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arxiv: 2607.01957 · v1 · pith:K4KCEJZHnew · submitted 2026-07-02 · 🧮 math.NT

The Eichler--Selberg trace formula for Hilbert cusp forms, the class numbers of quartic CM fields, and their distributions

Seiji Kuga , Andrei Seymour-Howell , Satoshi Wakatsuki This is my paper

Pith reviewed 2026-07-03 07:02 UTC · model grok-4.3

classification 🧮 math.NT
keywords Eichler-Selberg trace formulaHilbert modular formsHurwitz class numbersCM fieldsclass numbersHecke operatorsreal quadratic fields
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The pith

Generalized Hurwitz class numbers from quartic CM fields give an Eichler-Selberg trace formula for Hilbert cusp forms.

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

The paper introduces generalized Hurwitz class numbers for totally real number fields, defined using class numbers of quartic CM fields. These are used to establish an Eichler-Selberg trace formula for the space of holomorphic Hilbert cusp forms over real quadratic fields with narrow class number one. The authors apply the formula to study distributions of these class numbers, prove class number relations, and perform numerical computations of Hecke operator traces for the fields Q(sqrt(5)) and Q(sqrt(29)). A sympathetic reader would care because this provides an explicit arithmetic expression for traces that generalizes the classical case involving imaginary quadratic fields.

Core claim

Motivated by Su's Cohen-type Eisenstein series, we introduce generalized Hurwitz class numbers to totally real number fields. Using these, we establish an Eichler-Selberg trace formula for holomorphic Hilbert cusp forms over real quadratic fields of narrow class number one. The generalized Hurwitz class numbers in the formula are defined in terms of class numbers of quartic CM fields. We also study their distributions, prove class number relations, and compute traces numerically for specific quadratic fields.

What carries the argument

Generalized Hurwitz class numbers defined in terms of class numbers of quartic CM fields that enter the Eichler-Selberg trace formula for the space of holomorphic Hilbert cusp forms.

If this is right

  • The trace formula expresses traces of Hecke operators explicitly in terms of these class numbers.
  • Class number relations follow from the trace formula.
  • The distribution of the generalized Hurwitz class numbers can be analyzed using the formula.
  • Numerical values of traces can be computed for Q(sqrt(5)) and Q(sqrt(29)).

Where Pith is reading between the lines

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

  • Similar generalizations might apply to totally real fields of higher degree.
  • The quartic CM field connection may relate to other trace formulas or L-functions in arithmetic geometry.
  • The numerical computations could inspire conjectures on the average size or asymptotic behavior of these class numbers.

Load-bearing premise

The generalized Hurwitz class numbers constructed from class numbers of quartic CM fields correctly encode the data required for the Eichler-Selberg trace formula to hold for holomorphic Hilbert cusp forms over the given real quadratic fields.

What would settle it

Computing the dimension or a Hecke trace directly from the definition of the space of Hilbert cusp forms for a specific discriminant and level, then comparing it to the sum over the generalized Hurwitz class numbers, would test the formula; disagreement would disprove it.

Figures

Figures reproduced from arXiv: 2607.01957 by Andrei Seymour-Howell, Satoshi Wakatsuki, Seiji Kuga.

Figure 1
Figure 1. Figure 1: Sato–Tate plot for Q( √ 29) and weight (2, 2). This has 148, 837 totally positive prime elements in 45 bins [PITH_FULL_IMAGE:figures/full_fig_p021_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Sato–Tate plot for Q( √ 29) and weight (2, 6). This has 148, 837 totally positive prime elements (giving here 297, 674 data points as we can consider both Galois embeddings) in 65 bins. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Sato–Tate plot for Q( √ 5) and weight (8, 8). This has 148, 931 totally positive prime elements in 45 bins [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Sato–Tate plot for Q( √ 5) and weight (4, 8). This has 148, 931 totally positive prime elements (giving here 297, 862 data points as we can consider both Galois embeddings) in 65 bins. Acknowledgments. The second author has been supported by the National Research Founda￾tion of Korea (NRF) grant funded by the Korea government (MSIP) (No. RS-2024-00415601 (G-BRL)). The third author is partially supported by… view at source ↗
read the original abstract

Motivated by Su's construction of Cohen-type Eisenstein series of half-integral weight over totally real number fields \cite{Su16}, we introduce a generalization of Hurwitz class numbers to totally real number fields. Using these generalized Hurwitz class numbers, we establish an Eichler--Selberg trace formula for the space of holomorphic Hilbert cusp forms over real quadratic fields of narrow class number one. While the classical Hurwitz class numbers are defined in terms of class numbers of imaginary quadratic fields, the generalized Hurwitz class numbers appearing in our Eichler--Selberg trace formula are defined in terms of class numbers of quartic CM fields. For applications of this Eichler--Selberg trace formula, we study the distribution of the generalized Hurwitz class numbers, prove class number relations, and carry out numerical computations of traces of Hecke operators for $\mathbb{Q}(\sqrt{5})$ and $\mathbb{Q}(\sqrt{29})$.

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

1 major / 2 minor

Summary. The manuscript introduces generalized Hurwitz class numbers for totally real number fields, defined via class numbers of associated quartic CM fields and motivated by Su's Cohen-type Eisenstein series. It establishes an Eichler-Selberg trace formula for the space of holomorphic Hilbert cusp forms over real quadratic fields of narrow class number one, expressed in terms of these generalized class numbers. Applications include the distribution of the generalized class numbers, proofs of class number relations, and numerical computations of Hecke traces for the fields Q(sqrt(5)) and Q(sqrt(29)).

Significance. If the derivation holds, the work supplies an explicit arithmetic expression for traces of Hecke operators on Hilbert cusp forms, directly tying them to class numbers of quartic CM fields. This extends the classical Eichler-Selberg formula in a coherent way and supplies concrete numerical checks together with distribution results. The construction is parameter-free once the generalized class numbers are fixed, and the restriction to narrow class number one is stated explicitly as the setting in which the formula is proved.

major comments (1)
  1. [§2] §2, Definition of generalized Hurwitz class numbers: the claim that these numbers correctly capture the arithmetic data for the trace formula rests on the construction via quartic CM fields; an explicit verification that the definition reduces to the classical Hurwitz class number when the base field is Q (or a direct comparison with the imaginary quadratic case) is needed to confirm consistency of the generalization.
minor comments (2)
  1. [Introduction] Introduction: a short paragraph recalling the statement of the classical Eichler-Selberg formula over Q would help orient readers before the generalization is presented.
  2. [§5] §5, numerical examples: the precision and method used to compute the class numbers of the quartic CM fields for the traces on Q(sqrt(5)) and Q(sqrt(29)) should be stated explicitly to ensure reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and the constructive comment on the definition of the generalized Hurwitz class numbers. We address the point below.

read point-by-point responses

  1. Referee: [§2] §2, Definition of generalized Hurwitz class numbers: the claim that these numbers correctly capture the arithmetic data for the trace formula rests on the construction via quartic CM fields; an explicit verification that the definition reduces to the classical Hurwitz class number when the base field is Q (or a direct comparison with the imaginary quadratic case) is needed to confirm consistency of the generalization.

    Authors: We agree that an explicit verification of the reduction to the classical case strengthens the consistency of the generalization. In the revised manuscript we will add a remark (or short computation) in §2 showing that when the totally real base field F is ℚ the associated quartic CM fields reduce to imaginary quadratic fields and the generalized Hurwitz class numbers coincide with the classical Hurwitz class numbers, following directly from the class-number definition. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines generalized Hurwitz class numbers in terms of class numbers of quartic CM fields (motivated by Su's prior Eisenstein series construction) and then proves an Eichler-Selberg trace formula expressed using those numbers for Hilbert cusp forms over narrow class number one real quadratic fields. No quoted equations or steps reduce the trace formula to the definition by construction, nor does any load-bearing premise rely on self-citation chains or fitted inputs renamed as predictions. The restriction to narrow class number one fields and the listed applications (distributions, relations, numerical traces) are presented as downstream consequences rather than foundational inputs. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Review performed on abstract alone; full list of assumptions and definitions unavailable. The central objects are newly defined quantities whose properties are asserted to support the trace formula.

axioms (1)
  • domain assumption The construction of generalized Hurwitz class numbers is valid when motivated by Su's half-integral weight Eisenstein series over totally real fields.
    The paper states the motivation as the starting point for the definition used in the trace formula.
invented entities (1)
  • Generalized Hurwitz class numbers for totally real number fields no independent evidence
    purpose: To serve as the arithmetic side of the Eichler-Selberg trace formula
    Newly introduced quantities defined in terms of class numbers of quartic CM fields; no independent evidence supplied in abstract.

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