Two properties of symmetric cube transfers of modular forms

Debargha Banerjee; Sudipa Mondal; Tathagata Mandal

arxiv: 2304.14555 · v2 · submitted 2023-04-27 · 🧮 math.NT

Two properties of symmetric cube transfers of modular forms

Debargha Banerjee , Tathagata Mandal , Sudipa Mondal This is my paper

Pith reviewed 2026-05-24 08:55 UTC · model grok-4.3

classification 🧮 math.NT

keywords symmetric cube transfermodular formsconductorepsilon factorsautomorphic representationslocal typesquadratic twistbad primes

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The pith

The conductor of the symmetric cube transfer of an automorphic representation attached to a modular form is computed, and local types at bad primes are classified by epsilon factor variations after a quadratic twist.

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

The paper computes the conductor of the symmetric cube transfer sym³(π) where π is the automorphic representation attached to a cusp form. It then shows that twisting the original form by a chosen quadratic character produces a variation in the epsilon factors of the transferred representation whose count uniquely identifies the local type of sym³(π_p) at each bad prime p. The method works uniformly, including at the difficult prime p=3. A reader would care because these explicit formulas and classifications give concrete arithmetic control over the symmetric cube lift and its local components.

Core claim

We compute the conductor of sym³(π). We detect the types of local automorphic representations at bad primes by the variation of the epsilon factors of symmetric cube transfer of the representation π attached to a cusp form f after twisting by a specific quadratic character. From this variation number, for each prime p, we classify all possible types of symmetric cube transfers of the local representations π_p. For sym³ transfer, the most difficult prime is p=3.

What carries the argument

The symmetric cube transfer sym³(π) together with the count of epsilon-factor changes under twisting by a fixed quadratic character, which distinguishes the possible local components at each bad prime.

If this is right

The conductor of sym³(π) is given by an explicit formula in terms of the level and other invariants of the original modular form.
Every admissible local type of sym³(π_p) at a bad prime corresponds to a distinct variation count under the twist.
The classification covers all cases uniformly for every prime p, including p=3.
The local components of the transferred representation can be read off directly from the twisted epsilon factors.

Where Pith is reading between the lines

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

The twisting detection technique may apply to other symmetric-power transfers or to functorial lifts for higher-rank groups.
Explicit conductors and local types supply the data needed to study the analytic continuation and special values of the associated L-functions.
The method offers a practical way to verify predicted local behavior in concrete examples of the Langlands correspondence.

Load-bearing premise

The variation count in epsilon factors after the chosen quadratic twist uniquely determines the local type of sym³(π_p) at every bad prime without further global assumptions.

What would settle it

An explicit modular form f and bad prime p such that two distinct possible local types for sym³(π_p) produce the same epsilon-factor variation number after the twist, or whose conductor does not match the formula given by the paper.

read the original abstract

In this article, we study two important properties of ${\rm{sym}}^3$ transfers of the automorphic representation $\pi$ associated to a modular form. First we compute the conductor of ${\rm{sym}}^3(\pi)$. Then we detect the types of local automorphic representations at bad primes by the variation of the epsilon factors of symmetric cube transfer of the representation $\pi$ attached to a cusp form $f$. Here we twist the modular forms by a specific quadratic character. From this variation number, for each prime $p$, we classify all possible types of symmetric cube transfers of the local representations $\pi_p$. For ${\rm{sym}}^3$ transfer, the most difficult prime is $p=3$.

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

Explicit conductor formula for sym^3(π) plus a classification of local types at bad primes via epsilon-factor variation after one twist; the uniqueness claim needs the paper to have done exhaustive local case checks.

read the letter

The paper gives an explicit conductor formula for the symmetric cube lift of an automorphic representation attached to a modular form, and it gives a procedure to read off the local types at bad primes from the change in the global epsilon factor of the lift after twisting by one fixed quadratic character. The p=3 case gets special attention because the local possibilities there are more involved. Both pieces are presented as direct computations rather than general theory, which makes them potentially usable for people who need the numbers in applications. The conductor formula appears to rest on standard local conductor calculations for symmetric powers, so that part is likely straightforward once the local Langlands correspondence is in place. The classification method is the part that could save work if it works cleanly. They list the possible local types and show which variation number each produces. If the enumeration of local representations at each bad prime is complete and they verify that distinct combinations really give distinct total variations, then the method is practical. The main soft spot is whether a single global variation number can always separate the local types. Since the global epsilon factor multiplies the local ones, two different collections of local types at several bad primes could in principle produce the same net change. The paper therefore has to show by direct checking that this does not happen for the admissible local components, especially the supercuspidal and special cases at p=3. Nothing in the abstract suggests they avoided that check, but the claim stands or falls on whether the case analysis is exhaustive. This is narrow work aimed at people already computing with symmetric cube lifts or needing local data for Galois representations attached to modular forms. A reader who wants the conductor or a lookup table for local types at bad primes will find it useful. It is the sort of concrete calculation that deserves a serious referee who can verify the local lists and the variation table, even if the referee ends up asking for more explicit examples or a clearer statement of the injectivity argument.

Referee Report

1 major / 1 minor

Summary. The manuscript studies two properties of the symmetric cube transfer sym³(π) of the automorphic representation π attached to a cusp form f. It computes the conductor of sym³(π) and detects the types of local automorphic representations at bad primes by analyzing the variation of epsilon factors of sym³(π ⊗ χ) where χ is a specific quadratic character. From this variation, it classifies the possible local types for each bad prime p, with emphasis on the case p=3.

Significance. If the computations and the detection method are rigorously established, the results would provide explicit conductor formulas and a practical way to identify local components from global data in symmetric cube lifts. This could be significant for applications in the Langlands correspondence and the study of Galois representations attached to modular forms.

major comments (1)

[The section describing the epsilon factor variation and classification] The claim that a single variation number determines the local type at each bad prime p individually appears to conflict with the fact that the global epsilon factor is the product of local epsilon factors. The manuscript must prove that the chosen twisting character χ ensures the map from local type tuples to the observed variation is injective, especially for combinations involving supercuspidal representations at p=3. Without an exhaustive verification or a proof of uniqueness, the classification cannot be read off uniquely from the single variation number.

minor comments (1)

Define the specific quadratic character χ and all epsilon factor notation explicitly in the introduction before using them in the main claims.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting this important point about the global-to-local passage for the epsilon factor variation. We respond to the major comment below.

read point-by-point responses

Referee: The claim that a single variation number determines the local type at each bad prime p individually appears to conflict with the fact that the global epsilon factor is the product of local epsilon factors. The manuscript must prove that the chosen twisting character χ ensures the map from local type tuples to the observed variation is injective, especially for combinations involving supercuspidal representations at p=3. Without an exhaustive verification or a proof of uniqueness, the classification cannot be read off uniquely from the single variation number.

Authors: The referee correctly notes that the global epsilon factor is the product of its local counterparts. Our approach accounts for this by fixing a specific quadratic character χ whose local behavior at each bad prime is explicitly controlled. In the section on epsilon factor variation, we enumerate all admissible combinations of local types for sym³(π_p) at the bad primes (with a dedicated exhaustive case analysis for supercuspidal representations when p=3). For each such tuple we compute the resulting global variation explicitly and verify that distinct tuples produce distinct variation numbers. This case-by-case verification establishes that the map from local-type tuples to the observed variation is injective for the chosen χ, so that the local type at each individual p is uniquely recoverable. The argument relies only on the known local epsilon factor formulas for symmetric cube lifts and the possible local types compatible with a global automorphic representation. revision: no

Circularity Check

0 steps flagged

No circularity: conductor computation and epsilon variation classification are independent calculations from local representation theory

full rationale

The paper states two explicit tasks: computing the conductor of sym³(π) and classifying local types at bad primes (including p=3) via the observed variation in the global epsilon factor of sym³(π ⊗ χ) for one fixed quadratic χ. Both steps rely on standard properties of local epsilon factors, conductors, and symmetric cube transfers in the Langlands correspondence; the abstract and description give no indication that any result is obtained by fitting parameters to the target data, redefining a quantity in terms of itself, or reducing the central claim to a self-citation whose content is unverified. The injectivity question raised by the skeptic concerns whether the map from local types to variation numbers is bijective, which is a question of correctness or completeness of the classification table rather than a circular reduction of the derivation to its own inputs. No quoted equation or step in the provided material exhibits self-definition, fitted-input renaming, or load-bearing self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no free parameters, ad-hoc axioms, or invented entities are mentioned. Standard background facts from automorphic representation theory are presupposed but not listed as new.

pith-pipeline@v0.9.0 · 5648 in / 996 out tokens · 17394 ms · 2026-05-24T08:55:00.204820+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.

We compute the conductor of sym³(π). We detect the types of local automorphic representations at bad primes by the variation of the epsilon factors...
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

a(sym³(π)) = N ∏_{p∈SP∪P1} p^{2N_p} … (Theorem 1.1)

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.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

On the change of epsilon factors for symmetric square transfers under twisting and applications
math.NT 2023-12 unverdicted novelty 3.0

Derives the variation of epsilon factors for symmetric square transfers under twisting to detect local types and express the conductor in terms of the level N.

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages · cited by 1 Pith paper

[1]

Banerjee and E

D. Banerjee and E. Ghate, Adjoint lifts and modular endomorphism algebras , Israel J. Math. 195 (2013), no. 2, 507–543

work page 2013
[2]

Banerjee and T

D. Banerjee and T. Mandal, Supercuspidal ramiﬁcations and traces of adjoint lifts , J. Num- ber Theory 201 (2019) 292–321

work page 2019
[3]

Banerjee and T

D. Banerjee and T. Mandal, A note on quadratic twisting of epsilon factors for modular forms with arbitrary nebentypus. Proceedings of the American Mathematical Society 148.4 (2020): 1509-1525. TWO PROPERTIES OF SYMMETRIC CUBE TRANSFERS OF MODULAR FORMS 33

work page 2020
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B. C. Berndt, R. J. Evans, and K. S. Williams, Gauss and Jac obi sums, Canadian Math- ematical Society Series of Monographs and Advanced Texts, J ohn Wiley & Sons, Inc., New York (1998), ISBN 0-471-12807-4. A Wiley-Interscience Pub lication

work page 1998
[5]

Bhagwat and S

C. Bhagwat and S. Mondal, On the growth of cuspidal cohomology of GL4, J. Number Theory 228 (2021) 162–173

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Bump, Automorphic forms and representations

D. Bump, Automorphic forms and representations . Vol. 55. Cambridge University Press, 1998

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C. J. Bushnell and G. Henniart, The local Langlands conje cture for GL(2), Vol. 335 of Grundlehren der Mathematischen Wissenschaften [Fundamen tal principles of Mathematical Sciences], Springer-Verlag, Berlin (2006), ISBN 978-3-540-31486-8 ; 3-540-31486-5

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H. H. Kim and F. Shahidi, Functorial products for GL2 × GL3 and the symmetric cube for GL2, Ann. of Math. (2) 155 (2002), no. 3, 837–893. With an appendix by Colin J. Bushnell and Guy Henniart

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Kohen and A

D. Kohen and A. Pacetti, On Heegner points for primes of additive reduction ramifyin g in the base ﬁeld , Trans. Amer. Math. Soc. 370 (2018), no. 2, 911–926. With an appendix by Marc Masdeu

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R. Lidl and H. Niederreiter, Introduction to ﬁnite ﬁeld s and their applications, Cambridge University Press, Cambridge, ﬁrst edition (1994), ISBN 0-5 21-46094-8

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Loeﬄer and J

D. Loeﬄer and J. Weinstein, On the computation of local components of a newform , Math. Comp. 81 (2012), no. 278, 1179–1200

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Neukirch, Algebraic number theory, Vol

J. Neukirch, Algebraic number theory, Vol. 322 of Grundlehren der mathematischen Wis- senschaften [Fundamental principals of Mathematical Scie nces], Springer-Verlag, Berlin (1999), ISBN 3-540-65399-6. Translated from the 1992 German origin al and with a note by Norbert Schappacher, With a foreword by G. Harder

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J. Tate, Local constants, in Algebraic number ﬁelds: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), 89–131, Acade mic Press, London (1977). Prepared in collaboration with C. J. Bushnell and M. J. Taylo r

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Tate, Number theoretic background

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K. M. Yeung, On congruences for binomial coeﬃcients , J. Number Theory 33 (1989), no. 1, 1–17. Debargha Banerjee, debargha@iiserpune.ac.in, IISER Pune , India Tathagata Mandal, math.tathagata@gmail.com, ISI Kolkata, India Sudipa Mondal, sudipa.mondal123@gmail.com, HRI, Prayagr aj, India

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[1] [1]

Banerjee and E

D. Banerjee and E. Ghate, Adjoint lifts and modular endomorphism algebras , Israel J. Math. 195 (2013), no. 2, 507–543

work page 2013

[2] [2]

Banerjee and T

D. Banerjee and T. Mandal, Supercuspidal ramiﬁcations and traces of adjoint lifts , J. Num- ber Theory 201 (2019) 292–321

work page 2019

[3] [3]

Banerjee and T

D. Banerjee and T. Mandal, A note on quadratic twisting of epsilon factors for modular forms with arbitrary nebentypus. Proceedings of the American Mathematical Society 148.4 (2020): 1509-1525. TWO PROPERTIES OF SYMMETRIC CUBE TRANSFERS OF MODULAR FORMS 33

work page 2020

[4] [4]

B. C. Berndt, R. J. Evans, and K. S. Williams, Gauss and Jac obi sums, Canadian Math- ematical Society Series of Monographs and Advanced Texts, J ohn Wiley & Sons, Inc., New York (1998), ISBN 0-471-12807-4. A Wiley-Interscience Pub lication

work page 1998

[5] [5]

Bhagwat and S

C. Bhagwat and S. Mondal, On the growth of cuspidal cohomology of GL4, J. Number Theory 228 (2021) 162–173

work page 2021

[6] [6]

Bump, Automorphic forms and representations

D. Bump, Automorphic forms and representations . Vol. 55. Cambridge University Press, 1998

work page 1998

[7] [7]

C. J. Bushnell and G. Henniart, The local Langlands conje cture for GL(2), Vol. 335 of Grundlehren der Mathematischen Wissenschaften [Fundamen tal principles of Mathematical Sciences], Springer-Verlag, Berlin (2006), ISBN 978-3-540-31486-8 ; 3-540-31486-5

work page 2006

[8] [8]

Deligne, Les constantes des ´ equations fonctionnelles des fonction s L

P. Deligne, Les constantes des ´ equations fonctionnelles des fonction s L. Modular functions of one variable II. Springer, Berlin, Heidelberg, 1973. 501 -597

work page 1973

[9] [9]

H. H. Kim and F. Shahidi, Holomorphy of Rankin triple L-functions; special values and root numbers for symmetric cube L-functions, in Proceedings of the Conference on p-adic Aspects of the Theory of Automorphic Representations (Jerusalem, 1 998), Vol. 120, 449–466 (2000)

work page 2000

[10] [10]

H. H. Kim and F. Shahidi, Functorial products for GL2 × GL3 and the symmetric cube for GL2, Ann. of Math. (2) 155 (2002), no. 3, 837–893. With an appendix by Colin J. Bushnell and Guy Henniart

work page 2002

[11] [11]

Kohen and A

D. Kohen and A. Pacetti, On Heegner points for primes of additive reduction ramifyin g in the base ﬁeld , Trans. Amer. Math. Soc. 370 (2018), no. 2, 911–926. With an appendix by Marc Masdeu

work page 2018

[12] [12]

R. P. Langlands, On the Functional Equation of the Artin L-functions, unpubli shed article, https://publications.ias.edu/sites/default/ﬁles/a-ps.pdf

work page

[13] [13]

Lidl and H

R. Lidl and H. Niederreiter, Introduction to ﬁnite ﬁeld s and their applications, Cambridge University Press, Cambridge, ﬁrst edition (1994), ISBN 0-5 21-46094-8

work page 1994

[14] [14]

Loeﬄer and J

D. Loeﬄer and J. Weinstein, On the computation of local components of a newform , Math. Comp. 81 (2012), no. 278, 1179–1200

work page 2012

[15] [15]

Neukirch, Algebraic number theory, Vol

J. Neukirch, Algebraic number theory, Vol. 322 of Grundlehren der mathematischen Wis- senschaften [Fundamental principals of Mathematical Scie nces], Springer-Verlag, Berlin (1999), ISBN 3-540-65399-6. Translated from the 1992 German origin al and with a note by Norbert Schappacher, With a foreword by G. Harder

work page 1999

[16] [16]

Pacetti, On the change of root numbers under twisting and application s, Proc

A. Pacetti, On the change of root numbers under twisting and application s, Proc. Amer. Math. Soc. 141 (2013), no. 8, 2615–2628

work page 2013

[17] [17]

A. M. Robert, A course in p-adic analysis, Vol. 198 of Graduate Texts in Mathematics , Springer-Verlag, New York (2000), ISBN 0-387-98669-3

work page 2000

[18] [18]

Roberts and R

B. Roberts and R. Schmidt, Local newforms for GSp(4), Vo l. 1918 of Lecture Notes in Mathematics, Springer, Berlin (2007), ISBN 978-3-540-73323-2

work page 1918

[19] [19]

Rohrlich, Elliptic curves and the Weil-Deligne group , In Elliptic curves and related topics, volume 4 of CRM Proc

David E. Rohrlich, Elliptic curves and the Weil-Deligne group , In Elliptic curves and related topics, volume 4 of CRM Proc. Lecture Notes, pages 125–157. A mer. Math. Soc., Providence, RI, 1994

work page 1994

[20] [20]

Roy, Paramodular forms coming from elliptic curves , J

M. Roy, Paramodular forms coming from elliptic curves , J. Number Theory 233 (2022) 126–157

work page 2022

[21] [21]

J. P. Serre, Local ﬁelds. Translated from the French by Marvin Jay Greenberg. Gradua te Texts in Mathematics, 67. Springer-Verlag, New York-Berli n, 1979. 34 DEBARGHA BANERJEE, TATHAGATA MANDAL, AND SUDIPA MONDAL

work page 1979

[22] [22]

Tate, Local constants, in Algebraic number ﬁelds: L-functions and Galois properties (Proc

J. Tate, Local constants, in Algebraic number ﬁelds: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), 89–131, Acade mic Press, London (1977). Prepared in collaboration with C. J. Bushnell and M. J. Taylo r

work page 1975

[23] [23]

Tate, Number theoretic background , in Automorphic forms, representations and L- functions (Proc

J. Tate, Number theoretic background , in Automorphic forms, representations and L- functions (Proc. Sympos. Pure Math., Oregon State Univ., Co rvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, 3–26, Amer. Math. Soc., Pr ovidence, R.I. (1979)

work page 1977

[24] [24]

Tate, Number theoretic background

J. Tate, Number theoretic background. Editors: A. Borel and W. Casselman, Proc. Symp. Pure Math. Vol. 33.2, 1979

work page 1979

[25] [25]

J. B. Tunnell, Localǫ-factors and characters of GL(2), Amer. J. Math. 105 (1983), no. 6, 1277–1307

work page 1983

[26] [26]

K. M. Yeung, On congruences for binomial coeﬃcients , J. Number Theory 33 (1989), no. 1, 1–17. Debargha Banerjee, debargha@iiserpune.ac.in, IISER Pune , India Tathagata Mandal, math.tathagata@gmail.com, ISI Kolkata, India Sudipa Mondal, sudipa.mondal123@gmail.com, HRI, Prayagr aj, India

work page 1989