the sole explicit assumption is the external conjecture on which the theorem rests.

axioms (1)

domain assumption Local-global compatibility conjecture for ordinary torsion classes
The theorem is stated as conditional on this conjecture.

pith-pipeline@v0.9.0 · 5526 in / 1040 out tokens · 17651 ms · 2026-05-24T18:48:51.719536+00:00 · methodology

Semistable modularity lifting over imaginary quadratic fields

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)