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arxiv: 2606.30063 · v1 · pith:GTPJ7I4Lnew · submitted 2026-06-29 · 🧮 math.NT

Harder's conjecture and Hermitian automorphic forms

Hidenori Katsurada , Nobuki Takeda This is my paper

Pith reviewed 2026-06-30 05:27 UTC · model grok-4.3

classification 🧮 math.NT
keywords Harder's conjectureHermitian cusp formsKlingen-Eisenstein liftsspin liftscongruencesunitary groupsSiegel modular forms
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The pith

Under explicit arithmetic hypotheses on a congruence prime, a Hermitian cusp eigenform congruent to a Klingen-Eisenstein lift is the Hermitian spin lift of a Siegel cusp eigenform of weight det^k Sym^j, which yields the spinor L-polynomial

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

The paper studies congruences between the Hermitian Klingen-Eisenstein lift attached to a primitive elliptic cusp form f of weight 2k+j-2 and actual Hermitian cusp forms on the quasi-split unitary group U_{2,2}. Under explicit arithmetic hypotheses on the congruence prime, it proves that any such cusp eigenform must arise as the Hermitian spin lift of a Siegel cusp eigenform of weight det^k Sym^j. This identification directly implies that the spinor L-polynomials of the two forms agree, confirming the relation asserted by Harder's conjecture. The argument proceeds by combining Mok's endoscopic classification of automorphic representations, Skinner's construction of Galois representations for unitary groups, and vanishing results for associated Selmer groups.

Core claim

Let k ≥ 4 and j ≥ 2 be integers with j even, and let f be a primitive elliptic cusp form of weight 2k+j-2 for SL_2(Z). For a congruence prime satisfying the stated arithmetic hypotheses, every Hermitian cusp eigenform on U_{2,2} that is congruent to the Klingen-Eisenstein lift of f is the Hermitian spin lift of a Siegel cusp eigenform of weight det^k Sym^j; consequently the spinor L-polynomials of these forms satisfy the congruence predicted by Harder's conjecture.

What carries the argument

The Hermitian spin lift that maps Siegel cusp eigenforms of weight det^k Sym^j to Hermitian cusp eigenforms on U_{2,2}, together with the endoscopic classification, Galois representations, and Selmer-group vanishing used to identify the lift inside the congruence.

If this is right

  • The spinor L-polynomials of the Hermitian cusp form and the Siegel cusp form agree at all primes.
  • Harder's predicted L-polynomial congruence holds for all such congruent pairs under the given hypotheses.
  • Properties of Siegel forms that are preserved under spin lifting, such as certain Hecke eigenvalues, transfer to the corresponding Hermitian forms.

Where Pith is reading between the lines

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

  • If the arithmetic hypotheses on congruence primes can be verified for a larger class of forms, the method would produce many new instances of Harder's conjecture.
  • The same combination of endoscopic transfer and Selmer vanishing might apply to congruences on other unitary groups of higher rank.
  • Computational checks for small even j and moderate k could locate the first concrete primes where the hypotheses hold and the L-polynomial match can be verified directly.

Load-bearing premise

The explicit arithmetic hypotheses on the congruence prime are strong enough to guarantee that endoscopic classification, Galois representations, and Selmer vanishing together force the congruent cusp form to be the spin lift.

What would settle it

An explicit pair of weights k and j together with a prime p satisfying the arithmetic hypotheses, for which there exists a Hermitian cusp eigenform congruent to the Klingen-Eisenstein lift whose spinor L-polynomial modulo p differs from that of every Siegel form of weight det^k Sym^j.

read the original abstract

Let $k\ge4$ and $j\ge2$ be integers with $j$ even, and let $f$ be a primitive elliptic cusp form of weight $2k+j-2$ for $\mathrm{SL}_2(\mathbb{Z})$. We study congruences between a Hermitian Klingen--Eisenstein lift associated with $f$ and Hermitian cusp forms on the quasi-split unitary group $\mathrm{U}_{2,2}$. Under explicit arithmetic hypotheses on a congruence prime, we prove that the Hermitian cusp eigenform appearing in such a congruence is the Hermitian spin lift of a Siegel cusp eigenform of weight ${\det}^{k}\mathrm{Sym}^{j}$. As a consequence, we obtain the spinor $L$-polynomial congruence predicted by Harder's conjecture. The proof combines Mok's endoscopic classification, Skinner's Galois representations for unitary groups, and Selmer-group vanishing arguments.

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 / 2 minor

Summary. The paper studies congruences between Hermitian Klingen-Eisenstein lifts associated to primitive elliptic cusp forms f of weight 2k+j-2 (k≥4, j≥2 even) and Hermitian cusp forms on the quasi-split unitary group U(2,2). Under explicit arithmetic hypotheses on a congruence prime, it proves that any such Hermitian cusp eigenform is the Hermitian spin lift of a Siegel cusp eigenform of weight det^k Sym^j. As a consequence, the spinor L-polynomial congruence predicted by Harder's conjecture holds. The argument combines Mok's endoscopic classification, Skinner's Galois representations for unitary groups, and Selmer-group vanishing.

Significance. If the result holds, it advances Harder's conjecture by establishing the predicted lifting and L-polynomial congruence in the Hermitian setting, conditional on arithmetic hypotheses that enable the application of three independent prior tools. The approach is a direct and standard combination of endoscopic classification, Galois representations, and Selmer vanishing, yielding a falsifiable prediction as a consequence without introducing free parameters or self-referential definitions.

minor comments (2)
  1. The abstract refers to 'explicit arithmetic hypotheses on a congruence prime' without listing them; the introduction or §2 should state these hypotheses verbatim (including any conditions on the prime, the level, or the residual representation) so that the applicability of Mok, Skinner, and Selmer vanishing can be checked directly.
  2. Notation for the Hermitian spin lift and the Siegel form of weight det^k Sym^j should be introduced with a precise reference to the relevant endoscopic transfer map or L-parameter in the first section where they appear.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation of minor revision. The report accurately summarizes our conditional result combining Mok's endoscopic classification, Skinner's Galois representations, and Selmer vanishing to establish the Hermitian spin lift and the predicted spinor L-polynomial congruence under explicit arithmetic hypotheses. No major comments were provided.

Circularity Check

0 steps flagged

No circularity; derivation rests on independent external results

full rationale

The paper's central claim is a conditional lifting result obtained by combining three cited external theorems (Mok endoscopic classification, Skinner Galois representations, Selmer vanishing) whose statements are independent of the present work and do not depend on its conclusions. No equation or definition in the abstract reduces the predicted L-polynomial congruence to a fitted parameter, self-referential ansatz, or self-citation chain. The explicit arithmetic hypotheses are invoked as sufficient conditions rather than being derived from the target statement. This is the standard non-circular pattern for such conditional automorphic lifting arguments.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on three external theorems whose statements are taken as given; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract itself.

axioms (3)
  • domain assumption Mok's endoscopic classification of automorphic representations on unitary groups
    Invoked to identify the possible automorphic forms appearing in the congruence.
  • domain assumption Skinner's construction of Galois representations attached to Hermitian forms on unitary groups
    Used to translate the congruence into a statement about Galois representations.
  • domain assumption Vanishing of certain Selmer groups controlling the congruences
    Applied to deduce that the cusp form must be the indicated lift.

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