Semistable abelian varieties over $\mathbb{Q}$ with bad reduction at 19 only: an overview of the Fontaine--Schoof strategy

Francesco Campagna; Pip Goodman

arxiv: 2510.12625 · v2 · submitted 2025-10-14 · 🧮 math.NT

Semistable abelian varieties over mathbb{Q} with bad reduction at 19 only: an overview of the Fontaine--Schoof strategy

Francesco Campagna , Pip Goodman This is my paper

Pith reviewed 2026-05-18 07:47 UTC · model grok-4.3

classification 🧮 math.NT MSC 11G1011G18

keywords semistable abelian varietiesbad reductionFontaine-Schoof strategyGalois representationsisogeny classificationconductornumber theory

0 comments

The pith

The Fontaine-Schoof strategy classifies all semistable abelian varieties over Q with bad reduction only at 19 up to isogeny.

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

The paper supplies a self-contained overview of the strategy first outlined by Fontaine and later sharpened by Schoof for identifying abelian varieties over the rationals whose bad reduction occurs at only one given prime. It supplies proofs of several supporting facts about the associated Galois representations so that readers without prior exposure can follow the arguments. After working through the cases of bad reduction at 3 and at 5, the authors carry out the same classification when the single prime of bad reduction is 19; this last step constitutes the new contribution. Readers would care because the method yields an explicit list of all possible isogeny classes once the prime is fixed, and the 19-case shows that the approach remains workable for moderately large primes.

Core claim

Following the Fontaine-Schoof strategy, every semistable abelian variety over Q whose conductor is a power of 19 falls into one of a finite list of isogeny classes that can be determined explicitly from the possible Galois representations on its Tate module.

What carries the argument

The Fontaine-Schoof strategy, which uses the action of Gal(Qbar/Q) on the l-adic Tate module together with the semistable reduction condition and the conductor to bound the dimension and restrict the possible isogeny classes.

If this is right

Any semistable abelian variety over Q with bad reduction only at 19 must be isogenous to a product of elliptic curves or higher-dimensional varieties whose Tate modules realize one of the permitted Galois representations.
The same sequence of steps that works for 19 can be repeated for other primes once the necessary local and global representation constraints are checked.
The explicit classification for 19 supplies a concrete list of isogeny classes that can be compared against tables of abelian varieties of small conductor.

Where Pith is reading between the lines

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

The method may scale to larger primes provided the computational cost of enumerating possible representations stays manageable.
If the classification is complete, then every such variety for p=19 arises from base change or isogeny factors already visible in lower-dimensional cases.

Load-bearing premise

The background results on Galois representations and the Fontaine-Schoof machinery remain valid when specialized to the prime 19 and the semistable case.

What would settle it

An explicit semistable abelian variety over Q with conductor a power of 19 whose attached Galois representation lies outside the finite list of representations allowed by the classification.

read the original abstract

In this paper we provide an overview of a strategy pioneered by Fontaine and heavily refined by Schoof to classify abelian varieties with prescribed bad reduction. Throughout the overview, we prove various non-trivial background results turning it into an introduction for readers unacquainted with this topic. With the overview completed, we provide explicit examples of the strategy in action. At first we give introductory examples, classifying semistable abelian varieties over $\mathbb{Q}$ with bad reduction at exactly one of 3 or 5 up to isogeny over $\mathbb{Q}$. We then move onto a harder example, proving the analogous result for 19, which is new.

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

The paper gives a clear overview of the Fontaine-Schoof method and delivers a new explicit classification of semistable abelian varieties over Q with bad reduction only at 19.

read the letter

The core takeaway is that this work spells out the Fontaine-Schoof strategy for classifying semistable abelian varieties with conductor a single prime, proves some supporting results along the way, and then carries the method through to a complete list up to isogeny for the prime 19. That last part is new and not just a rehash of the 3 and 5 cases already in the literature. The overview itself is written to be usable by readers who have not seen the machinery before, which is a practical contribution on its own. The introductory examples for 3 and 5 are straightforward and help set up the harder 19 case without obvious gaps in the logic presented in the abstract. The background results on Galois representations and inertia action are stated to be proved or recalled in the paper, so the classification rests on that foundation rather than on unexamined assumptions. The main soft spot is the usual one with overview-plus-example papers: for a larger prime like 19 the number of possible mod-ℓ representations grows, and it is not obvious from the abstract alone whether every semistable inertia type at 19 is explicitly ruled in or out. If the full case analysis is written out with the same care as the smaller examples, this is minor; if some representations are dispatched by general lemmas without checking the 19-specific weights or levels, that would need tightening. Overall the work is aimed at people who want concrete lists or who are building larger programs around effective Shafarevich theorems or modularity lifting. It is the sort of explicit arithmetic result that deserves a serious referee rather than a desk rejection, even if the final list turns out to be short.

Referee Report

1 major / 1 minor

Summary. The manuscript provides an overview of the Fontaine-Schoof strategy for classifying semistable abelian varieties over Q with bad reduction at a single prime. It proves various background results on Galois representations, presents introductory examples for bad reduction exactly at 3 or 5, and claims to prove the analogous classification for bad reduction only at 19 as a new result.

Significance. If the central classification for p=19 holds with complete case analysis, the work supplies a self-contained introduction to the method together with an explicit new example, which would be useful for further applications of the strategy to small conductors.

major comments (1)

[section on the 19 example] The section applying the strategy to p=19: the claim that every irreducible mod-ℓ Galois representation arising from a semistable abelian variety of conductor exactly 19 is accounted for (or ruled out) is load-bearing for the new classification result, yet the overview does not explicitly enumerate the possible inertia actions at 19 or verify that no Serre weights or level structures fall outside the handled cases.

minor comments (1)

Notation for the conductor and the precise statement of the main theorem for p=19 could be stated more explicitly in the abstract and introduction to make the scope of the new result immediately clear.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the importance of making the completeness of the p=19 classification fully transparent. We address the major comment below and will incorporate the suggested clarifications in a revised version.

read point-by-point responses

Referee: [section on the 19 example] The section applying the strategy to p=19: the claim that every irreducible mod-ℓ Galois representation arising from a semistable abelian variety of conductor exactly 19 is accounted for (or ruled out) is load-bearing for the new classification result, yet the overview does not explicitly enumerate the possible inertia actions at 19 or verify that no Serre weights or level structures fall outside the handled cases.

Authors: We agree that an explicit enumeration of the possible inertia actions at 19, together with a verification that all relevant Serre weights and level structures are covered, would strengthen the presentation and make the load-bearing claim easier to verify. In the revised manuscript we will add a short dedicated paragraph (or subsection) at the beginning of the p=19 section. This paragraph will list the admissible inertia types at 19 for semistable abelian varieties (using the standard classification of tame inertia actions compatible with conductor 19), and will confirm that the cases subsequently treated in the Galois-representation analysis exhaust these possibilities. No new theorems are required; the addition is expository and draws only on material already present in the background sections. revision: yes

Circularity Check

0 steps flagged

No circularity: overview applies independently proved background results to obtain new classification at 19.

full rationale

The paper frames itself as an overview of the Fontaine-Schoof strategy, explicitly states that it proves various non-trivial background results on Galois representations and the machinery, and then applies the strategy to obtain classifications for bad reduction at 3, 5, and the new case at 19. No step reduces a claimed prediction or classification to a fitted parameter, self-defined quantity, or load-bearing self-citation by the present authors; the cited prior work is by Fontaine and Schoof (distinct authors), and the 19 result is presented as independent and new. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities. The strategy presumably relies on standard facts from p-adic Hodge theory and Galois representations, but none are itemized here.

pith-pipeline@v0.9.0 · 5643 in / 1141 out tokens · 38217 ms · 2026-05-18T07:47:16.531128+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1: Any semistable abelian variety over Q that has good reduction outside of 19 is isogenous to a power of J_0(19).
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Classification of simple objects in C_19 via ramification constraints and class-field computations (Section 4).

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

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S. Bosch, W. L¨ utkebohmert, and M. Raynaud.N´ eron models. Vol. 21. Ergeb- nisse der Mathematik und ihrer Grenzgebiete (3). Springer-Verlag, Berlin, 1990 (cit. on pp. 5, 14)

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Abhyankar’s lemma and the class group

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Les sch´ emas de modules de courbes ellip- tiques

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G. Poitou. “Sur les petits discriminants”. fr. In:S´ eminaire Delange-Pisot- Poitou. Th´ eorie des nombres18.1 (1976), pp. 1–17 (cit. on p. 11)

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Sch´ emas en groupes de type (p, . . . , p)

M. Raynaud. “Sch´ emas en groupes de type (p, . . . , p)”. In:Bull. Soc. Math. France102 (1974), pp. 241–280 (cit. on pp. 7, 12, 13, 19)

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Abelian varieties overQwith bad reduction in one prime only

R. Schoof. “Abelian varieties overQwith bad reduction in one prime only”. In:Compos. Math.141.4 (2005), pp. 847–868 (cit. on pp. 1, 17, 21, 26)

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Abelian varieties over cyclotomic fields with good reduction ev- erywhere

R. Schoof. “Abelian varieties over cyclotomic fields with good reduction ev- erywhere”. In:Math. Ann.325.3 (2003), pp. 413–448 (cit. on pp. 12, 18, 22, 25). 28 REFERENCES

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On the modular curveX 0(23)

R. Schoof. “On the modular curveX 0(23)”. In:Geometry and arithmetic. EMS Ser. Congr. Rep. Eur. Math. Soc., Z¨ urich, 2012, pp. 317–345 (cit. on p. 1)

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Semistable abelian varieties with good reduction outside 15

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Serre.Local fields

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Finite flat group schemes

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Grothendieck topologies, fibered categories and descent theory

A. Vistoli. “Grothendieck topologies, fibered categories and descent theory”. In:Fundamental algebraic geometry. Vol. 123. Math. Surveys Monogr. Amer. Math. Soc., Providence, RI, 2005, pp. 1–104 (cit. on p. 14)

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work page 1979