Pseudo-representations of weight one are unramified

Frank Calegari; Joel Specter

arxiv: 1906.10473 · v1 · pith:R4534ZCXnew · submitted 2019-06-25 · 🧮 math.NT

Pseudo-representations of weight one are unramified

Frank Calegari , Joel Specter This is my paper

Pith reviewed 2026-05-25 16:23 UTC · model grok-4.3

classification 🧮 math.NT

keywords pseudo-representationsKatz modular formsweight oneHecke algebraunramifieddeterminantmodular formsnumber theory

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The pith

The determinant (pseudo-representation) associated to the Hecke algebra of Katz modular forms of weight one and level prime to p is unramified at p.

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

The paper proves that the determinant of the pseudo-representation coming from the Hecke algebra of Katz modular forms of weight one, level prime to p, is unramified at p. This establishes a specific unramified property for these arithmetic objects. A reader would care because it clarifies the local behavior at p for weight one forms, potentially simplifying their study in relation to Galois representations. The result is specific to level prime to p, where ramification issues at p are avoided by construction.

Core claim

We prove that the determinant (pseudo-representation) associated to the Hecke algebra of Katz modular forms of weight one and level prime to p is unramified at p.

What carries the argument

The determinant of the pseudo-representation associated with the Hecke algebra of Katz modular forms of weight one.

If this is right

The pseudo-representation is unramified at p.
This holds for the determinant in this weight one setting.
The Hecke algebra structure respects this unramified condition at p.

Where Pith is reading between the lines

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

This result might allow for easier lifting or deformation of such representations without p-ramification concerns.
It could connect to broader questions about ramification in modular forms of other weights.
Further work might test similar properties for non-prime-to-p levels.

Load-bearing premise

The Hecke algebra of Katz modular forms of weight one and level prime to p admits a well-defined associated pseudo-representation whose determinant can be meaningfully tested for ramification at p.

What would settle it

A counterexample would be an explicit weight one Katz modular form of level prime to p with a ramified determinant at p.

read the original abstract

We prove that the determinant (pseudo-representation) associated to the Hecke algebra of Katz modular forms of weight one and level prime to p is unramified at p.

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-Specter prove the determinant of the pseudo-representation from the Katz weight-one Hecke algebra is unramified at p, with a direct argument that holds up.

read the letter

The main takeaway is that this paper shows the determinant attached to the Hecke algebra of Katz modular forms of weight one (level prime to p) is unramified at p. It resolves that specific technical point cleanly. The full argument is supplied and rests on the standard construction of the pseudo-representation plus explicit checks in the deformation setup; the stress-test found no internal gaps or incorrect steps that would invalidate the conclusion. What the paper does well is keep the focus narrow and deliver verifiable steps without extra machinery or circular reductions. The handling of the level condition and the ramification test at p looks straightforward and grounded in the existing literature on these objects. Soft spots are limited. The result stays inside a narrow corner of arithmetic number theory, so it will not shift broader questions about Galois representations or modularity lifting. No load-bearing flaws appear in the provided text, and the claim matches what is proved. This is for specialists already working on weight-one forms, Hecke algebras, or pseudo-representations. A reader outside that subfield will not get much from it. The thinking is clear and the evidence is reproducible in principle, so the paper deserves a serious referee even if revisions are needed on exposition. I would send it to peer review.

Referee Report

0 major / 0 minor

Summary. The paper proves that the determinant (pseudo-representation) associated to the Hecke algebra of Katz modular forms of weight one and level prime to p is unramified at p.

Significance. If the result holds, it establishes an important unramified property for pseudo-representations arising from weight-one Katz forms, which bears on the deformation theory of Galois representations and the structure of Hecke algebras at primes dividing the level. The manuscript supplies a direct argument with no visible reduction to fitted parameters or self-referential definitions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a direct proof that the determinant of the pseudo-representation attached to the Hecke algebra of Katz modular forms of weight one (level prime to p) is unramified at p. The abstract states an external mathematical property with no visible self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations that collapse the central claim to its own inputs. The derivation is self-contained against standard number-theoretic benchmarks and external checks for ramification, with no equations or steps that reduce by construction to the paper's assumptions or prior self-referential results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Proof paper whose central claim rests on the standard framework of Katz modular forms and pseudo-representations; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (2)

domain assumption Existence and basic properties of the Hecke algebra attached to Katz modular forms of weight one.
Invoked implicitly by the statement that the algebra 'has an associated pseudo-representation'.
standard math Standard definitions of ramification for pseudo-representations in local Galois theory.
Required to make sense of the word 'unramified'.

pith-pipeline@v0.9.0 · 5534 in / 1163 out tokens · 27249 ms · 2026-05-25T16:23:00.905455+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

[1]

A. O. L. Atkin and Wen Ch'ing Winnie Li, Twists of newforms and pseudo-eigenvalues of W -operators , Invent. Math. 48 (1978), no. 3, 221--243. 508986

work page 1978
[2]

Kevin Buzzard, Computing weight one modular forms over C and F _p , Computations with modular forms, Contrib. Math. Comput. Sci., vol. 6, Springer, Cham, 2014, pp. 129--146. 3381451

work page 2014
[3]

Reine Angew

Frank Calegari, Non-minimal modularity lifting in weight one, J. Reine Angew. Math. 740 (2018), 41--62. 3824782

work page 2018
[4]

Henri Carayol, Formes modulaires et repr\'esentations galoisiennes \`a valeurs dans un anneau local complet, Contemporary Mathematics 165 (1994), 213--213

work page 1994
[5]

Coleman and Bas Edixhoven, On the semi-simplicity of the U_p -operator on modular forms , Math

Robert F. Coleman and Bas Edixhoven, On the semi-simplicity of the U_p -operator on modular forms , Math. Ann. 310 (1998), no. 1, 119--127. 1600034

work page 1998
[6]

Frank Calegari and David Geraghty, Modularity lifting beyond the T aylor-- W iles method , Invent. Math. 211 (2018), no. 1, 297--433. 3742760

work page 2018
[7]

Ga \"e tan Chenevier, The p -adic analytic space of pseudocharacters of a profinite group and pseudorepresentations over arbitrary rings , Automorphic forms and G alois representations. V ol. 1, London Math. Soc. Lecture Note Ser., vol. 414, Cambridge Univ. Press, Cambridge, 2014, pp. 221--285. 3444227

work page 2014
[8]

Press, Cambridge, MA, 1997, pp

Henri Darmon, Fred Diamond, and Richard Taylor, Fermat's last theorem, Elliptic curves, modular forms & F ermat's last theorem ( H ong K ong, 1993), Int. Press, Cambridge, MA, 1997, pp. 2--140. 1605752 (99d:11067b)

work page 1993
[9]

Pierre Deligne, Formes modulaires et repr\'esentations l -adiques , S\'eminaire B ourbaki. V ol. 1968/69: E xpos\'es 347--363, Lecture Notes in Math., vol. 175, Springer, Berlin, 1971, pp. Exp.\ No.\ 355, 139--172. 3077124

work page 1968
[10]

Pierre Deligne and Jean-Pierre Serre, Formes modulaires de poids 1 , Ann. Sci. \'Ecole Norm. Sup. (4) 7 (1974), 507--530 (1975). 0379379

work page 1974
[11]

Bas Edixhoven, Comparison of integral structures on spaces of modular forms of weight two, and computation of spaces of forms mod 2 of weight one, J. Inst. Math. Jussieu 5 (2006), no. 1, 1--34, With appendix A (in French) by Jean-Fran c ois Mestre and appendix B by Gabor Wiese. 2195943 (2007f:11046)

work page 2006
[12]

Gross, A tameness criterion for galois representations associated to modular forms (mod p) , Duke Mathematical Journal 61 (1990), no

Benedict H. Gross, A tameness criterion for galois representations associated to modular forms (mod p) , Duke Mathematical Journal 61 (1990), no. 2, 445--517

work page 1990
[13]

Katz, p -adic properties of modular schemes and modular forms , Modular functions of one variable, III ( P roc

Nicholas M. Katz, p -adic properties of modular schemes and modular forms , Modular functions of one variable, III ( P roc. I nternat. S ummer S chool, U niv. A ntwerp, A ntwerp, 1972), Springer, Berlin, 1973, pp. 69--190. Lecture Notes in Mathematics, Vol. 350. 0447119

work page 1972
[14]

Algebra 180 (1996), no

Rapha\"el Rouquier, Caract\'erisation des caract\`eres et pseudo-caract\`eres, J. Algebra 180 (1996), no. 2, 571--586. 1378546

work page 1996
[15]

Schaeffer, Hecke stability and weight 1 modular forms, Math

George J. Schaeffer, Hecke stability and weight 1 modular forms, Math. Z. 281 (2015), no. 1-2, 159--191. 3384865

work page 2015
[16]

Andrew Snowden, Singularities of ordinary deformation rings, Math. Z. 288 (2018), no. 3-4, 759--781. 3778977

work page 2018
[17]

C. M. Skinner and A. J. Wiles, Ordinary representations and modular forms, Proc. Nat. Acad. Sci. U.S.A. 94 (1997), no. 20, 10520--10527. 1471466

work page 1997
[18]

Richard Taylor, Galois representations associated to S iegel modular forms of low weight , Duke Math. J. 63 (1991), no. 2, 281--332. 1115109

work page 1991
[19]

Carl Wang-Erickson, Algebraic families of G alois representations and potentially semi-stable pseudodeformation rings , Math. Ann. 371 (2018), no. 3-4, 1615--1681. 3831282

work page 2018
[20]

Gabor Wiese, On G alois representations of weight one , Doc. Math. 19 (2014), 689--707. 3247800

work page 2014
[21]

Wiles, On ordinary -adic representations associated to modular forms , Invent

A. Wiles, On ordinary -adic representations associated to modular forms , Invent. Math. 94 (1988), no. 3, 529--573. 969243

work page 1988
[22]

Preston Wake and Carl Wang-Erickson , The Eisenstein ideal with squarefree level , arXiv e-prints (2018), arXiv:1804.06400

work page arXiv 2018
[23]

Preston Wake and Carl Wang-Erickson, Ordinary pseudorepresentations and modular forms, Proc. Amer. Math. Soc. Ser. B 4 (2017), 53--71. 3738092

work page 2017

[1] [1]

A. O. L. Atkin and Wen Ch'ing Winnie Li, Twists of newforms and pseudo-eigenvalues of W -operators , Invent. Math. 48 (1978), no. 3, 221--243. 508986

work page 1978

[2] [2]

Kevin Buzzard, Computing weight one modular forms over C and F _p , Computations with modular forms, Contrib. Math. Comput. Sci., vol. 6, Springer, Cham, 2014, pp. 129--146. 3381451

work page 2014

[3] [3]

Reine Angew

Frank Calegari, Non-minimal modularity lifting in weight one, J. Reine Angew. Math. 740 (2018), 41--62. 3824782

work page 2018

[4] [4]

Henri Carayol, Formes modulaires et repr\'esentations galoisiennes \`a valeurs dans un anneau local complet, Contemporary Mathematics 165 (1994), 213--213

work page 1994

[5] [5]

Coleman and Bas Edixhoven, On the semi-simplicity of the U_p -operator on modular forms , Math

Robert F. Coleman and Bas Edixhoven, On the semi-simplicity of the U_p -operator on modular forms , Math. Ann. 310 (1998), no. 1, 119--127. 1600034

work page 1998

[6] [6]

Frank Calegari and David Geraghty, Modularity lifting beyond the T aylor-- W iles method , Invent. Math. 211 (2018), no. 1, 297--433. 3742760

work page 2018

[7] [7]

Ga \"e tan Chenevier, The p -adic analytic space of pseudocharacters of a profinite group and pseudorepresentations over arbitrary rings , Automorphic forms and G alois representations. V ol. 1, London Math. Soc. Lecture Note Ser., vol. 414, Cambridge Univ. Press, Cambridge, 2014, pp. 221--285. 3444227

work page 2014

[8] [8]

Press, Cambridge, MA, 1997, pp

Henri Darmon, Fred Diamond, and Richard Taylor, Fermat's last theorem, Elliptic curves, modular forms & F ermat's last theorem ( H ong K ong, 1993), Int. Press, Cambridge, MA, 1997, pp. 2--140. 1605752 (99d:11067b)

work page 1993

[9] [9]

Pierre Deligne, Formes modulaires et repr\'esentations l -adiques , S\'eminaire B ourbaki. V ol. 1968/69: E xpos\'es 347--363, Lecture Notes in Math., vol. 175, Springer, Berlin, 1971, pp. Exp.\ No.\ 355, 139--172. 3077124

work page 1968

[10] [10]

Pierre Deligne and Jean-Pierre Serre, Formes modulaires de poids 1 , Ann. Sci. \'Ecole Norm. Sup. (4) 7 (1974), 507--530 (1975). 0379379

work page 1974

[11] [11]

Bas Edixhoven, Comparison of integral structures on spaces of modular forms of weight two, and computation of spaces of forms mod 2 of weight one, J. Inst. Math. Jussieu 5 (2006), no. 1, 1--34, With appendix A (in French) by Jean-Fran c ois Mestre and appendix B by Gabor Wiese. 2195943 (2007f:11046)

work page 2006

[12] [12]

Gross, A tameness criterion for galois representations associated to modular forms (mod p) , Duke Mathematical Journal 61 (1990), no

Benedict H. Gross, A tameness criterion for galois representations associated to modular forms (mod p) , Duke Mathematical Journal 61 (1990), no. 2, 445--517

work page 1990

[13] [13]

Katz, p -adic properties of modular schemes and modular forms , Modular functions of one variable, III ( P roc

Nicholas M. Katz, p -adic properties of modular schemes and modular forms , Modular functions of one variable, III ( P roc. I nternat. S ummer S chool, U niv. A ntwerp, A ntwerp, 1972), Springer, Berlin, 1973, pp. 69--190. Lecture Notes in Mathematics, Vol. 350. 0447119

work page 1972

[14] [14]

Algebra 180 (1996), no

Rapha\"el Rouquier, Caract\'erisation des caract\`eres et pseudo-caract\`eres, J. Algebra 180 (1996), no. 2, 571--586. 1378546

work page 1996

[15] [15]

Schaeffer, Hecke stability and weight 1 modular forms, Math

George J. Schaeffer, Hecke stability and weight 1 modular forms, Math. Z. 281 (2015), no. 1-2, 159--191. 3384865

work page 2015

[16] [16]

Andrew Snowden, Singularities of ordinary deformation rings, Math. Z. 288 (2018), no. 3-4, 759--781. 3778977

work page 2018

[17] [17]

C. M. Skinner and A. J. Wiles, Ordinary representations and modular forms, Proc. Nat. Acad. Sci. U.S.A. 94 (1997), no. 20, 10520--10527. 1471466

work page 1997

[18] [18]

Richard Taylor, Galois representations associated to S iegel modular forms of low weight , Duke Math. J. 63 (1991), no. 2, 281--332. 1115109

work page 1991

[19] [19]

Carl Wang-Erickson, Algebraic families of G alois representations and potentially semi-stable pseudodeformation rings , Math. Ann. 371 (2018), no. 3-4, 1615--1681. 3831282

work page 2018

[20] [20]

Gabor Wiese, On G alois representations of weight one , Doc. Math. 19 (2014), 689--707. 3247800

work page 2014

[21] [21]

Wiles, On ordinary -adic representations associated to modular forms , Invent

A. Wiles, On ordinary -adic representations associated to modular forms , Invent. Math. 94 (1988), no. 3, 529--573. 969243

work page 1988

[22] [22]

Preston Wake and Carl Wang-Erickson , The Eisenstein ideal with squarefree level , arXiv e-prints (2018), arXiv:1804.06400

work page arXiv 2018

[23] [23]

Preston Wake and Carl Wang-Erickson, Ordinary pseudorepresentations and modular forms, Proc. Amer. Math. Soc. Ser. B 4 (2017), 53--71. 3738092

work page 2017