Discrete series multiplicities for classical groups over Z and level 1 algebraic cusp forms

Ga\"etan Chenevier; Olivier Ta\"ibi

arxiv: 1907.08783 · v1 · pith:QOEMK5OXnew · submitted 2019-07-20 · 🧮 math.NT · math.AG· math.RT

Discrete series multiplicities for classical groups over Z and level 1 algebraic cusp forms

Ga\"etan Chenevier , Olivier Ta\"ibi This is my paper

Pith reviewed 2026-05-24 19:00 UTC · model grok-4.3

classification 🧮 math.NT math.AGmath.RT

keywords level one automorphic formsclassical groupsGL_ncuspidal representationsSiegel modular formstrace formulaWeil explicit formuladiscrete series multiplicities

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The pith

The Weil explicit formula determines multiplicities of discrete series for level one forms on classical groups and classifies algebraic cusp forms on GL_n up to motivic weight 24.

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

This paper introduces a method to compute the multiplicity of any given discrete series in the space of level one automorphic forms on a split classical group over the integers. The method relies on the Weil explicit formula to produce vanishing results that fix the elliptic contribution to the trace formula for an appropriate test function. The same vanishing results yield a complete classification of level one cuspidal algebraic automorphic representations of GL_n over Q whose motivic weight is at most 24. These computations recover all previously known dimensions of spaces of Siegel modular cusp forms on Sp_{2g}(Z) and extend them in a uniform manner, with explicit numerical results for groups of absolute rank at most 8.

Core claim

A suitable choice of test function in the Weil explicit formula produces sufficiently strong vanishing statements to determine the elliptic part of the geometric side of the trace formula, which in turn classifies all level one cuspidal algebraic automorphic representations of GL_n over Q with motivic weight at most 24 and gives the multiplicities of discrete series in level one automorphic forms on split classical groups G over Z.

What carries the argument

The Weil explicit formula applied with a suitable test function to produce vanishing results that determine the elliptic contribution in the trace formula.

If this is right

Dimensions of spaces of Siegel modular cusp forms for Sp_{2g}(Z) are obtained for all g such that the absolute rank is at most 8.
All previously known dimensions for these Siegel spaces are recovered without using earlier case-by-case arguments.
Multiplicities are obtained for every discrete series in the level one spectrum of classical groups of absolute rank at most 8.
The classification exhausts all possibilities for level one algebraic cusp forms on GL_n of any rank when the motivic weight is at most 24.

Where Pith is reading between the lines

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

Refinements of the test function could push the classification to higher motivic weights.
The vanishing technique might adapt to compute multiplicities for non-split forms or other reductive groups over Z.
The classification imposes concrete constraints on the possible Galois representations or motives attached to these automorphic forms.

Load-bearing premise

The Weil explicit formula can be applied with a suitable test function to produce vanishing results that are strong enough to determine the elliptic contribution in the trace formula for the groups considered.

What would settle it

Existence of a level one algebraic cusp form on GL_n with motivic weight at most 24 whose parameters lie outside the listed classification, or a computed multiplicity for a discrete series in a classical group of rank at most 8 that differs from an independent count.

read the original abstract

The aim of this paper is twofold. First, we introduce a new method for evaluating the multiplicity of a given discrete series in the space of level $1$ automorphic forms of a split classical group $G$ over $\mathbb{Z}$, and provide numerical applications in absolute rank $\leq 8$. Second, we prove a classification result for the level one cuspidal algebraic automorphic representations of ${\rm GL}_n$ over $\mathbb{Q}$ ($n$ arbitrary) whose motivic weight is $\leq 24$. In both cases, a key ingredient is a classical method based on the Weil explicit formula, which allows to disprove the existence of certain level one algebraic cusp forms on ${\rm GL}_n$, and that we push further on in this paper. We use these vanishing results to obtain an arguably ``effortless'' computation of the elliptic part of the geometric side of the trace formula of $G$, for an appropriate test function. Thoses results have consequences for the computation of the dimension of the spaces of (possibly vector-valued) Siegel modular cuspforms for ${\rm Sp}_{2g}(\mathbb{Z})$: we recover all the previously known cases without relying on any, and go further, by a unified and ``effortless'' method.

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

Chenevier-Taibi classify level-1 GL_n algebraic cusp forms up to motivic weight 24 via stronger Weil vanishing and give a unified multiplicity method for classical groups up to rank 8.

read the letter

The main news is that they classify all level-1 cuspidal algebraic automorphic representations of GL_n over Q with motivic weight at most 24, and they supply a new route to the multiplicities of discrete series in level-1 forms on split classical groups, with concrete numbers through absolute rank 8. They recover and extend the known dimensions of Siegel cusp forms on Sp(2g,Z) without leaning on earlier case-by-case work. The approach rests on pushing the Weil explicit formula to produce vanishing results strong enough to kill off most candidates, then feeding those vanishings into the trace formula so the elliptic term becomes almost immediate. Existence for the surviving cases is taken from prior constructions, so the argument splits cleanly into non-existence from the formula and existence from the literature. This produces a single framework that matches everything previously known and reaches further, up to weight 24. The numerical output for the classical groups looks reproducible once the test functions are fixed. A minor point worth checking in the full text is how tightly the error terms are controlled at the upper end of the weight range and for larger n; the abstract claims the choices work, but the details of the test-function support and the resulting bounds would need a close look. Nothing indicates circularity or dependence on parameters fitted inside the paper itself. The citation pattern is standard and draws on the usual trace-formula and Weil-formula references. This is for people who need explicit dimensions or multiplicity tables in the Langlands program, especially those computing L-functions or looking for Galois representations attached to Siegel forms. A reader who wants concrete counts rather than general existence theorems will find direct use in the results. It deserves a serious referee because the classification is a definite extension of the known range and the method is grounded in classical tools applied in a new way.

Referee Report

1 major / 2 minor

Summary. The paper introduces a method using the Weil explicit formula to compute multiplicities of discrete series representations in the space of level-1 automorphic forms on split classical groups over Z (with numerical results for absolute rank ≤8) and proves a classification of all level-1 cuspidal algebraic automorphic representations of GL_n/Q (n arbitrary) with motivic weight ≤24. Vanishing results from the formula are used to simplify the elliptic contribution to the geometric side of the trace formula, yielding dimensions of Siegel cusp forms on Sp_{2g}(Z) that recover all prior cases and extend them via a unified approach.

Significance. If the vanishing results hold with the stated rigor, the classification supplies a complete, finite list of such GL_n forms up to weight 24 and furnishes an efficient, parameter-free route to Siegel-form dimensions that does not rely on case-by-case prior computations. The numerical applications in rank ≤8 and the explicit use of standard tools (Weil formula plus trace formula) constitute concrete, verifiable strengths.

major comments (1)

[Abstract and key ingredient section] Abstract / key-ingredient section: the claim that the Weil explicit formula produces vanishing results 'strong enough to determine the elliptic contribution' for arbitrary n and weights ≤24 requires an explicit statement of the test-function family, its support/decay properties, and a uniform error bound that remains valid as n grows; without these the translation from vanishing to the trace-formula simplification is not yet load-bearing.

minor comments (2)

[Abstract] Abstract: 'Thoses results' is a typographical error.
[Abstract] The phrase 'arguably effortless' is informal; a brief sentence quantifying the reduction in computational steps relative to earlier methods would improve precision.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and the constructive comment on the presentation of the key ingredient. We address the point below.

read point-by-point responses

Referee: [Abstract and key ingredient section] Abstract / key-ingredient section: the claim that the Weil explicit formula produces vanishing results 'strong enough to determine the elliptic contribution' for arbitrary n and weights ≤24 requires an explicit statement of the test-function family, its support/decay properties, and a uniform error bound that remains valid as n grows; without these the translation from vanishing to the trace-formula simplification is not yet load-bearing.

Authors: We agree that an explicit description of the test-function family would strengthen the exposition in the abstract and key-ingredient section. In the revised manuscript we will insert a short dedicated paragraph (or subsection) that states the precise family of test functions used, records that they have compact support in an interval whose length is independent of n, gives the decay estimates, and confirms that the resulting error term in the Weil formula is uniform for all n when the motivic weight is at most 24. These properties are inherited from the standard application of the Weil explicit formula to the logarithmic derivative of the completed L-function and do not require new estimates; the uniformity in n follows directly from the fixed weight bound. With this addition the passage from the vanishing statements to the simplification of the elliptic term in the trace formula will be fully explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation relies on applying the classical Weil explicit formula with suitable test functions to obtain vanishing results that rule out most level-1 algebraic cusp forms on GL_n (motivic weight ≤24), then feeding those non-existence statements into the elliptic term of the trace formula for classical groups. Both the explicit formula and the trace formula are external, independently established tools; the paper's contribution is extending the range of applicable test functions and performing the resulting computations, without defining any quantity in terms of itself or renaming a fitted parameter as a prediction. Existence statements for the surviving representations are imported from prior independent constructions rather than derived internally. No load-bearing step reduces to a self-citation chain or to an ansatz smuggled from the authors' own earlier work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard domain assumptions from automorphic forms theory; no free parameters or invented entities are indicated in the abstract.

axioms (2)

domain assumption The Weil explicit formula applies to the groups and test functions chosen to yield the stated vanishing results
Key ingredient used to disprove existence of forms
domain assumption The trace formula holds with the chosen test function for the split classical groups over Z
Used to obtain the elliptic part of the geometric side

pith-pipeline@v0.9.0 · 5768 in / 1210 out tokens · 24737 ms · 2026-05-24T19:00:30.385056+00:00 · methodology

Discrete series multiplicities for classical groups over Z and level 1 algebraic cusp forms

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)