Root numbers for twisted Fermat quotient curves II
Pith reviewed 2026-05-09 23:47 UTC · model grok-4.3
The pith
The root number of the curve y to the power ell^N equals x^r times (delta minus x)^s is computed when ell to the N-1 exactly divides r.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
In this sequel, the author computes the root number of the Fermat quotient curve y^{ℓ^N}=x^r(δ-x)^s under the condition that ℓ^{N-1} exactly divides r and ℓ does not divide stδ, where ℓ is an odd prime, δ is ℓ^N-th power free, and r+s+t=ℓ^N.
What carries the argument
The root number, obtained as the product of local epsilon factors, under the exact divisibility condition ℓ^{N-1} || r with ℓ not dividing stδ.
If this is right
- The sign in the functional equation of the L-function attached to the curve is now known explicitly in this case.
- The root number is determined for the curve whenever an odd prime divides exactly one of the three exponents to the precise power N-1.
- This supplies the missing piece needed to know the root number for all such curves when the prime divides the exponents in limited ways.
- The formula permits direct verification of the parity of the order of vanishing at s=1 for the L-functions of these curves in explicit families.
Where Pith is reading between the lines
- The method of handling this partial divisibility could be adapted to compute root numbers when the prime divides r to a higher power than N-1.
- Numerical checks for small values of ℓ, N, r, s, t and δ would provide independent evidence by comparing the formula against direct local computations.
- This step brings the full determination of root numbers for the entire family of twisted Fermat quotient curves closer to completion.
Load-bearing premise
The technical assumptions that ℓ is an odd prime, δ is ℓ^N-th-power-free, and the exact divisibility ℓ^{N-1} || r with ℓ not dividing stδ hold.
What would settle it
An independent computation of the root number via local epsilon factors for a concrete small example satisfying ℓ^{N-1} || r and ℓ not dividing stδ that differs from the paper's explicit value would show the claim is incorrect.
read the original abstract
This is a sequel to the previous work of the author Yanagihara (2025). Let $\ell$ be an odd prime, let $N \geq 1$ be an integer, and let $\delta \geq 1$ be an $\ell^N$-th-power-free integer. Let $r,s,t>0$ be integers satisfying $r+s+t=\ell^N$. In Yanagihara (2025), the author computed the root number of the Fermat quotient curve $y^{\ell^N}=x^r(\delta-x)^s$ under the assumptions that $\ell\nmid rst$ and that $\operatorname{ord}_{\ell}(\delta)=0$ or $\ell\nmid \operatorname{ord}_{\ell}(\delta)$. In this paper, we study the case where the technical assumption $\ell\nmid rst$ is dropped. As one such case, we compute the root number when $\ell^{N-1}\| r$ and $\ell\nmid st\delta$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This sequel to Yanagihara (2025) computes the global root number of the Fermat quotient curve y^{ℓ^N} = x^r (δ - x)^s for odd prime ℓ, integer N ≥ 1, ℓ^N-power-free δ ≥ 1, and positive integers r, s, t with r + s + t = ℓ^N, specifically in the case ℓ^{N-1} || r and ℓ ∤ stδ. The work drops the prior assumption ℓ ∤ rst and provides an explicit root-number formula under these arithmetic conditions.
Significance. If the local computations and global product formula are correct, the result supplies an explicit, case-by-case root number for a family of curves that previously fell outside the treated range. This strengthens the arithmetic toolkit for these curves and supports applications to L-functions, parity conjectures, and potential BSD verifications in the presence of ℓ-adic torsion or ramification.
minor comments (3)
- The abstract states the result clearly but the manuscript should include a brief comparison table (or explicit statement) of how the new root-number formula differs from the one in Yanagihara (2025) when ℓ divides r.
- Notation for the local root numbers at primes dividing ℓ should be introduced once in §2 or §3 and used consistently; the current text appears to switch between W_ℓ and w_ℓ without a single definition.
- The assumption that δ is ℓ^N-power-free is used repeatedly; a short remark confirming that this implies the model is minimal at ℓ would improve readability.
Simulated Author's Rebuttal
We thank the referee for the positive summary, recognition of the significance of extending the root-number computation by removing the assumption ℓ ∤ rst, and the recommendation of minor revision. The manuscript focuses on the explicit case ℓ^{N-1} || r with ℓ ∤ stδ for the curve y^{ℓ^N} = x^r (δ - x)^s.
Circularity Check
Minor self-citation to prior work; central computation independent
full rationale
The paper is explicitly a sequel to Yanagihara (2025) by the same author, extending the root number computation by dropping the assumption ℓ ∤ rst and addressing the specific case ℓ^{N-1} || r with ℓ ∤ stδ. The abstract and structure present this as a direct, targeted computation under delimited arithmetic conditions (odd prime ℓ, δ ℓ^N-power-free, exact valuation, etc.), building on but not defined by the prior paper. No equations reduce by construction to fitted inputs, no predictions are statistically forced from subsets, and no uniqueness theorems or ansatzes are smuggled via self-citation. The self-citation provides context and possibly auxiliary results but is not load-bearing for the new central claim, which remains an independent local computation. This qualifies as minor self-citation without significant circularity.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption ℓ is an odd prime
- domain assumption δ is ℓ^N-th-power-free
Reference graph
Works this paper leans on
-
[1]
Some congruences for binomial coefficients. , author =. Michigan Mathematical Journal , volume =. 1977 , publisher =
work page 1977
- [2]
-
[3]
Journal of Number Theory , volume =
On norm residue symbols and conductors , author =. Journal of Number Theory , volume =. 2001 , publisher =
work page 2001
-
[4]
Coleman, Robert and McCallum, William , journal =
-
[5]
Illinois journal of mathematics , volume =
Root numbers of Jacobi-sum Hecke characters , author =. Illinois journal of mathematics , volume =. 1992 , publisher =
work page 1992
- [6]
-
[7]
Inventiones mathematicae , volume =
Fr. Inventiones mathematicae , volume =. 1973 , publisher =
work page 1973
- [8]
- [9]
-
[10]
Zhi-Wei Sun , journal =
-
[11]
Gross, Benedict H and Rohrlich, David E , journal =. 1978 , publisher =
work page 1978
- [12]
- [13]
- [14]
- [15]
- [16]
-
[17]
Dokchitser, Vladimir and Maistret, C\'eline , TITLE =. Proc. Lond. Math. Soc. (3) , FJOURNAL =. 2023 , NUMBER =. doi:10.1112/plms.12545 , URL =
-
[18]
Grothendieck, Alexandre , journal=. Groupes de monodromie en g. 1973 , publisher=
work page 1973
-
[19]
On the arithmetic of abelian varieties , author=. 1972 , publisher=
work page 1972
-
[20]
Coleman, Robert F , journal=
-
[21]
Zeta and L-functions of varieties and motives , author=. 2020 , publisher=
work page 2020
- [22]
-
[23]
Functiones et Approximatio Commentarii Mathematici , volume=
Fleck's congruence, associated magic squares and a zeta identity , author=. Functiones et Approximatio Commentarii Mathematici , volume=. 2011 , publisher=
work page 2011
-
[24]
Zhi-Wei Sun, Daqing Wan , journal =. On Fleck quotients , url =
-
[25]
The PUMP Journal of Undergraduate Research , volume=
An Elementary Proof of Weisman's Congruence When p= 2 , author=. The PUMP Journal of Undergraduate Research , volume=
-
[26]
Finite Fields and Their Applications , volume=
Combinatorial congruences and -operators , author=. Finite Fields and Their Applications , volume=. 2006 , publisher=
work page 2006
-
[27]
The Fibonacci Quarterly , volume=
On the order of Stirling numbers and alternating binomial coefficient sums , author=. The Fibonacci Quarterly , volume=. 2001 , publisher=
work page 2001
-
[28]
Journal of Number Theory , volume=
A divisibility property for Stirling numbers , author=. Journal of Number Theory , volume=. 1978 , publisher=
work page 1978
-
[29]
On higher-dimensional Fibonacci numbers, Chebyshev polynomials and sequences of vector convergents , author=. Journal de th
-
[30]
Lucas’ Theorem Modulo p2 , volume=
Rowland, Eric , year=. Lucas’ Theorem Modulo p2 , volume=. The American Mathematical Monthly , publisher=. doi:10.1080/00029890.2022.2038004 , number=
-
[31]
Arithmetic properties of binomial coefficients, I: Binomial coefficients modulo prime powers , booktitle=. 1997 , organization=
work page 1997
- [32]
-
[34]
Journal of the London Mathematical Society , volume=
Conductor and discriminant of Picard curves , author=. Journal of the London Mathematical Society , volume=. 2020 , publisher=
work page 2020
-
[35]
I. I. Bouw, A. Koutsianas, J. Sijsling, and S. Wewers. Conductor and discriminant of picard curves. Journal of the London Mathematical Society , 102(1):368--404, 2020
work page 2020
-
[36]
R. Coleman and W. McCallum. Stable reduction of Fermat curves and Jacobi sum Hecke characters . J. reine angew. Math , 385(41):101, 1988
work page 1988
-
[37]
R. F. Coleman. Torsion points on abelian \'e tale coverings of ^1-\ 0, 1, \ . Transactions of the American Mathematical Society , 311(1):185--208, 1989
work page 1989
-
[38]
I. B. Fesenko and S. V. Vostokov. Local fields and their extensions , volume 121. American Mathematical Soc., 2002
work page 2002
-
[39]
D. E. Rohrlich. Root Numbers of Hecke L-Functions of CM Fields . American Journal of Mathematics , 104(3):517--543, 1982
work page 1982
-
[40]
D. E. Rohrlich. Root numbers of jacobi-sum hecke characters. Illinois journal of mathematics , 36(1):155--176, 1992
work page 1992
-
[41]
R. T. Sharifi. On norm residue symbols and conductors. Journal of Number Theory , 86(2):196--209, 2001
work page 2001
-
[42]
J. Shu. Root numbers for the Jacobian varieties of Fermat curves . Journal of Number Theory , 226:243--270, 2021
work page 2021
-
[43]
R. Yanagihara. Root numbers for twisted fermat quotient curves. arXiv preprint arXiv:2503.22991 , 2025. https://arxiv.org/abs/2503.22991
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.