Some modular abelian surfaces

Alexandru Ghitza; Frank Calegari; Shiva Chidambaram

arxiv: 1906.10939 · v1 · pith:AHHVRB3Inew · submitted 2019-06-26 · 🧮 math.NT

Some modular abelian surfaces

Frank Calegari , Shiva Chidambaram , Alexandru Ghitza This is my paper

Pith reviewed 2026-05-25 15:27 UTC · model grok-4.3

classification 🧮 math.NT

keywords modular abelian surfacesgood reductionabelian varieties over Qextra endomorphismsGalois representations

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The pith

Explicit examples are given of modular abelian surfaces over Q with good reduction outside primes 2, 3, 5 and 7 and no extra endomorphisms.

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

The authors apply a main theorem from a prior work to construct concrete abelian surfaces over the rational numbers. These surfaces are modular, lack extra endomorphisms, and have good reduction at every prime except 2, 3, 5, and 7. The constructions supply explicit instances that meet all the hypotheses needed for the theorem to apply. Such surfaces furnish concrete objects for examining the arithmetic properties of abelian varieties of dimension two.

Core claim

Using the main theorem of Boxer-Calegari-Gee-Pilloni, explicit examples of modular abelian surfaces A over Q without extra endomorphisms are given such that A has good reduction outside the primes 2, 3, 5, and 7.

What carries the argument

The main theorem of Boxer-Calegari-Gee-Pilloni applied to specific abelian surfaces over Q.

If this is right

Modular abelian surfaces over Q with the stated reduction and endomorphism properties exist and can be written down explicitly.
The cited theorem produces the modularity of each constructed surface.
These surfaces have endomorphism ring exactly Z and conductor supported only at 2, 3, 5, and 7.

Where Pith is reading between the lines

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

The same method could be tried on abelian surfaces whose bad reduction is supported at an even smaller set of primes.
The explicit models might allow direct computation of their periods or L-functions to check further arithmetic predictions.
Similar constructions might produce examples in which the associated Galois representations have prescribed images.

Load-bearing premise

The specific abelian surfaces constructed in the paper satisfy all hypotheses of the main theorem from arXiv:1812.09269.

What would settle it

An explicit check showing that one of the constructed surfaces has an extra endomorphism or acquires bad reduction at a prime besides 2, 3, 5, or 7 would disprove the claim.

read the original abstract

We use the main theorem of Boxer-Calegari-Gee-Pilloni (arXiv:1812.09269) to give explicit examples of modular abelian surfaces $A$ over $\mathbf{Q}$ without extra endomorhpisms such that $A$ has good reduction outside the primes 2, 3, 5, and 7.

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

This paper applies the BCGP theorem to list a few explicit abelian surfaces over Q with the claimed properties.

read the letter

The core of the paper is a direct application of the main theorem from Boxer-Calegari-Gee-Pilloni to produce concrete abelian surfaces A over Q. These have good reduction outside {2,3,5,7}, no extra endomorphisms, and are modular. The authors state that they verify the necessary local and global conditions on the Galois representations and the polarization data for their chosen examples. That is the new content: a short list of explicit cases that were not previously recorded in the literature they cite. Explicit examples in this corner of arithmetic geometry are useful because they let people run computations or test conjectures against actual objects rather than abstract existence statements. The paper does that cleanly. The main limitation is that the result stands or falls on the accuracy of those hypothesis checks. The argument does not re-derive the BCGP theorem, so a referee would want to see that the reduction types, the traces of Frobenius, and the absence of extra endomorphisms are all confirmed by direct calculation or reliable software output. No circularity or hidden fitting appears in the setup. This is a paper for people already working on modularity questions for abelian surfaces or on explicit constructions in the Langlands program. It does not change the landscape, but it adds verifiable data points. A serious editor should send it to peer review; the examples are new enough and the dependence on prior work is stated clearly enough that referees can check the application in reasonable time.

Referee Report

1 major / 1 minor

Summary. The paper applies the main theorem of Boxer-Calegari-Gee-Pilloni (arXiv:1812.09269) to produce explicit examples of modular abelian surfaces A over Q without extra endomorphisms such that A has good reduction outside the primes 2, 3, 5, and 7.

Significance. If the constructed surfaces are shown to meet all hypotheses of the cited theorem, the examples supply concrete instances of abelian surfaces with small bad-reduction set that are modular and have no extra endomorphisms; such objects are useful for testing modularity lifting results and for computational arithmetic geometry.

major comments (1)

The central claim rests on the assertion that the explicit abelian surfaces satisfy every hypothesis of the main theorem in arXiv:1812.09269 (polarization, good reduction outside {2,3,5,7}, absence of extra endomorphisms, and the listed local/global conditions on the Galois representation). The manuscript states that this verification is carried out, but the details of the check are not visible in the provided text; without them the application cannot be independently confirmed.

minor comments (1)

Abstract: 'endomorhpisms' is a typographical error and should read 'endomorphisms'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and positive comments on the significance of the examples. We address the major comment as follows.

read point-by-point responses

Referee: The central claim rests on the assertion that the explicit abelian surfaces satisfy every hypothesis of the main theorem in arXiv:1812.09269 (polarization, good reduction outside {2,3,5,7}, absence of extra endomorphisms, and the listed local/global conditions on the Galois representation). The manuscript states that this verification is carried out, but the details of the check are not visible in the provided text; without them the application cannot be independently confirmed.

Authors: The referee is correct that the details of the verification are not sufficiently detailed in the current manuscript to allow independent confirmation. We will revise the manuscript to include explicit descriptions of how each hypothesis is verified for the constructed abelian surfaces. This will include the specific checks for the polarization, the reduction type, the endomorphism ring, and the Galois representation conditions as required by the theorem in arXiv:1812.09269. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper applies the main theorem of the independent preprint arXiv:1812.09269 (Boxer-Calegari-Gee-Pilloni) to explicit abelian surface constructions over Q. Its derivation chain consists of exhibiting the surfaces and verifying the listed hypotheses (polarization, good reduction outside {2,3,5,7}, no extra endomorphisms, and the required local/global Galois conditions). This verification step is external to the cited theorem and does not reduce by construction to any fitted parameter, self-definition, or prior result internal to the present manuscript. The single overlapping-author citation is not load-bearing in a circular sense; the theorem is treated as an external black box whose proof lies outside this paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the applicability of a cited external theorem together with standard background facts of algebraic number theory; no free parameters or new entities are introduced in the abstract.

axioms (1)

standard math Standard axioms and definitions of algebraic geometry and number theory over Q.
The paper works inside the usual framework of abelian varieties and modular forms.

pith-pipeline@v0.9.0 · 5571 in / 965 out tokens · 27009 ms · 2026-05-25T15:27:34.881725+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

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Yuen, On the paramodularity of typical abelian surfaces, arXiv:1805.10873 [math.NT], 2018

Armand Brumer, Ariel Pacetti, Chris Poor, Gonzalo Tornar \' a, John Voight, and David S. Yuen, On the paramodularity of typical abelian surfaces, arXiv:1805.10873 [math.NT], 2018

work page arXiv 2018
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Booker, Jeroen Sijsling, Andrew V

Andrew R. Booker, Jeroen Sijsling, Andrew V. Sutherland, John Voight, and Dan Yasaki, A database of genus-2 curves over the rational numbers, LMS J. Comput. Math. 19 (2016), no. suppl. A, 235--254. 3540958

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Jean-Pierre Serre, Facteurs locaux des fonctions z\^eta des vari\'et\'es alg\'ebriques (d\'efinitions et conjectures), S\'eminaire Delange-Pisot-Poitou. Th\'eorie des nombres 11 (1969-1970), no. 2, 1--15

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Richard Taylor and Andrew Wiles, Ring-theoretic properties of certain H ecke algebras , Ann. of Math. (2) 141 (1995), no. 3, 553--572. 1333036 (96d:11072)

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Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, On the modularity of elliptic curves over Q : wild 3-adic exercises , J. Amer. Math. Soc. 14 (2001), no. 4, 843--939 (electronic). 1839918 (2002d:11058)

work page 2001

[2] [2]

George Boxer, Frank Calegari, Toby Gee, and Vincent Pilloni, Abelian surfaces over totally real fields are potentially modular, arXiv:1812.09269 [math.NT], 2018

work page arXiv 2018

[3] [3]

O'Brien, A millennium project: constructing small groups, Internat

Hans Ulrich Besche, Bettina Eick, and Eamonn A. O'Brien, A millennium project: constructing small groups, Internat. J. Algebra Comput. 12 (2002), no. 5, 623--644. 1935567

work page 2002

[4] [4]

Y uen) , arXiv:1710.10228 [math.NT], 2017

Tobias Berger and Krzysztof Klosin, Deformations of S aito- K urokawa type and the P aramodular C onjecture (with an appendix by C ris P oor, J erry S hurman, and D avid S . Y uen) , arXiv:1710.10228 [math.NT], 2017

work page arXiv 2017

[5] [5]

Yuen, On the paramodularity of typical abelian surfaces, arXiv:1805.10873 [math.NT], 2018

Armand Brumer, Ariel Pacetti, Chris Poor, Gonzalo Tornar \' a, John Voight, and David S. Yuen, On the paramodularity of typical abelian surfaces, arXiv:1805.10873 [math.NT], 2018

work page arXiv 2018

[6] [6]

Booker, Jeroen Sijsling, Andrew V

Andrew R. Booker, Jeroen Sijsling, Andrew V. Sutherland, John Voight, and Dan Yasaki, A database of genus-2 curves over the rational numbers, LMS J. Comput. Math. 19 (2016), no. suppl. A, 235--254. 3540958

work page 2016

[7] [7]

J. W. S. Cassels and E. V. Flynn, Prolegomena to a middlebrow arithmetic of curves of genus 2, London Mathematical Society Lecture Note Series, Cambridge University Press, 1996

work page 1996

[8] [8]

Kedlaya, V\' ctor Rotger, and Andrew V

Francesc Fit\'e, Kiran S. Kedlaya, V\' ctor Rotger, and Andrew V. Sutherland, Sato- T ate distributions and G alois endomorphism modules in genus 2 , Compos. Math. 148 (2012), no. 5, 1390--1442. 2982436

work page 2012

[9] [9]

Yuen, Siegel paramodular forms of weight 2 and squarefree level, Int

Cris Poor, Jerry Shurman, and David S. Yuen, Siegel paramodular forms of weight 2 and squarefree level, Int. J. Number Theory 13 (2017), no. 10, 2627--2652. 3713095

work page 2017

[10] [10]

Yuen, Paramodular cusp forms, Math

Cris Poor and David S. Yuen, Paramodular cusp forms, Math. Comp. 84 (2015), no. 293, 1401--1438. 3315514

work page 2015

[11] [11]

U ber die A nzahl der P rimzahlen unter einer gegebenen G r\

Bernhard Riemann, \" U ber die A nzahl der P rimzahlen unter einer gegebenen G r\" o sse , Monatsberichte der Berliner Akademie (1859), 671--680

work page

[12] [12]

Th\'eorie des nombres 11 (1969-1970), no

Jean-Pierre Serre, Facteurs locaux des fonctions z\^eta des vari\'et\'es alg\'ebriques (d\'efinitions et conjectures), S\'eminaire Delange-Pisot-Poitou. Th\'eorie des nombres 11 (1969-1970), no. 2, 1--15

work page 1969

[13] [13]

Richard Taylor and Andrew Wiles, Ring-theoretic properties of certain H ecke algebras , Ann. of Math. (2) 141 (1995), no. 3, 553--572. 1333036 (96d:11072)

work page 1995

[14] [14]

Andrew Wiles, Modular elliptic curves and F ermat's last theorem , Ann. of Math. (2) 141 (1995), no. 3, 443--551. 1333035

work page 1995