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arxiv: 2312.04505 · v2 · submitted 2023-12-07 · 🧮 math.NT

On the change of epsilon factors for symmetric square transfers under twisting and applications

Tathagata Mandal , Sudipa Mondal This is my paper

Pith reviewed 2026-05-24 04:42 UTC · model grok-4.3

classification 🧮 math.NT
keywords epsilon factorssymmetric squaretwistingWeil-Deligne representationsconductorsmodular formsautomorphic representationslocal types
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The pith

The epsilon factor of the symmetric square transfer varies under twisting according to local Weil-Deligne data at each prime.

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

The paper studies how the epsilon factor of the symmetric square of an automorphic representation π changes when twisted by characters. It gives this variation explicitly in terms of the local Weil-Deligne representation at each prime p. This variation is then used to identify the type of the local symmetric square representation. The conductor of the symmetric square is also computed in terms of the level N of the original modular form.

Core claim

We study the variation of the epsilon factor of sym²(π) under twisting in terms of the local Weil-Deligne representation at each prime p. As an application, we detect the possible types of the symmetric square transfer of the local representation at p. Furthermore, as the conductor of sym²(π) is involved in the variation number, we compute it in terms of N.

What carries the argument

The local Weil-Deligne representation at each prime, which determines the change in the epsilon factor of sym²(π) under twisting.

If this is right

  • The type of the symmetric square transfer at a prime p can be detected from the epsilon factor variation.
  • The conductor of sym²(π) can be computed in terms of N.
  • This provides a method to study local components of the symmetric square lift via twisting.

Where Pith is reading between the lines

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

  • This technique might apply to other functorial lifts to determine their local types.
  • Global consequences for the conductors of symmetric square L-functions could be explored further.
  • Testing against known examples of modular forms could verify the formulas for specific cases.

Load-bearing premise

The symmetric square transfer of π exists as an automorphic representation whose local components are the symmetric squares of the local Weil-Deligne representations of π.

What would settle it

A specific modular form f of level N and a character χ such that the epsilon factor of sym²(π ⊗ χ) at some prime p does not match the predicted variation from the local Weil-Deligne representation.

read the original abstract

Let us consider the symmetric square transfer of the automorphic representation $\pi$ associated to a modular form $f \in S_k(N,\epsilon)$. In this article, we study the variation of the epsilon factor of ${\mathrm{sym}}^2(\pi)$ under twisting in terms of the local Weil-Deligne representation at each prime $p$. As an application, we detect the possible types of the symmetric square transfer of the local representation at $p$. Furthermore, as the conductor of ${\mathrm{sym}}^2(\pi)$ is involved in the variation number, we compute it in terms of $N$.

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.

Referee Report

0 major / 3 minor

Summary. The paper studies the variation of the epsilon factor of the symmetric square transfer sym²(π) of an automorphic representation π associated to a newform f ∈ S_k(N, ε) under twisting, expressed explicitly in terms of the local Weil-Deligne representation at each prime p. As an application it identifies the possible local types of sym²(π_p); it also derives an explicit formula for the conductor of sym²(π) in terms of the level N of f.

Significance. The explicit local formulas for the change in epsilon factors under twisting, together with the resulting conductor expression in terms of N, supply concrete computational tools for the symmetric-square lift. These rest on the standard identification of local components via the Gelbart–Jacquet lift and the usual Artin-conductor definition of local epsilon factors; when the derivations are verified they constitute a useful reference for applications involving conductors or local type detection.

minor comments (3)
  1. [Introduction / §1] The global statement of the main variation formula (presumably Theorem 1 or its local analogue) should be stated separately from the local computations so that the dependence on the Weil-Deligne parameters is immediately visible.
  2. [Notation section] Notation for the twisting character χ and the nebentypus ε should be distinguished more clearly when both appear in the same local formula.
  3. [Conductor computation] The conductor formula in terms of N is stated as a corollary; a short table or explicit list of the possible local contributions at each prime dividing N would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our work on the variation of epsilon factors under twisting for symmetric square transfers and the resulting conductor formula. The recommendation for minor revision is noted. No specific major comments appear in the report, so we provide no point-by-point responses below.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard local representation theory

full rationale

The paper studies variation of epsilon factors for sym²(π) under twisting via local Weil-Deligne representations, detects types of the lift at p, and computes the conductor of sym²(π) in terms of N. These rest on the known Gelbart–Jacquet lift (whose local components are Sym² of the WD representations) and the standard definition of local epsilon factors via Artin conductors and root numbers. No self-definitional equations, fitted inputs renamed as predictions, or load-bearing self-citations appear in the abstract or program. The central claims are explicit local computations that do not reduce to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only; no free parameters, invented entities, or ad-hoc axioms are mentioned. The work relies on standard background facts about epsilon factors and Weil-Deligne representations.

axioms (1)
  • standard math Epsilon factors and conductors of automorphic representations are determined by their local Weil-Deligne representations at each prime.
    Invoked when the variation under twisting is expressed in local terms and when the conductor is computed in terms of N.

pith-pipeline@v0.9.0 · 5626 in / 1366 out tokens · 28797 ms · 2026-05-24T04:42:15.830271+00:00 · methodology

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Reference graph

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