On the Lang--Trotter conjecture for Siegel modular forms

Ariel Weiss; Arvind Kumar; Moni Kumari

arxiv: 2201.09278 · v2 · pith:GDRQPEYHnew · submitted 2022-01-23 · 🧮 math.NT

On the Lang--Trotter conjecture for Siegel modular forms

Arvind Kumar , Moni Kumari , Ariel Weiss This is my paper

Pith reviewed 2026-05-24 12:36 UTC · model grok-4.3

classification 🧮 math.NT

keywords Siegel modular formsGalois representationsopen image theoremHecke eigenvaluesLang-Trotter conjectureadelic groupsgenus two

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The pith

An adelic open image theorem for Galois representations attached to genus two Siegel modular eigenforms implies upper bounds on the number of primes where a fixed Hecke eigenvalue occurs.

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

The paper proves an adelic open image theorem for the compatible system of Galois representations attached to a genus two cuspidal Siegel modular eigenform. This generalizes the corresponding results of Ribet and Momose that were known for elliptic modular forms. The theorem is applied to control the distribution of the Hecke eigenvalues a_p and to produce explicit upper bounds on the size of the set of primes p up to x at which a_p equals any fixed complex number a. A reader would care because the bounds give concrete information on how often these eigenvalues can repeat, in the same spirit as the Lang-Trotter conjecture for elliptic curves.

Core claim

We prove an adelic open image theorem for the compatible system of Galois representations associated to f, generalising the results of Ribet and Momose for elliptic modular forms. Using this result, we investigate the distribution of the Hecke eigenvalues a_p of f, and obtain upper bounds for the sizes of the sets {p ≤ x : a_p = a} for fixed a in C, in the spirit of the Lang-Trotter conjecture for elliptic curves.

What carries the argument

The adelic open image theorem asserting that the image of the Galois representation attached to the Siegel form is open inside the adelic points of the associated algebraic group.

If this is right

The Galois image attached to any such Siegel form is open in its adelic completion.
For every fixed complex number a the set of primes p ≤ x with a_p(f) = a has size bounded above by a quantity smaller than x.
The same method yields distribution results for the eigenvalues in the spirit of the Lang-Trotter conjecture.
The open-image statement applies uniformly to all genus two cuspidal Siegel eigenforms.

Where Pith is reading between the lines

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

The same adelic-open-image strategy could be tested on Siegel forms of genus greater than two once the corresponding Galois representations are constructed.
Numerical checks of the eigenvalue bounds could be performed for low-weight explicit Siegel forms by computing Hecke eigenvalues at many primes.
The technique may adapt to other families of automorphic forms whose Galois representations are known to form compatible systems.

Load-bearing premise

A compatible system of Galois representations with the expected properties exists for every genus two cuspidal Siegel modular eigenform.

What would settle it

An explicit genus two cuspidal Siegel modular eigenform whose Galois image fails to be open in the adelic sense, or for which the count of p ≤ x with a_p equal to a fixed a exceeds the derived upper bound.

read the original abstract

Let $f$ be a genus two cuspidal Siegel modular eigenform. We prove an adelic open image theorem for the compatible system of Galois representations associated to $f$, generalising the results of Ribet and Momose for elliptic modular forms. Using this result, we investigate the distribution of the Hecke eigenvalues $a_p$ of $f$, and obtain upper bounds for the sizes of the sets $\{p \le x : a_p = a\}$ for fixed $a\in\mathbf{C}$, in the spirit of the Lang--Trotter conjecture for elliptic curves.

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

The paper generalizes the adelic open image theorem to genus-2 Siegel forms and derives Lang-Trotter bounds, but the scope depends on unstated conditions for the Galois representations.

read the letter

The main takeaway is that this work extends Ribet-Momose style adelic open image results from elliptic modular forms to the four-dimensional Galois representations attached to genus two cuspidal Siegel eigenforms, then uses the image control to bound the number of primes p ≤ x with a fixed Hecke eigenvalue a_p = a. The new part is the move to the GSp(4) setting and the resulting distribution statements, which do not reduce directly to the elliptic case. The paper follows the standard adelic framework and applies it cleanly to obtain the upper bounds in the spirit of the Lang-Trotter conjecture. That part of the argument looks straightforward once the open image theorem is in hand. The citation pattern is appropriate, pointing back to the elliptic precedents without circularity. The soft spot is the existence and basic properties of the compatible system ρ_f. The abstract invokes it without qualification, but for Siegel forms this ultimately traces to Weissauer's construction, which requires k ≥ 3 or additional endoscopic assumptions in some weight ranges. If the paper does not explicitly restrict the statement to those cases or supply a new existence proof, the open image theorem and the eigenvalue bounds are conditional on an external result whose range may not match the claimed generality. That needs checking in the body. The work is aimed at arithmetic geometers and automorphic forms people who care about higher-genus Galois images and eigenvalue distributions. A reader already comfortable with the elliptic case will see the value in the extension. It deserves a serious referee because the result is a targeted but genuine step forward in the subfield, even if revisions are needed on the scope of the hypotheses.

Referee Report

2 major / 1 minor

Summary. The paper proves an adelic open-image theorem for the compatible system of Galois representations attached to a genus-two cuspidal Siegel modular eigenform f, generalizing Ribet–Momose results for elliptic modular forms. It then derives upper bounds on the cardinality of {p ≤ x : a_p(f) = a} for fixed a ∈ ℂ, in the direction of the Lang–Trotter conjecture.

Significance. If the stated theorems hold under precisely the hypotheses where the underlying Galois representations are known to exist, the work supplies a concrete higher-genus extension of the Ribet–Momose open-image machinery and yields the first explicit Lang–Trotter-type bounds for Siegel forms. The generalization itself is a clear technical advance.

major comments (2)

[Abstract] Abstract (and presumably §1): the statement “let f be a genus two cuspidal Siegel modular eigenform” and “the compatible system of Galois representations associated to f” invokes existence without qualification. Weissauer’s construction of the 4-dimensional compatible system ρ_f : Gal(ℚ̄/ℚ) → GSp_4(𝔸_f) is known to require k ≥ 3 (or endoscopic/non-endoscopic restrictions) in certain weight ranges; the paper must either restrict the main theorems to those cases or prove existence anew. This directly affects the scope of both the open-image theorem and the subsequent distribution bounds.
[Main theorems (location not specified in abstract)] The open-image theorem is presented as unconditional for arbitrary such f, yet the subsequent Lang–Trotter bounds inherit the same hypothesis. If the paper does not add an explicit hypothesis list (e.g., “assume f is of weight k ≥ 3 and non-endoscopic”) at the beginning of the main theorems, the claimed generality is overstated.

minor comments (1)

Notation for the adelic image and the precise target group GSp_4(𝔸_f) should be introduced with a short reminder of the determinant and Hodge–Tate weight conditions that are used in the open-image argument.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments regarding the hypotheses on the existence of the Galois representations. We agree that the statements of the main results require explicit qualification to align with the known range of Weissauer's construction, and we will revise the manuscript to incorporate the necessary assumptions.

read point-by-point responses

Referee: [Abstract] Abstract (and presumably §1): the statement “let f be a genus two cuspidal Siegel modular eigenform” and “the compatible system of Galois representations associated to f” invokes existence without qualification. Weissauer’s construction of the 4-dimensional compatible system ρ_f : Gal(ℚ̄/ℚ) → GSp_4(𝔸_f) is known to require k ≥ 3 (or endoscopic/non-endoscopic restrictions) in certain weight ranges; the paper must either restrict the main theorems to those cases or prove existence anew. This directly affects the scope of both the open-image theorem and the subsequent distribution bounds.

Authors: We agree that the existence of the compatible system relies on Weissauer's results, which impose conditions such as weight k ≥ 3 (with possible endoscopic restrictions). The manuscript does not prove existence anew. In the revised version we will add an explicit hypothesis statement at the beginning of the main theorems, restricting the results to the cases where the representations are known to exist. revision: yes
Referee: [Main theorems (location not specified in abstract)] The open-image theorem is presented as unconditional for arbitrary such f, yet the subsequent Lang–Trotter bounds inherit the same hypothesis. If the paper does not add an explicit hypothesis list (e.g., “assume f is of weight k ≥ 3 and non-endoscopic”) at the beginning of the main theorems, the claimed generality is overstated.

Authors: We concur that an explicit hypothesis list is needed to avoid any overstatement of generality. The revised manuscript will include such a list (specifying weight k ≥ 3 and non-endoscopic, as appropriate) at the start of the statements of both the adelic open-image theorem and the ensuing distribution bounds. revision: yes

Circularity Check

0 steps flagged

No circularity: theorem generalizes independent prior results of Ribet and Momose

full rationale

The paper proves an adelic open-image theorem for the Galois representations attached to genus-2 Siegel eigenforms and derives Lang-Trotter-type bounds on Hecke eigenvalue multiplicities. Both steps are framed as direct generalizations of the independent theorems of Ribet and Momose; the derivation chain invokes the existence of the compatible system from external literature (Weissauer) but does not reduce the new open-image statement or the distribution bounds to any fitted parameter, self-definition, or load-bearing self-citation. No equation or claim collapses to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; the paper relies on the standard existence of Galois representations for Siegel modular forms but provides no further detail on free parameters or invented entities.

axioms (1)

domain assumption Existence of a compatible system of Galois representations attached to a genus two cuspidal Siegel modular eigenform
Invoked in the abstract to state the open image theorem; treated as background from prior literature.

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discussion (0)

Forward citations

Cited by 1 Pith paper

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

Bounds for the distribution of the Frobenius traces associated to a generic abelian variety
math.NT 2022-07 unverdicted novelty 5.0

Under GRH, the count of primes p ≤ x with Frobenius trace a_{1,p}(A) = t is ≪ x to a power strictly less than 1, yielding that |a_{1,p}(A)| exceeds p to a positive power for almost all p.

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages · cited by 1 Pith paper

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MR 3135650 ↑2 [AS06] Mahdi Asgari and Freydoon Shahidi, Generic transfer from GSp(4) to GL(4), Compos

Orthogonal and symplectic groups. MR 3135650 ↑2 [AS06] Mahdi Asgari and Freydoon Shahidi, Generic transfer from GSp(4) to GL(4), Compos. Math. 142 (2006), no. 3, 541–550. MR 2231191 ↑2 [BPP+19] Armand Brumer, Ariel Pacetti, Cris Poor, Gonzalo Tornar ía, John Voight, and David S. Yuen, On the paramodularity of typical abelian surfaces , Algebra Number Theo...

work page arXiv 2006

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MR 0568299 ↑1 [Mit14] Howard H

Distrib ution of Frobenius automorphisms in GL2- extensions of the rational numbers. MR 0568299 ↑1 [Mit14] Howard H. Mitchell, The subgroups of the quaternary abelian linear group , Trans. Amer. Math. Soc. 15 (1914), no. 4, 379–396. MR 1500986 ↑13 [MMS88] M. Ram Murty, V. Kumar Murty, and N. Saradha, Modular forms and the Chebotarev density theorem , Amer...

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MR 0453647 ↑3 [Rib80] , Twists of modular forms and endomorphisms of abelian variet ies, Math. Ann. 253 (1980), no. 1, 43–62. MR 594532 ↑3, 7 [Rib85] , On l-adic representations attached to modular forms. II , Glasgow Math. J. 27 (1985), 185–

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MR 819838 ↑2, 3, 7, 11, 14, 15 [Ser18] Jean-Pierre Serre, On the mod p reduction of orthogonal representations , Lie groups, geometry, and representation theory, 2018, pp. 527–540. MR 3890220 ↑6, 8 [Ser81] , Quelques applications du théorème de densité de Chebotarev , Inst. Hautes Études Sci. Publ. Math. 54 (1981), 323–401. MR 644559 ↑2, 4, 5, 16, 17, 19 ...

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MR 1484415 ↑2, 3, 4, 13, 14 [Tay91] Richard Taylor, Galois representations associated to Siegel modular forms of low weight , Duke Math

With the collaboration o f Willem Kuyk and John Labute, Revised reprint of the 1968 original. MR 1484415 ↑2, 3, 4, 13, 14 [Tay91] Richard Taylor, Galois representations associated to Siegel modular forms of low weight , Duke Math. J. 63 (1991), no. 2, 281–332. MR 1115109 ↑6 [Tay93] , On the l-adic cohomology of Siegel threefolds , Invent. Math. 114 (1993)...

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