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Computing Hilbert modular forms as orthogonal modular forms

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Pith's one-line read Even-weight Hilbert modular newforms can be computed through ternary quadratic forms, with each Atkin–Lehner sign sector obtained by a radical character.

desk verdict Strong algorithmic paper with a genuinely new idea (the radical character), but the one-sentence proof of Theorem 7.5 leaves a real gap in the newform theory that needs to be closed before the main theorem is established. read the letter →

arxiv 2506.21981 v1 pith:J6AH73DP submitted 2025-06-27 math.NT

Jeffery Hein , Gonzalo Tornaria , John Voight This is my paper · ORCID
classification math.NT MSC 11F4111F2711R5211E1211Y16
keywords HilbertmodularformsorthogonalternaryquadraticevenCliffordalgebraquaternionordersAtkin-Lehneroperatorsp-neighborsradicalcharacter
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that Hilbert modular forms of even weight and trivial central character can be computed by converting them into orthogonal modular forms on ternary quadratic forms, and it gives an explicit algorithm that does this for every totally real field except the parity case where the field degree is odd and the level is a square. The core of the method is a Hecke-equivariant bijection between cuspidal orthogonal modular forms for a ternary lattice with a prescribed radical character and quaternionic cusp forms with matching Atkin–Lehner eigenvalues; the Eichler–Shimizu–Jacquet–Langlands correspondence then carries these to Hilbert newforms. If the claims are correct, computing a new space is reduced to enumerating isometry classes of ternary lattices and applying p-neighbor operators, with running time roughly linear in the output dimension after a precomputation.

What carries the argument

The even Clifford algebra functor $\Lambda \mapsto \mathrm{Clf}^0(\Lambda)$ from integral ternary quadratic lattices to quaternion orders, which identifies the class set of $\Lambda$ with the type set of $\mathcal{O}$; and the radical character $\nu_p : SO(\Lambda_p) \to \{\pm 1\}$, defined by the action of an isometry on the line $\mathrm{rad}(\Lambda_p)/2N\Lambda_p$ and extended multiplicatively to $\nu_M$. The functor makes the orthogonal and quaternionic Hecke modules canonically isomorphic, and the radical character selects the Atkin–Lehner eigenspace that the Hilbert newforms occupy. On the computational side, Kneser's $p$-neighbor method and reduction of ternary quadratic forms supply the Hecke operators.

What would settle it

For a composite squarefree level such as $N=33$ over $\mathbb{Q}$, compute the degeneracy matrix between the radical-character component $\nu_r$ on $\mathrm{Cl}(\Lambda)$ and the component for a super-lattice $\Lambda'$ via the pullback defined in Section 7; if the matrix is not diagonal with respect to the Atkin–Lehner eigenvalues, the new subspace obtained by restricting Corollary 6.12 is not the Hilbert new space. A simpler check: the multiplicities from Proposition 4.20 predict the dimensions of $S_k^{\mathrm{new}}(\Gamma_0(33))_\varepsilon$, and comparing those dimensions with the output of an orthogonal implementation for each $\varepsilon$ would settle the one-sentence step in Theorem 7.5.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the missing half of Birch's method is supplied by a character: for each Atkin–Lehner sign vector $\varepsilon$ the paper defines a radical character $\nu_\varepsilon$ on the special orthogonal group of a ternary lattice, and proves (Theorem 1.4, generalized in Theorem 7.5) a Hecke-equivariant bijection between the cuspidal orthogonal modular forms $S_k(SO(\Lambda), \nu_\varepsilon)$ and the quaternionic forms $S_k(\widehat{O})_\varepsilon$ with Atkin–Lehner signs $\varepsilon$. Combined with the Eichler–Shimizu–Jacquet–Langlands correspondence (Theorem 4.10), this yields all spaces $S_k^{\mathrm{new}}(\Gamma_0(N))_\varepsilon$ whenever a suitable quaternion order exists, i.e., unless $[F:\mathbb{Q}]$ is odd and $N$ is a square. The bijection is canonical because the even Clifford algebra functor sends class sets of ternary lattices to type sets of quaternion orders, so the two function spaces are literally identified; the radical character files the classes according to the action on the one-dimensional radical of the $p$-adic lattice, which matches the Atkin–Lehner signs.

Load-bearing premise

The load-bearing premise is one sentence in the proof of Theorem 7.5: restricting the bijection of Corollary 6.12 to new subspaces preserves the radical-character labeling, so that the orthogonal degeneracy maps behave exactly like the quaternionic ones; if those maps mix the $\nu_\varepsilon$ components, the computed new space would be the wrong Hecke module.

Editorial extensions

If this is right

  • For any totally real field $F$, even weight $k$, factored level $N$ with $[F:\mathbb{Q}]$ even or $N$ nonsquare, and any sign vector $\varepsilon$, the new space $S_k^{\mathrm{new}}(\Gamma_0(N))_\varepsilon$ is exhibited as a Hecke module by matrices obtained from ternary lattice classes.
  • The Hecke matrices are sparse, and the running time for $T_p$ is $\widetilde{O}(d^2 \operatorname{Nm}(p))$ for fixed $F,N$, reducing to $\widetilde{O}(d \operatorname{Nm}(p))$ after lattice setup, so the cost is roughly linear in the dimension $d$ of the output space.
  • The two deficiencies Birch identified—retrieving only half the forms and losing information when the level is not squarefree—are both removed: every Atkin–Lehner eigenspace is obtained, for arbitrary level outside the square-odd-degree exception.
  • The dimension of each $\varepsilon$-eigenspace is asymptotically $2^{-r} \operatorname{Nm}(N) |k-1|$ by the mass formula, so splitting the new space by sign vectors does not make any individual piece asymptotically negligible.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The radical-character construction is likely to adapt to nontrivial central characters or bounded level: the paper's framework already handles residually unramified orders, and its Remark 4.9 points toward residually ramified orders as a further extension.
  • Because the spinor norm has the explicit formula $\theta(\sigma)=1+\operatorname{tr}(\sigma)$ in dimension three, the radical character can be evaluated directly from the trace of an isometry, which may simplify implementations and extend the method to lattices whose Clifford order is not residually unramified.
  • A natural test beyond the paper is to compare the orthogonal-side output with quaternionic or modular-symbol computations at composite levels where the old subspace has several layers; the multiplicity formula in Proposition 4.20 predicts exactly which newforms appear in each component.
  • The speed demonstrated on $F=\mathbb{Q}$ suggests that sampling Hecke eigenvalues in a fixed Atkin–Lehner eigenspace could make searches for elliptic curves of moderately large conductor practical, an application the authors hint at but do not develop.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper presents an algorithm to compute Hilbert modular forms of even weight and trivial central character as orthogonal modular forms on a totally definite ternary quadratic space, extending Birch's method to totally real fields, nonsquare levels, and all Atkin-Lehner eigenspaces. The main theoretical result, Theorem 1.4, is a Hecke-equivariant bijection between cuspidal orthogonal modular forms with a radical character and quaternionic cusp forms with specified Atkin-Lehner eigenvalues; Theorem 7.5 upgrades this to new subspaces. The paper further gives a complexity analysis, an algorithm for constructing suitable lattices and orders, and reports on implementations in Magma and C++. The main advertised computational output is Theorem 1.3: for a totally real field F, even weight k, factored level N with [F:Q] even or N nonsquare, and sign vector ε, the new space S_new_k(bΓ0(N))_ε can be computed as a Hecke module with running time eO(d^2 Nm(p)) for T_p, where d is the dimension of the output space.

Significance. If the main theorems are correct, this is a substantial advance: it gives a uniform, practical method for computing Hilbert modular forms in cases where modular symbols are expensive or infeasible, and it resolves both known limitations of Birch's ternary-form method by introducing the radical character. The even Clifford algebra approach is conceptually transparent and provides a canonical, Hecke-equivariant correspondence without analytic theta-series arguments. The paper also contains an explicit categorical inverse to the even Clifford functor (Appendix A) and concrete computational examples, including a large-level example that outperforms modular symbols. These are real strengths. However, the proof of the newform-theoretic bridge, Theorem 7.5, is currently a one-sentence assertion, and this is load-bearing for the advertised output; the complexity statement also contains an internal inconsistency. The underlying strategy appears defensible, but the manuscript as submitted does not yet establish the central claim.

major comments (3)
  1. [§7, Theorem 7.5] The proof of Theorem 7.5 is a single sentence: 'We restrict Corollary 6.12 to the new subspace on quaternionic modular forms; by definition, this corresponds to the new subspace on orthogonal modular forms.' This is not sufficient. Corollary 6.12 gives a Hecke-equivariant bijection between the full spaces M_k(SO(bΛ), νM) and M_k(bO)_{εM}. The orthogonal old subspace, by Definition 7.4, is generated by pullbacks α_Λ' from lattices Λ' corresponding to superorders O' ⊇ O, with SO(bΛ) ≤ SO(bΛ'). To restrict Corollary 6.12 to new subspaces, one must prove that these degeneracy maps are compatible with the radical character: for each such Λ' there should be a character ν'_M on SO(bΛ') whose restriction to SO(bΛ) equals νM, and the induced map should land in M_k(SO(bΛ), νM). The manuscript does not define such characters on the superorder lattices, does not prove that νM extends or restricts in the required way, and does not show that the orthogonal old subspace corresponds, under Corollary 6.12, to the quaternionic old subspace generated by pullbacks from O' ⊇ O. Without this compatibility, the orthogonal complement computed by the algorithm is not identified with S_new_k(bΓ0(N))_ε. This is the only place where the space actually computed (orthogonal new space with radical character) is connected to the space advertised in Theorem 1.3 (Hilbert new space with Atkin-Lehner signs), so the gap is load-bearing.
  2. [§1, Theorem 1.3 and following paragraph] The complexity statement is internally inconsistent. Theorem 1.3 states that, for fixed F and N, the algorithm takes eO(d^2 Nm(p)) bit operations to compute T_p, where d = dim_C S_new_k(bΓ0(N))_ε. The paragraph immediately after Theorem 1.3 claims: 'After a precomputation (to set up the lattice), the running time becomes eO(d Nm(p))'. These two displayed statements cannot both be true with the same meaning of d. Theorem 8.5 gives yet another expression, eO(Nm(p) H_{3n}(∥Λ∥ d^2)), where d = #Cl(Λ), which is not obviously reconciled with either statement in §1. The authors should state clearly the worst-case cost, the amortized cost after precomputation, and the precise role of the output dimension d in each formula.
  3. [§7, Definition 7.4 and §6, Corollary 6.12] Related to the gap in Theorem 7.5, the definition of the orthogonal old subspace is not extended to the radical-character twisted setting. Definition 7.4 defines the old subspace inside M_W(SO(bΛ)) using degeneracy maps α_Λ' that pull back untwisted functions from SO(bΛ')-invariant classes. But the spaces in Corollary 6.12 and Theorem 7.5 are the twisted spaces M_k(SO(bΛ), νM). The paper never states what the degeneracy maps are on these twisted spaces, which character on the superorder lattice Λ' is used, or why the images land in the νM-isotypic component rather than mixing different radical characters. This is not just a presentational gap: if the radical character mixes under pullback, the orthogonal complement of the old subspace in M_k(SO(bΛ), νM) need not be the νM-component of the new space, and Theorem 7.5 would fail. A complete proof should include a commutative diagram for the degeneracy maps on both sides.
minor comments (4)
  1. [§3, Remark 3.9] The phrase 'isotopic decomposition' should be 'isotypic decomposition'.
  2. [§6, equation (6.3)] The symbol O is overloaded: on the right-hand side of (6.3), O(rad_p(Λ)/2NΛ_p) denotes the orthogonal group, while elsewhere O denotes a quaternion order. Using a different symbol, such as Aut or Orth, would avoid confusion.
  3. [§8, Theorem 8.5] The role of d = #Cl(Λ) in the Hermite normal form complexity H_{3n}(∥Λ∥ d^2) is not explained; in particular, it is unclear whether the factor d^2 is a worst-case bound for the size of the lattice entries after reduction, and how this relates to the dimension d of the output space in Theorem 1.3.
  4. [§1, theorem statement] Theorem 1.3 promises to 'return for each ideal n coprime to N a matrix [T_n]', but the algorithms and complexity analysis in §8 only discuss the prime Hecke operator T_p. The authors should state how composite T_n are obtained (e.g., by the standard recurrence from the T_p for p not dividing N), or otherwise clarify the scope of the theorem.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the radical character is proved to match the Atkin–Lehner sign decomposition, and the algorithm's output is not fitted to its input.

full rationale

The derivation is self-contained in the relevant sense. The radical character νp is defined canonically from the lattice in (6.1)–(6.3), and Proposition 6.8 proves that the pairing with the normalizer is perfect and identifies the radical character with the Atkin–Lehner involution signs; Corollary 6.12 then gives the Hecke-equivariant bijection with the quaternionic ε-eigenspace. No parameter is fitted to any data set, and the sign vector ε is an input, not a quantity inferred from the output. The main structural input, the even-Clifford bijection Cl Λ ↔ Typ O, is imported from Voight's monograph [Voi21] and developed further in Appendix A; although Voight is an author, these are published theorem references with independent mathematical content, so the self-citations are not load-bearing in a circular sense. The one genuinely weak passage is the proof of Theorem 7.5, which says: 'We restrict Corollary 6.12 to the new subspace on quaternionic modular forms; by definition, this corresponds to the new subspace on orthogonal modular forms.' That is an omitted proof that the orthogonal degeneracy maps preserve the νM-isotypic components, not a circular reduction: the conclusion is not assumed in Definition 7.4, and a failure there would make the advertised new-space identification false rather than tautologically true. The implementation is also checked against known newforms in Section 9, providing external, non-circular evidence. I therefore find no definitional or fitted-input circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No numerical free parameters are used: the algorithm is exact, with probabilistic choices only in Algorithm 8.1. The radical character is a mathematical construct derived from the lattice, not a fitted or postulated physical entity. The main extra-mathematical inputs are the standard theorems listed in the axioms.

assumptions (4)
  • standard math Eichler-Shimizu-Jacquet-Langlands correspondence for suitable quaternion orders (Theorem 4.10).
    Invoked without proof, citing Jacquet-Langlands and others; it connects quaternionic cusp forms to Hilbert cusp forms and is essential for the output of the algorithm to be Hilbert modular forms.
  • standard math Even Clifford algebra functor gives bijections between ternary lattices and Gorenstein quaternion orders (Theorem 5.7, Corollary 5.8, Appendix A).
    The paper cites Voight and gives a proof sketch in Appendix A; it underpins the bijection between orthogonal and quaternionic modular forms.
  • standard math Kneser's method of p-neighbors computes the class set and the Hecke action (Section 2, Section 8).
    Assumed from the literature ([Kne57], [Voi23]); it provides the concrete combinatorial engine for the algorithm.
  • domain assumption The input must satisfy that either [F:Q] is even or N is nonsquare, so a suitable quaternion algebra exists (Proposition 4.5).
    Outside this domain the theorem does not apply; the authors prove existence precisely in this case, but the algorithm's scope is bounded by this condition.

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Pith. "Pith review of Computing Hilbert modular forms as orthogonal modular forms." pith.science (2026). https://pith.science/paper/J6AH73DP

@misc{pith2026250621981,
  author       = {Pith},
  title        = {Pith review of: Computing Hilbert modular forms as orthogonal modular forms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J6AH73DP}},
  note         = {Machine review of arXiv:2506.21981}
}
read the original abstract

We show how to efficiently compute Hilbert modular forms as orthogonal modular forms, generalizing and expanding upon the method of Birch.

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Forward citations

Cited by 1 Pith paper

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

  1. An orthogonal perspective on Gauss composition

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    Clifford and norm functors give a discriminant-preserving equivalence between binary quadratic modules and pseudoregular modules over quadratic algebras, recovering and generalizing Gauss composition over any base scheme.

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