All modular forms of weight 2 can be expressed by Eisenstein series

Jiacheng Xia; Martin Raum

REVIEW 2 major objections 3 minor 23 references

All modular forms of weight 2 can be expressed by Eisenstein series

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Every modular form of integral weight greater than 1 is a linear combination of products of at most two cusp expansions of Eisenstein series.

desk verdict Genuinely new result that removes the central L-value obstruction for weight 2, but the proof rests on an unproved key lemma that a referee must check. read the letter →

arxiv 1908.03616 v1 pith:M5MKMWZA submitted 2019-08-09 math.NT

Martin Raum , Jiacheng Xia This is my paper · ORCID

classification math.NT MSC 11F1111F6711F25

keywords modularformsEisensteinseriescuspexpansionscentralL-valuesquadratictwistsvector-valuedHeckeoperatorspairingformulacongruencesubgroups

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 proves that every elliptic modular form of integral weight greater than 1—including weight 2, the case earlier methods could not handle—is a linear combination of products of at most two cusp expansions of Eisenstein series, where 'cusp expansions' means the Fourier expansions of Eisenstein series at every cusp, not just the one at infinity. For weights greater than 2, products of exactly two such expansions suffice. Previous results of this kind required the central L-value of a weight-2 newform to be nonzero, an obstruction that excludes forms attached to elliptic curves of positive rank. The proof removes that obstruction by using nonvanishing of quadratic twists instead, and it specifies explicit levels at which the Eisenstein factors may be chosen.

What carries the argument

The carrying object is $E_k(N)$, the space of Fourier expansions at every cusp of weight-k Eisenstein series of level N; unlike the space of expansions at infinity alone, it carries an action of $\mathrm{SL}_2(\mathbb{Z})$, which lets the proof use representation theory. Three mechanisms do the work: vector-valued Hecke operators, which move expansions from one level to another; a pairing formula (Lemma 3.1) that evaluates the inner product of a product of Eisenstein expansions with a cusp form f as a nonzero constant times $L(f^c,k+l-1)L(f^c\otimes\psi,l)$; and a representation-theoretic comparison of $T$-fixed vectors that shows weight-$(k+l)$ Eisenstein spaces are contained in products of weight-k and weight-l Eisenstein spaces. The vector-valued setup lets the weights k and l stay fixed while the levels vary, so the special L-values that must be shown nonzero are quadratic twists rather than the central value itself.

What would settle it

Take a weight-2 newform f whose central L-value vanishes and compute both sides of the pairing formula in Lemma 3.1 for a product g constructed from weight-1 Eisenstein expansions with an odd character ψ≠χ. If the identity fails for any admissible triple, the proof of Theorem 3.4—and with it Theorem I—would break; if it holds numerically, that supports the omitted proof of the lemma.

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Extended reading notes

Core claim

The central discovery is Theorem I: for any positive integers k, l, and N, some positive integer N0 exists with $M_{k+l}(\Gamma(N)) \subseteq E_{k+l}(N) + E_k(N_0)\cdot E_l(N_0)$, where $E_k(N)$ is the space of Fourier expansions at all cusps of weight-k Eisenstein series of level N. Whenever $k+l \geq 3$, an explicit N0 and another integer N1 give $M_{k+l}(\Gamma(N)) \subseteq E_k(\mathrm{lcm}(N_0, N N_1))\cdot E_l(\mathrm{lcm}(N_0, N_1))$, with no weight-$(k+l)$ Eisenstein term on the right. In words, every modular form of integral weight at least 2 is captured by products of at most two Eisenstein expansions, and every form of weight at least 3 by products of exactly two. The same statement is proved for $\Gamma_1(N)$, $\Gamma_0(N)$, Dirichlet-character forms, and vector-valued modular forms. This is the first version of the statement that includes weight-2 newforms whose central L-value vanishes, the case ruled out in previous work.

Load-bearing premise

The whole cusp-form half of the proof rests on a pairing formula (Lemma 3.1) that is quoted from earlier work without proof; if that formula does not extend to the fixed weights, levels, and exceptional character conditions used here, the inclusion of cusp forms in Theorem I collapses.

Editorial extensions

If this is right

Weight-2 newforms with vanishing central L-values, previously excluded, are now covered, so the main obstruction to Eisenstein-product expressions is removed.
For every modular form of weight at least 3, the expression uses exactly two Eisenstein factors with explicit levels, not a sum over intermediate weights.
The inclusion holds for Γ1(N), Γ0(N), Dirichlet characters, and vector-valued modular forms, not only for principal congruence subgroups.
Because the right-hand side's action of SL2(Z) is explicit, Fourier expansions at all cusps can in principle be computed from Eisenstein data, including cusps not mapped to infinity by the usual algorithms.
The only unproved case of the refined 'exactly two' statement is weight 2 with both factors of weight 1; the paper leaves it open with numerical evidence suggesting it holds.

Reading between the lines

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

If the explicit level bounds N0 and N1 are made effective and small, the theorem turns into an algorithm for computing all cusp expansions of modular forms, a task current modular-symbol methods only do at infinity.
The same vector-valued Hecke mechanism is not tied to two factors, so a similar proof may express weight-2 forms with prescribed ramification using longer products of Eisenstein expansions.
The pairing-formula route suggests a general principle: a modular form can be built from Eisenstein data exactly when the associated family of L-values has some nonzero twist, which reframes the vanishing-central-value obstruction as a nonvanishing problem in a larger family.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Let me give you the quick version. This paper proves that every modular form of integral weight >1, including weight 2 with vanishing central L-values, is a linear combination of products of at most two cusp expansions of Eisenstein series. For weight >2, products of exactly two suffice. That is a genuine advance over Kohnen–Zagier, Borisov–Gunnells, Dickson–Neururer, and Raum's earlier paper. The obstruction that blocked weight 2 in all previous work is gone.

What's new is mostly in the architecture. The authors work with the full space of cusp expansions Ek(N) instead of just the expansion at infinity, fix the weights of the Eisenstein factors, and vary their levels using vector-valued Hecke operators. That gives enough room to avoid the central L-value obstruction. The proof of the 'exactly two factors' part (Theorem 4.2) is a clever representation-theoretic constant-term argument, and the level bounds are explicit except in a couple of corners.

The soft spots are real but not fatal. Lemma 3.1 is the load-bearing analytic step: it gives the Petersson pairing between a product of Eisenstein series and a newform as a product of two L-values. The proof is omitted, with 'mutatis mutandis' pointing to Dickson–Neururer. That would be fine for a routine variant, but the cited setting is more restrictive, and the exceptional-character conditions (ψ≠1 for k=2, ψ≠χ for l≤2) are exactly what the later nonvanishing argument needs. The paper does not verify those conditions in the k=l cases. A referee should ask for a complete proof of Lemma 3.1 or a precise derivation from [11]. It is very likely true, but as written it is an unverified assertion at a critical juncture.

The case k=l=1 is honestly flagged as incomplete: N0 exists only non-constructively, and Theorem 4.2 excludes it. That is a limitation, not a flaw.

Overall, this is a serious paper with a new theorem, a plausible but partly hidden proof, and some genuinely nice ideas. It deserves peer review. I would send it to a good number theory journal and ask the referee to focus on Lemma 3.1.

Referee Report

2 major / 3 minor

Summary. The paper proves that every holomorphic modular form of integral weight greater than 1 can be expressed as a linear combination of products of at most two cusp expansions of Eisenstein series; for weights greater than 2, products of exactly two suffice. The proof combines vector-valued Hecke operators, a Petersson product formula relating products of Eisenstein series to products of L-values, and nonvanishing results for twisted L-values due to Waldspurger, Kohnen-Zagier, and Ono-Skinner. Explicit levels for the Eisenstein factors are given in most cases, except the weight (1,1) case where only an existence statement is provided. The main theorem is stated for principal congruence subgroups, with extensions to Γ0(N), Γ1(N), Dirichlet characters, and vector-valued modular forms.

Significance. If the proof is completed, the result is significant: it removes the central L-value obstruction that limited all earlier works on products of Eisenstein series, and it yields new explicit descriptions of spaces of modular forms in terms of Eisenstein products, with algorithmic consequences for computing expansions at all cusps. The strategy of fixing the Eisenstein weights and varying their levels is a genuine advance over the earlier approach that varied the weight. The paper also provides substantial technical machinery, including a careful treatment of the passage between classical and vector-valued modular forms.

major comments (2)

[Section 3, Lemma 3.1] Lemma 3.1 is the only mechanism in the proof of Theorem 3.4 that produces a nonzero Petersson pairing between an arbitrary newform f and a product of Eisenstein series, so every inclusion in (3.2) rests on formula (3.1). The proof is omitted with the comment 'mutatis mutandis' from Corollary 4.2 of [11], but the cited result is established under a more restrictive level assumption (as the paper itself states in the introduction, Eq. (0.1), the level in [11] is a product of two prime powers and a square-free integer). The exact form of the factorization in (3.1), with no imprimitive or local factors and with the stated exclusions ψ≠1 at k=2 and ψ≠χ at l≤2, is critical because the subsequent nonvanishing arguments are tuned to the values at the central point and at the abscissa of convergence. A complete proof of Lemma 3.1 for the stated level of generality, or a precise reference containing exactly this statement, must be supplied before the main theorem can be regarded as established.
[Section 3, proof of Theorem 3.4, cases k=l≥2 and k=l=1] The applications of Lemma 3.1 to twisted newforms do not verify the lemma's hypotheses. In the case k=l≥2, after choosing D with L(f^c⊗ε_D,l)≠0, the text applies Lemma 3.1 to f⊗ε_Dψ for 'a suitable Dirichlet character ψ mod 16' and asserts that L(f^c⊗ε_Dψ^2,l)≠0; however, the reason why ψ^2 is a permissible twist (for instance, ψ^2=1) is not stated, and for k=l=2 the condition ψ≠χ is not checked. In the case k=l=1, a second application of Ono-Skinner provides a discriminant D' with L(f^c⊗ε_D⊗ε_{D'},1)≠0, but the condition ψ≠χ for ψ=ε_{D'} is not verified; the character of the twisted form is χ·ε_D·ψ, so this requires χ·ε_D≠1. Without these verifications, the conclusion ⟨g∞,f⟩≠0 does not follow from Lemma 3.1.

minor comments (3)

[Theorem 4.2] There are typos in the statement and proof: 'intergers' should be 'integers', and 'irreducibe' should be 'irreducible'. The displayed formula for N0 in Theorem 3.4 would be clearer if the exponent 4 were set off typographically, for example N(16B(k+1/2,N))^4 B(k+l,(16B(k+1/2,N))^4).
[Introduction, Section 2.2] The key notion 'cusp expansions of Eisenstein series' is used throughout but not defined in the introduction. The distinction between E_k(N)_∞ and E_k(N) is central to the argument, and a short explanation of why the space of cusp expansions at all cusps, rather than only at infinity, is needed for the representation-theoretic method would improve readability.
[Section 3, proof of Theorem 3.4, k=l≥2 case] The use of the half-integral weight Sturm bound to select a fundamental discriminant D < B(k+1/2,N) with nonvanishing L^1/2 value is not documented. The standard Sturm bound applies to all Fourier coefficients, whereas the desired conclusion requires a nonzero coefficient at a fundamental discriminant. A reference or a short justification for this variant would remove a potential gap.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the main inclusions are obtained from external Rankin–Selberg pairings and independent nonvanishing results; the only flagged issue is an omitted 'mutatis mutandis' proof of Lemma 3.1, which is a correctness gap, not circularity.

full rationale

The derivation chain is not circular. The target inclusions (0.2)/(0.3) are not definitions: the spaces E_k(N), E_l(N_0) are spaces of Fourier expansions of Eisenstein series, defined independently of the modular forms being represented, and the theorem asserts a nontrivial span inclusion. In Theorem 3.4 the only bridge from products to arbitrary cusp forms is Lemma 3.1, a Petersson-pairing identity quoted from Dickson–Neururer [11]: 'We omit the proof, which is mutatis mutandis the one of Corollary 4.2 of [11].' This is an internal completeness risk, because every inclusion of newforms in S_{k+l}(χ) depends on the exact factorization (3.1), including the exceptional-character exclusions ψ≠1 for k=2 and ψ≠χ for l≤2. But it is not circular: [11] is an external Rankin–Selberg computation, and the nonvanishing of the two L-factors is obtained from Deligne's proof of the Ramanujan conjecture, Waldspurger/Kohnen–Zagier, and Ono–Skinner, none of which presuppose Theorem I. The paper's extensive use of [22] (Prop. 2.5, 2.10, 2.17, 2.19, Thm. 2.8; induction maps §2.1) is self-citation, since [22] is the first author's previous work, but those cited statements concern vector-valued Hecke operators and induction isomorphisms, general structural facts that are not equivalent to the theorem being proved. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported to force a choice, and no known result is merely re-labelled. Hence the finding is no significant circularity; the score of 2 reflects only the load-bearing omitted proof and the heavy reliance on [22], which are correctness concerns rather than a circular reduction.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper's central claim rests on several external results, including a Rankin-Selberg pairing formula asserted without proof in Lemma 3.1, Deligne's Ramanujan conjecture, and nonvanishing theorems of Waldspurger, Kohnen-Zagier, and Ono-Skinner. These are mathematical theorems rather than fitted parameters; no data were fitted.

assumptions (7)

standard math All congruence types are unitarizable.
Section 1.1 states this without proof; it is used to take orthogonal complements and to apply Lemma 2.1.
standard math Frobenius reciprocity for induction from Γ1(N) and Γ0(N).
Used in Lemma 2.1 and in the decomposition of induced representations in Section 1.1.
domain assumption Lemma 3.1: Petersson pairing formula ⟨g,f⟩ = c L(f^c,k+l−1) L(f^c⊗ψ,l) for suitable Eisenstein products g.
Stated without proof; text says proof is mutatis mutandis from Corollary 4.2 of [11]. This identity is the only source of nonzero pairings in Theorem 3.4.
domain assumption Deligne's proof of the Ramanujan conjecture for holomorphic modular forms.
Used in Theorem 3.4 to guarantee nonvanishing of L-values on the abscissa of convergence, via Iwaniec-Kowalski Lemma 5.9.
domain assumption Waldspurger and Kohnen-Zagier: central L-values of quadratic twists are Fourier coefficients of half-integral weight forms, with nonvanishing for some D below the Sturm bound.
Used in Theorem 3.4 to produce a nonvanishing central value L(f^c⊗ε_D,l) for k=l.
domain assumption Ono-Skinner theorem: infinitely many quadratic twists of weight 2 newforms have nonvanishing L(1).
Used twice in the k=l=1 case of Theorem 3.4 to obtain two nonvanishing twists.
standard math Sturm bounds for integral and half-integral weight modular forms.
Used to make N0 explicit in Theorem 3.4 and to find a nonvanishing Fourier coefficient of a half-integral weight form.

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Pith. "Pith review of All modular forms of weight 2 can be expressed by Eisenstein series." pith.science (2026). https://pith.science/paper/M5MKMWZA

@misc{pith2026190803616,
  author       = {Pith},
  title        = {Pith review of: All modular forms of weight 2 can be expressed by Eisenstein series},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M5MKMWZA}},
  note         = {Machine review of arXiv:1908.03616}
}

abstract

We show that every elliptic modular form of integral weight greater than $1$ can be expressed as linear combinations of products of at most two cusp expansions of Eisenstein series. This removes the obstruction of nonvanishing central $\mathrm{L}$-values present in all previous work. For weights greater than $2$, we refine our result further, showing that linear combinations of products of exactly two cusp expansions of Eisenstein series suffice.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [11]

La conjecture de Weil. II

P . Deligne. “La conjecture de Weil. II”. Inst. Hautes Études Sci. Publ. Math. 52 (1980)

work page 1980

[2] [1]

4 Eisenstein series In preparation for the proof of this section’ s main theorem, we determine the space of cusp expan- sions of Eisenstein series

This concludes the proof. 4 Eisenstein series In preparation for the proof of this section’ s main theorem, we determine the space of cusp expan- sions of Eisenstein series. Lemma 4.1. Fix an irreducible congruence type ρ with T -ﬁxed vectors and an integer k such that ρ = ρpar(k). Then we have the inclusions of SL2(Z)-representations ρ ,→ Ek(ρ) if k ≥ 3....

work page

[3] [2]

Modular forms in Pari/GP

K. Belabas and H. Cohen. “Modular forms in Pari/GP”. Res. Math. Sci. 5.3 (2018)

work page 2018

[4] [3]

Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves

M. Bhargava and A. Shankar. “Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves”. Ann. of Math. (2) 181.1 (2015)

work page 2015

[5] [4]

A positive proportion of elliptic curves over Q have rank one

M. Bhargava and C. Skinner. “A positive proportion of elliptic curves over Q have rank one”. J. Ramanujan Math. Soc. 29.2 (2014)

work page 2014

[6] [5]

Toric modular forms and nonvanishing of L-functions

L. A. Borisov and P . E. Gunnells. “Toric modular forms and nonvanishing of L-functions”. J. Reine Angew. Math. 539 (2001). – 13 – All modular forms of weight 2 can be expressed by Eisenstein series M. Raum, J. Xia

work page 2001

[7] [6]

Toric modular forms of higher weight

L. A. Borisov and P . E. Gunnells. “Toric modular forms of higher weight”. J. Reine Angew. Math. 560 (2003)

work page 2003

[8] [7]

Toric varieties and modular forms

L. A. Borisov and P . E. Gunnells. “Toric varieties and modular forms”. Invent. Math. 144.2 (2001)

work page 2001