Epsilon factors of symplectic type characters in the wild case

Sazzad Ali Biswas

REVIEW 3 major objections 3 minor 13 references

Epsilon factors of symplectic type characters in the wild case

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Wildly ramified quadratic extensions now have explicit signs for the epsilon factors of all symplectic type characters, with conductor $2t+1$ or even $>2(t+1)$.

desk verdict Genuinely open wild-case computation with a sound conductor dichotomy, but Theorem 1.2 as stated is false: unit scaling of psi breaks the claimed psi-independence. read the letter →

arxiv 1908.05353 v2 pith:GETBTOP2 submitted 2019-08-14 math.NT

Sazzad Ali Biswas This is my paper · ORCID

classification math.NT MSC 11S3722E50

keywords localfieldsepsilonfactorssymplectictypecharacterswildlyramifiedquadraticextensionsLamprecht-TateformulaGausssumscharacterconductorsp-adic

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper determines the explicit sign of the local epsilon factor for every character of $K^{\times}$ of symplectic type, where $K/F$ is a wildly ramified quadratic extension of a $p$-adic field. A symplectic type character is one whose restriction to $F^{\times}$ is the quadratic character attached to $K$ by class field theory. The paper proves that such a character has conductor either $2t+1$ or an even integer larger than $2(t+1)$, with $t$ the ramification break of $K/F$ (Theorem 1.1). When the conductor is an even number $2d$, the epsilon factor is $\chi(-1)^d$ for a conductor-zero additive character trivial on $F$ (Theorem 1.2); when the conductor is $2t+1$, it is $\chi(-1)^l$ times a normalized Gauss sum $G(Q)$ (Theorem 1.3). Together these fill the wild-case gap left in earlier explicit computations of these signs for unramified and tamely ramified quadratic extensions.

What carries the argument

The load-bearing mechanism is the Lamprecht-Tate formula (Theorem 2.2 and Corollary 2.3), which replaces an epsilon factor by $\chi(c)$ times a normalized sum over unit classes once a suitable $c$ satisfies $\chi(1+y)=\psi(c^{-1}y)$ on a deep congruence subgroup. The proof combines this with two structural inputs: the conductor analysis of Theorem 1.1, which uses the norm map's behavior on unit groups in wildly ramified extensions, and the explicit choices $c'=\pi_K^{2d}$ for even conductor and $c'=\pi_K^{2(t+l+1)}$ for odd conductor. These choices force $\psi(c'^{-1})=1$ and $\chi(c')=\chi(-1)^d$ in the even case; in the odd case the surviving finite sum is the Gauss sum $G(Q)$.

What would settle it

Take $F=\mathbb{Q}_2$ and $K=F(\sqrt{\pi_F})$, choose a symplectic type character $\chi$ of even conductor $8$ and a conductor-zero additive character $\psi$ trivial on $F$, and compute $\epsilon(\chi,\psi)$ from formula (2.1), comparing it with $\chi(-1)^4$. Also check whether $\chi(1+x)=\psi(\pi_K^{-8}x)$ holds for all $x$ with $v_K(x)\ge 4$; a mismatch in the first comparison would refute Theorem 1.2, while a failure of the identity would show the proof's central reduction does not hold.

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Extended reading notes

Core claim

The central discovery is that the conductor and epsilon factor of a symplectic type character in a wildly ramified quadratic extension are controlled by the single ramification break $t$. The conductor is either $2t+1$ or an even integer strictly larger than $2(t+1)$. In the even case, the Lamprecht-Tate formula with $c'=\pi_K^{2d}$ reduces the epsilon factor to $\chi(\pi_K^{2d})\psi(\pi_K^{-2d})$; because $\psi$ is trivial on $F$, $\pi_F=N_{K/F}(\pi_K)=-\pi_K^2$, and $\pi_F$ lies in the norm group, this becomes $\chi(-1)^d$. In the odd conductor case $2t+1$, the same reduction leaves a normalized finite sum $G(Q)$ over $P_K^t/P_K^{t+1}$, giving $\epsilon(\chi,\psi)=\chi(-1)^l G(Q)$, where $l$ is fixed by $n(\psi)=2l+1$. The paper conjectures that $G(Q)=\pm 1$, which would make every wild symplectic epsilon factor equal to $\pm 1$.

Load-bearing premise

The load-bearing premise is an identity stated without proof in the proof of Theorem 1.2: the chosen additive character must satisfy $\chi(1+x)=\psi(\pi_K^{-2d}x)$ for all $x$ with valuation at least $d$ (equation (3.5)); if that identity fails, the reduction $\epsilon(\chi,\psi)=\chi(-1)^d$ collapses.

Editorial extensions

If this is right

In the even wild case the epsilon factor is read off directly from the character: $\epsilon(\chi,\psi)=\chi(-1)^d$, so no integral needs to be evaluated in applications.
In the odd wild case the remaining object is a single normalized finite sum $G(Q)$; if the paper's conjecture $G(Q)=\pm 1$ is correct, every wild symplectic epsilon factor is $\pm 1$.
The conductor classification $a(\chi)=2t+1$ or even $>2(t+1)$ gives a necessary condition on symplectic type characters in the wild case.
Together with the unramified and tame formulas, the wild formulas complete the explicit description of symplectic epsilon factor signs for all quadratic extensions of $p$-adic fields.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The proof's reliance on equation (3.5) suggests that Theorem 1.2 is established for those additive characters for which that identity holds; extending it to all conductor-zero $\psi$ trivial on $F$ would require an additional independence argument.
Remark 3.4 identifies $G(Q)$ as a quadratic Gauss sum over the residue field and an 8th root of unity, so the conjecture $G(Q)=\pm 1$ could be settled by a finite residue-field computation of the associated quadratic character's discriminant.
Explicit local signs of this form are natural inputs for global arguments that need root numbers of symplectic representations, so the wild-case formulas may feed into multiplicity and central-value questions beyond the local statement.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper attacks a genuine gap in Prasad's explicit sign program for symplectic type characters: the wildly ramified quadratic case. Theorem 1.1's conductor dichotomy (odd conductor is exactly 2t+1, even conductor is > 2(t+1)) is new, and the argument using class field theory and norm filtration is mostly sound. The strategy of plugging into the Lamprecht–Tate formula is also sensible, and the paper is clearly written.

The trouble is Theorem 1.2 as stated. The proof picks c' = pi_K^{2d} and asserts (3.5) with 'Here we see that'—no verification. That relation is load-bearing, and it cannot hold for every psi in the stated family. If a is a unit of F with omega_{K/F}(a) = -1 (which exists in the wild case), then psi_a(x) = psi(ax) still has conductor 0 and is trivial on F, so it belongs to the same family. But the standard scaling law gives epsilon(chi, psi_a) = omega(a)^{-1} epsilon(chi, psi) = -epsilon(chi, psi). If Theorem 1.2 held for all such psi, it would force epsilon(chi, psi_a) = epsilon(chi, psi), a contradiction. So the theorem can only hold for a distinguished psi, and that psi is never specified. The proof's claim 't is odd and t < 2e' is also false; K = Q_2(sqrt(2)) has t = 2, e = 1.

The odd conductor case, Theorem 1.3, has a related issue: the same asserted relation picks a specific c', and the conclusion still contains an unevaluated Gauss sum G(Q). The abstract says 'explicit signs,' but this is only a reduction to a Gauss sum that is conjectured to be ±1. In fact Conjecture 1 follows immediately from Prasad's theorem that epsilon is ±1, so it is not an independent open problem.

If the author revises to fix the psi quantification, supply the missing verification of (3.5) for a distinguished psi, and correct the auxiliary claims, the even-conductor theorem could become a real contribution. As it stands, the main theorem is not correct in the stated generality. Still, the underlying idea and the conductor result merit referee attention—this is not a desk-reject. I would encourage a revision, with a careful statement of the chosen psi and a proof of the unit-part compatibility.

Referee Report

3 major / 3 minor

Summary. This paper studies multiplicative characters χ of K^× for a wildly ramified quadratic extension K/F of a p-adic field of characteristic zero, with χ|_{F^×}=ω_{K/F} (symplectic type). Theorem 1.1 states that the conductor of such χ is either 2t+1 or an even integer strictly larger than 2(t+1), where t is the ramification break of K/F. Theorem 1.2 claims that for even conductor 2d (d>2) and any additive character ψ of K of conductor 0 that is trivial on F, the epsilon factor ε(χ,ψ) equals χ(-1)^d. Theorem 1.3 claims that for odd conductor 2t+1 and any ψ of conductor 2l+1 trivial on F, ε(χ,ψ)=χ(-1)^l G(Q), where G(Q) is a finite Gauss sum. The proofs use the Lamprecht-Tate formula, local duality, and class field theory, with the goal of completing Prasad's explicit sign computation in the wild case.

Significance. The topic is appropriate for a number theory journal: explicit epsilon factors for symplectic-type characters have applications to Tunnell-type results and to arithmetic invariants. The conductor restriction in Theorem 1.1 is useful and appears to follow from standard filtrations. The paper also correctly identifies the Lamprecht-Tate formula as the right tool. However, the main theorems as stated are too strong: the epsilon factor has a well-known dependence on the scaling of the additive character, and the proofs never control the unit part in the Lamprecht-Tate parameter. In consequence the central even-conductor claim is false over the stated family of additive characters, and the odd-conductor formula inherits the same defect. The positive core—the conductor bound and the possibility of choosing a distinguished additive character so that a formula of this shape holds—may well be salvageable, but it is not what the manuscript currently proves.

major comments (3)

[Section 3, proof of Theorem 1.2, equation (3.5)] The proof of Theorem 1.2 asserts without demonstration that c'=π_K^{2d} satisfies χ(1+x)=(c'^{-1}ψ)(x) for all x with v_K(x)≥d. This is exactly the Lamprecht-Tate relation that fixes the unit part of c', and it is not a consequence of n(ψ)=0 and ψ|_F=1. The theorem's quantification over all such ψ is in fact false: for any unit a∈O_F^×, the character ψ_a(x)=ψ(ax) belongs to the same family (Tr(ac)=0 and n(ψ_a)=0), while the standard scaling law gives ε(χ,ψ_a)=ω_{K/F}(a)ε(χ,ψ). In the wild case there is a unit a with ω_{K/F}(a)=-1, so the claimed ψ-independent value χ(-1)^d cannot hold for both ψ and ψ_a. Thus either Theorem 1.2 must be restricted to a distinguished additive character satisfying (3.5), or the formula must be corrected by the factor χ(c') built from the actual Lamprecht-Tate parameter.
[Section 3, Theorem 1.3] The same defect is present in Theorem 1.3. The proof fixes c'=π_K^{2(t+l+1)} and uses ψ(c'^{-1})=1 because c'^{-1}∈F. But for a scaled ψ_a with a∈O_F^×, the admissible Lamprecht-Tate parameter is ac', and then χ(ac')=ω_{K/F}(a)χ(c'), while G(Q) is unchanged when c' is replaced by ac'. Hence the asserted identity ε(χ,ψ)=χ(-1)^lG(Q) is not valid for every ψ of conductor 2l+1 in the family; it can hold only after the unit part of c' is selected to satisfy the Lamprecht-Tate relation, and the statement needs to make that selection explicit.
[Section 3, Conjecture 1] Conjecture 1 is not an open problem given the paper's other claims. If Prasad's theorem, cited in the Introduction, says ε(χ,ψ)=±1 for symplectic-type characters and Theorem 1.3 is true, then G(Q)=χ(-1)^l ε(χ,ψ)∈{±1}. The paper therefore either has a theorem, not a conjecture, after Theorem 1.3, or the proof of Theorem 1.3 has not actually evaluated the sign; in the latter case the advertised completion of Prasad's explicit sign determination is incomplete. Remark 3.4, which identifies G(Q) with the 8th-root-valued gamma factor γ(Q), makes this gap concrete and needs to be reconciled with Conjecture 1.

minor comments (3)

[Proof of Theorem 1.2, paragraph after Theorem 1.1] The assertion that 't is odd and t<2e' is false; for K=Q_2(√2) one has t=2 and e=1. The conclusion d>2 needed in the proof follows already from d>t+1 and t≥1.
[Proof of Theorem 1.3] The displayed computation ε(χ,ψ)=χ(c')ψ^{-1}(c'^{-1})·G(Q) contains an inverse on ψ; Corollary 2.3(2) gives ψ(c'^{-1}), not ψ^{-1}(c'^{-1}).
[Remark 3.5] The application in equation (3.9) does not specify in which ramification context the displayed formula is asserted; please clarify whether it is intended for the wild case or for the tame/unramified discussion that precedes it.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the sign computations are derived from independent external results (Lamprecht–Tate and Prasad) and do not reduce to their own inputs.

full rationale

The derivation chain is not circular. Theorem 1.1 uses local class field theory and Serre's norm filtration to constrain the conductor; Theorems 1.2 and 1.3 apply the external Lamprecht–Tate formula (Theorem 2.2, Corollary 2.3) with a c' chosen in K, computing ε(χ,ψ) as χ(c')ψ(c'^{-1}) or as χ(c')ψ(c'^{-1})G(Q). The target signs χ(-1)^d and χ(-1)^lG(Q) are not inserted as hypotheses; they are outputs of evaluating those external formulas. There are no fitted constants, no known results renamed, and no load-bearing self-citations: [7] and [9] are independent published inputs. The paper's own text flags a weakness at equation (3.5), where it says "Here we see that our c' = π_K^{2d} satisfies equation (3.5)" without proof; that is an unproved assertion (and a possible correctness gap), but it is not circular, since the proof would be a genuine derivation if (3.5) were supplied as a lemma. Likewise the claim "t is odd" is false in some wildly ramified extensions, but this is an error, not a self-referential reduction. The scaling objection to the universal quantification over ψ is a correctness concern, not circularity. Conjecture 1 being a consequence of Theorem 1.3 plus Prasad's theorem is a presentation oddity, not circular reasoning. Hence the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted numerical parameters: choices such as π_K, ψ_F and ψ are normalizations fixed by the theorems, and the dualizing element c' is determined by the formula ν_K(c') = a(χ) + n(ψ). No new entities are postulated; G(Q) is a defined Gauss sum built from χ and ψ, not a new object. The main assumptions are standard background facts plus one asserted, unproven character-relation (3.5) that carries the weight of both theorems.

assumptions (6)

standard math Local class field theory: a unique quadratic character ω_{K/F} of F^× trivial on N_{K/F}(K^×) exists and parametrizes the extension K/F.
Invoked throughout, including Section 2 and the proof of Theorem 1.1, to define symplectic type characters and to justify ω_{K/F}(π_F) = 1 for norms.
standard math The Lamprecht-Tate formula (Theorem 2.2) and its corollaries (2.3), cited from Tate [9] and Langlands [4], correctly evaluate ε(χ,ψ) as the stated sums.
Central computational tool; the paper adds no proof of the formula and relies on it verbatim in both main theorems.
standard math Norm images of unit groups: N_{K/F}(U_K^{Ψ(n)}) = U_F^n for n > t and N_{K/F}(U_K^{Ψ(n)+1}) = U_F^{n+1} for n > t, as cited from Serre [8].
Used in the proof of Theorem 1.1 to fix the conductor of ω_{K/F} as t+1.
standard math Hasse-Arf: ramification breaks of an abelian extension are integers (Fesenko-Vostokov [2]).
Used to treat the single integer break t of the quadratic extension as a well-defined quantity.
standard math Prasad's theorem [7]: the epsilon factor of a symplectic type character is ±1 in general.
Background that fixes the target value; the paper does not reprove it, and it makes Conjecture 1 a consequence of Theorem 1.3.
ad hoc to paper The chosen additive character ψ of conductor 0 and trivial on F satisfies χ(1+x) = ψ(π_K^{-2d}x) for all x with v_K(x) ≥ d, i.e., equation (3.5) holds for c' = π_K^{2d}.
Asserted without proof in Theorem 1.2 ('Here we see that') and again for Theorem 1.3. It is load-bearing for applying Corollary 2.3, and its validity for every ψ in the stated family is not established and is in tension with the ψ-dependence of epsilon factors.

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Cite this review

Pith. "Pith review of Epsilon factors of symplectic type characters in the wild case." pith.science (2026). https://pith.science/paper/GETBTOP2

@misc{pith2026190805353,
  author       = {Pith},
  title        = {Pith review of: Epsilon factors of symplectic type characters in the wild case},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GETBTOP2}},
  note         = {Machine review of arXiv:1908.05353}
}

abstract

By work of John Tate we can associate an epsilon factor with every multiplicative character of a local field. In this paper we determine the explicit signs of the epsilon factors for symplectic type characters of $K^\times$, where $K/F$ is a wildly ramified quadratic extension of a non-Archimedean local field $F$ of characteristic zero.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

P. Deligne, Les constantes des ´ equations fonctionnelle des fonctionsL, in Modular functions of one variable II, Lecture Notes in Mathematics 349 (1972), 501-597, Springer-Verlag, Berlin-Heidelberg-New York

work page 1972
[2]

Fesenko, S.V

I.B. Fesenko, S.V. Vostokov, Local ﬁelds and their extensions, second edition, Translation of Mathematical Monographs, Vol. 121, American Mathematical Society, 2002

work page 2002
[3]

Lectures Notes in Mathematics 462, Berlin 1975

Paul Gerardin, Construction de series discrete p-adiques. Lectures Notes in Mathematics 462, Berlin 1975

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[4]

Langlands, On the functional equation of the Artin L-function, https://publications.ias.edu/sites/ default/files/a-ps.pdf

R. Langlands, On the functional equation of the Artin L-function, https://publications.ias.edu/sites/ default/files/a-ps.pdf

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[5]

R. Lidl, H. Niederreiter, Finite ﬁelds, Encyclopedia of Mathematics and its applications, Cambridge Uni- versity press 2000

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Neukirch, Algebraic Number Theory, Springer-Verlag Berlin Heidelberg, 1999

J. Neukirch, Algebraic Number Theory, Springer-Verlag Berlin Heidelberg, 1999

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Prasad, On an extension of a theorem of Tunnell, Compositio Math

D. Prasad, On an extension of a theorem of Tunnell, Compositio Math. 94 (1994), 19-28

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[8]

Serre, Local ﬁelds, Graduate texts in mathematics; 67, 1979, Springer-Verlag New York Inc

J-P. Serre, Local ﬁelds, Graduate texts in mathematics; 67, 1979, Springer-Verlag New York Inc

work page 1979

Show all 13 references

[9]

Tate, Local Constants, Algebraic Number Fields (L-functions and Galois properties), Proceedings of Symposium, Edited by A

J. Tate, Local Constants, Algebraic Number Fields (L-functions and Galois properties), Proceedings of Symposium, Edited by A. Fr´ ohlich, 1977, pp. 89-131

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[10]

Tate, Number theoretic background, Proceedings of Symposia in Pure Mathematics, Vol

J. Tate, Number theoretic background, Proceedings of Symposia in Pure Mathematics, Vol. 33 (1979), Part 2, pp.3-26

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M. J. Taylor, On Fr¨ ohlich’s conjecture for rings of intergers of tame extensions, Invent. Math. 63 (1981)

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Tunnell, Local epsilon factors and characters of GL(2), American Journal of Math

J. Tunnell, Local epsilon factors and characters of GL(2), American Journal of Math. 105 (1983), 1277-1307

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[13]

Weil, Basic Number Theory, Third edition, Springer-Verlag, 1974

A. Weil, Basic Number Theory, Third edition, Springer-Verlag, 1974. Einstein Institute of Mathematics, Hebrew University of Jerusalem, Givat Ram, Jerusalem, 91904, Israel Email address : sazzad.biswas@mail.huji.ac.il, sazzad.jumath@gmail.com

1974

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

P. Deligne, Les constantes des ´ equations fonctionnelle des fonctionsL, in Modular functions of one variable II, Lecture Notes in Mathematics 349 (1972), 501-597, Springer-Verlag, Berlin-Heidelberg-New York

work page 1972

[2] [2]

Fesenko, S.V

I.B. Fesenko, S.V. Vostokov, Local ﬁelds and their extensions, second edition, Translation of Mathematical Monographs, Vol. 121, American Mathematical Society, 2002

work page 2002

[3] [3]

Lectures Notes in Mathematics 462, Berlin 1975

Paul Gerardin, Construction de series discrete p-adiques. Lectures Notes in Mathematics 462, Berlin 1975

work page 1975

[4] [4]

Langlands, On the functional equation of the Artin L-function, https://publications.ias.edu/sites/ default/files/a-ps.pdf

R. Langlands, On the functional equation of the Artin L-function, https://publications.ias.edu/sites/ default/files/a-ps.pdf

work page

[5] [5]

R. Lidl, H. Niederreiter, Finite ﬁelds, Encyclopedia of Mathematics and its applications, Cambridge Uni- versity press 2000

work page 2000

[6] [6]

Neukirch, Algebraic Number Theory, Springer-Verlag Berlin Heidelberg, 1999

J. Neukirch, Algebraic Number Theory, Springer-Verlag Berlin Heidelberg, 1999

work page 1999

[7] [7]

Prasad, On an extension of a theorem of Tunnell, Compositio Math

D. Prasad, On an extension of a theorem of Tunnell, Compositio Math. 94 (1994), 19-28

work page 1994

[8] [8]

Serre, Local ﬁelds, Graduate texts in mathematics; 67, 1979, Springer-Verlag New York Inc

J-P. Serre, Local ﬁelds, Graduate texts in mathematics; 67, 1979, Springer-Verlag New York Inc

work page 1979