Defining newforms in characteristic $p$

Daniel R. Johnston

REVIEW 3 major objections 3 minor 11 references

Defining newforms in characteristic $p$

T0 review · 3 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Cusp forms with Dirichlet character over characteristic-p coefficient rings have the same ℓ-new subspaces whether newness is defined by the U_ℓ-eigenvalue operator or by the kernels of trace operators.

desk verdict A genuinely useful write-up of Deo–Medvedovsky plus a plausible but under-proved new chapter on character-twisted newforms; send it to a referee who will ask for the old-space gap to be filled. read the letter →

arxiv 2412.20606 v1 pith:X4DZCXTT submitted 2024-12-29 math.NT

Daniel R. Johnston This is my paper · ORCID

classification math.NT MSC 11F1111F2511F33

keywords newformscharacteristicpmodularformswithcharacterAtkin-LehneroperatorFrickeU-ell-newtraceDirichlet

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 thesis addresses a basic problem: in characteristic p, the classical decomposition of cusp forms into old and new parts breaks because of congruences, and the inner-product definition of newform no longer behaves. The paper takes up an algebraic way of defining newforms, using kernels of operators built from Hecke and Atkin-Lehner (and later Fricke) operators, and asks whether the two main algebraic notions agree for forms with Dirichlet character. Its central result is that they do: whenever ℓ divides the level exactly once and the character is induced from level $N/\ell$, the space $S^\chi(N,B)^{U^\chi_\ell\text{-new}}$ equals $S^\chi(N,B)^{\mathrm{Tr}^\chi_\ell\text{-new}}$ for any coefficient domain $B$ of characteristic $p$ containing $1/N$ and the values of $\chi$. In characteristic zero the same algebraic notions are shown to reproduce the classical newforms with character, and the trace definition is extended to levels where ℓ divides N more than once by replacing the Atkin-Lehner operator with the scaled Fricke operator. A sympathetic reader should care because this supplies a definition of newform that survives in characteristic p and is available for a wider class of levels.

What carries the argument

The load-bearing objects are the modified Hecke operator $D^\chi_\ell = \ell^2 U_\ell^2 - \chi'(\ell)\ell^k$ and the trace operators $\mathrm{Tr}^\chi_\ell(f) = f + \ell^{1-k}U_\ell W_\ell f$ (when $\ell$ divides $N$ exactly once) and $\mathrm{Tr}^\chi_\ell(f) = \chi(-1)\ell N^{-k}H_{N/\ell}U_\ell H_N f$ (in general), where $W_\ell = \ell^{k/2}w_\ell$ is the scaled Atkin-Lehner operator and $H_N = N^{k/2}h_N$ is the scaled Fricke operator. The character enters through the twisted eigenvalue condition $U_\ell^2 = \chi'(\ell)\ell^{k-2}$ and through the operator identity $\ell T_\ell = \ell U_\ell + \chi(\ell)W_\ell$ on forms coming from level $N/\ell$; these identities are what make the kernels of $D^\chi_\ell$, $\mathrm{Tr}^\chi_\ell$, and $\mathrm{Tr}^\chi_\ell W_\ell$ comparable. The p-adic lifting argument is the engine that transfers kernel equalities from characteristic zero down to characteristic $p$.

What would settle it

To test the central equality directly, compute the dimensions of $\ker(D^\chi_\ell)$ and $\ker(\mathrm{Tr}^\chi_\ell) \cap \ker(\mathrm{Tr}^\chi_\ell W_\ell)$ inside $S^\chi_k(N,\mathbb{F}_p)$ for a small case such as $N=22$, $\ell=2$, $\chi$ induced by the quadratic character mod $11$, and $p=3$, in a weight where the space is nonzero; if the two dimensions differ, Theorem 6.18 is false, and if they agree, the theorem survives this test.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is that 'new' can be defined purely algebraically in characteristic p, and that the two algebraic definitions coincide for cusp forms with character. The U definition declares $U^\chi_\ell$-new to be the kernel of $D^\chi_\ell = \ell^2 U_\ell^2 - \chi'(\ell)\ell^k$, so a form is new when $U_\ell^2 f = \chi'(\ell)\ell^{k-2}f$; the trace definition declares $\mathrm{Tr}^\chi_\ell$-new to be the simultaneous kernel of $\mathrm{Tr}^\chi_\ell$ and $\mathrm{Tr}^\chi_\ell W_\ell$. Theorem 6.18 proves these agree on $S^\chi(N,B)$ whenever $\chi$ is induced by a character mod $N/\ell$ and $B$ is a $Z[1/N,\chi]$-domain of characteristic $p$. The proof reduces the field case to $B = \mathbb{F}_p$, lifts a form in the $U$-kernel to $Z_p[\chi]$ with a $p$-adic decomposition into old and new parts, clears denominators, and pushes the kernel condition down modulo powers of $p$. The paper also shows in characteristic zero that the algebraic definitions agree with the classical $\ell$-new spaces, and it extends the trace definition to non-squarefree levels using the scaled Fricke operator, where the characteristic-zero agreement with classical newforms is proved via Li's theorem.

Load-bearing premise

The argument assumes, on the authority of an outside reference rather than a proof, that the scaled Fricke operator $H_N$ preserves coefficient rings of the form $Z[1/N,\zeta_d,\chi]$; if that integrality fails, the extension to non-squarefree levels and the equality in Theorem 6.21 lose their base ring.

Editorial extensions

If this is right

In squarefree level, cusp forms with character have a Hecke-theoretic definition of $\ell$-newform that is independent of the Petersson inner product, with the $U_\ell$ and trace formulations coinciding in characteristic $p$.
The trace/Fricke formulation gives a candidate definition of $\ell$-newform at non-squarefree levels; in characteristic zero it provably matches the classical $\ell$-new spaces, so characteristic-$p$ computations have a known characteristic-zero benchmark.
The explicit description of old forms lying in the $U$/trace kernels, via the equations $\ell T_\ell f = -(\ell+1)S_\ell g$ and $\ell T_\ell g = -(\ell+1)\chi'(\ell)f$, makes the failure of the naive old/new decomposition mod $p$ controllable.
Because the equality is proved by flatness and kernel preservation, it passes to every $Z[1/N,\chi]$-domain of characteristic $p$, so the algebraic theory is base-change friendly.
In characteristic zero, the algebraic definitions with character agree with the classical $\ell$-new spaces even when $\ell^2$ divides $N$, extending Li's characterization to modules over $Z[1/N,\zeta_d,\chi]$-domains.

Reading between the lines

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

If the missing Fricke integrality is supplied, the same p-adic lifting proof should yield the non-squarefree characteristic-$p$ equality, making the definition of newform uniform across all levels.
The equality of kernels suggests a canonical metric-free 'new' submodule of the Hecke module $S^\chi(N,B)$, which could serve as the correct input for studying Hecke-stable filtrations in characteristic $p$.
A computational scan of small $N$, $\ell$, $p$, and characters could map where the inclusion $S^\chi(N,B)^{\ell\text{-new}} \subseteq S^\chi(N,B)^{U^\chi_\ell\text{-new}}$ is strict, linking strictness to the congruence constant $(\ell+1)\ell^{(k-2)/2}\chi'(\ell)$.
The reliance on an outside reference for Fricke integrality points to a concrete missing lemma: proving $H_N$ integrality on $q$-expansion bases over $Z[1/N,\zeta_d,\chi]$ would put the non-squarefree story on the same footing as the squarefree case.

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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

Chapters 3–5 are a dependable, fully written-out account of Deo and Medvedovsky’s algebraic newforms in squarefree level and characteristic p. The exposition is careful, and it catches a real oversight in DM19: in weight 2 the Eisenstein space can make Uℓ-new and Trℓ-new disagree with the classical ℓ-new space (Example 3.18). That part is solid and useful on its own.

Chapter 6 is the original contribution: modular forms with character, the operators D^χ_ℓ and Tr^χ_ℓ, the equality S^χ(N,B)^{U^χ_ℓ-new} = S^χ(N,B)^{Tr^χ_ℓ-new} in characteristic p (Theorem 6.18), and a Fricke-operator version aimed at non-squarefree levels (Section 6.5). The author says at the start of Chapter 6 that most proofs will be short; that warning is accurate, and it marks the vulnerable spot.

The stress-test note is right to separate Theorem 6.18 from the Fricke integrality. Theorem 6.18 does not use H_N. It needs a “slight modification of Proposition 5.9” for the character-twisted ℓ-old subspace. That proposition describes elements f + W_ℓ(g) as sitting in the old space, so it presupposes S^χ_k(N,B)_ℓ-old = S^{χ′}_k(N/ℓ,B) ⊕ W_ℓ(S^{χ′}_k(N/ℓ,B)). For nontrivial χ, Section 6.1 defines the old space by extension of scalars from Z[χ]-integral oldforms. The direct-sum description is proved only in the trivial-character case (Proposition 5.5) using flatness over Z[1/ℓ], and that argument does not carry to characteristic-p coefficient rings. If the decomposition fails for some nontrivial character, the modified Proposition 5.9 and the p-adic lifting argument do not apply to the tensor-defined old subspace, and Theorem 6.18 is not established. This is a real gap in the central new claim, not a stylistic quibble.

The other issue is smaller: the integrality of the scaled Fricke operator in Section 6.5 is cited to a MathOverflow answer [Loe]. That underpins Definition 6.20 and Theorem 6.21, not 6.18, so it is not what threatens the main theorem, but it is still a shaky foundation for a claimed generalization.

The citation pattern is fair; DM19 and Li are used appropriately, and the thesis is explicit about which parts are expository and which parts are new. There is no machine-checked proof or reusable code here, so the claims stand on ordinary prose proofs, which is exactly why the sketched parts matter.

The audience is clear: graduate students and researchers using mod p Hecke algebras, Monsky-style filtrations, or newform congruences will get real value from Chapters 3–5 and from Chapter 6 as a roadmap. I would send it to a serious referee, with the expectation that the author fills in the character-twisted old-space proof and replaces the MathOverflow citation with a proof. Conditional acceptance, not desk reject.

Referee Report

3 major / 3 minor

Summary. The manuscript is an MSc thesis that develops algebraic definitions of newforms in characteristic p, following Deo and Medvedovsky. Chapters 1–3 review classical Atkin–Lehner theory and the squarefree-level Uℓ-new and Trℓ-new notions. Chapter 4 sets up modular forms over commutative rings and proves that in characteristic zero the algebraic notions coincide with classical newforms. Chapter 5 treats characteristic p and proves (Theorem 5.10) that for a Z[1/ℓ]-domain B of characteristic p, S(N,B)ℓ-new ⊆ S(N,B)Uℓ-new = S(N,B)Trℓ-new. Chapter 6 generalizes to modular forms with Dirichlet character, defining Uχℓ-new and Trχℓ-new and claiming equality in characteristic p (Theorem 6.18); it also proposes a Fricke-operator variant for non-squarefree levels (Theorem 6.21). The new part of the thesis is Chapter 6, where several proofs are presented only as sketches.

Significance. If the claims are correct, Theorem 6.18 is a natural extension of Deo–Medvedovsky's theorem to forms with character, and the framework provides useful algebraic replacements for classical newforms in characteristic p. The thesis is valuable as an exposition: it fills in many details of [DM19], gives explicit computations and worked examples, and clearly identifies the places where classical theory breaks down. However, the genuinely new part of Chapter 6 rests on several proof sketches and on an externally cited integrality statement, so the significance is conditional on completing those arguments.

major comments (3)

[§6.4, Theorem 6.18] The proof of Theorem 6.18 is not complete as written. It invokes 'a slight modification of Proposition 5.9' for character-twisted spaces, but that proposition presupposes a direct-sum description Sχ_k(N,B)ℓ-old = Sχ′_k(N/ℓ,B) ⊕ Wℓ(Sχ′_k(N/ℓ,B)), or at least trivial intersection of the two summands. Section 6.1 defines Sχ_k(N,B)ℓ-old by base change from Z[χ]-integral oldforms, and the only direct-sum statements proved earlier (Proposition 4.5 and Proposition 5.5(b)) are for the trivial character. No character-twisted analogue of Lemma 4.6/Proposition 4.5 is stated or proved. If the intersection is nontrivial for some nontrivial χ, the displayed description of (Sχ_k(N,B)ℓ-old)Uχℓ−new is unjustified and the p-adic lifting argument in Theorem 6.18 has no foundation. This is load-bearing because Theorem 6.18 is the central new result.
[§6.3–6.4, Theorems 6.13 and 6.17] The characteristic-zero identifications Sχ(N,B)ℓ-new = Sχ(N,B)Uχℓ−new and Sχ(N,B)ℓ-new = Sχ(N,B)Trχℓ−new are asserted by saying that one repeats the proofs of Theorems 3.9/3.14 'whilst keeping track of character'. In Theorem 6.13 the reduction to B=C is justified only by flatness of B over Z[χ]; flatness alone does not transfer an equality of kernels unless one has already established the relevant base-change compatibility and Z[χ]-integral bases for the spaces and kernels. Proposition 6.2 provides bases, but the kernel/base-change step is not written. Since Theorem 6.18's p-adic argument uses the characteristic-zero equality over Q_p[χ] via its lifting step, this omitted verification is also load-bearing for the central claim.
[§6.5, Definition 6.20 and Theorem 6.21] Property (iv) of the Fricke operator — that h_N and H_N extend to Mχ_k(N,B) for any Z[1/N,ζ_d,χ]-algebra B — is cited only to a MathOverflow answer [Loe] and is not proved in the thesis. This is not a purely cosmetic gap: Definition 6.20 and Theorem 6.21 are formulated in terms of Trχℓ and H_N, and the equality Sχ(N,B)ℓ-new = Sχ(N,B)Trχℓ−new′ depends on the extension of H_N to the coefficient ring B. The footnote in §6.2 even records that the author is unaware of a proof of the expected stronger Z[1/ℓ,ζ_m]-integrality. For a journal submission, the integrality statement cited to [Loe] needs to be either proved or replaced by a complete peer-reviewed reference. I note that this gap does not bear on Theorem 6.18, since that theorem avoids H_N.

minor comments (3)

[§5.2] The displayed formula 'Tℓ = Uℓ + ℓSℓWℓf' appears to be a typo; the subsequent argument in Proposition 5.9 uses the correct relation ℓUℓ = ℓTℓ − Wℓ, consistent with Tℓ = Uℓ + ℓ^{-1}Wℓ. Please correct the formula.
[§6.2, footnote (i)] The acknowledgement that the expected Z[1/ℓ,ζ_m]-integrality of the Atkin–Lehner operator is unproved should be moved into the main text, since it directly affects the scope and reliability of the non-squarefree generalization in Section 6.5.
[Notation throughout] The notation S(N,B) is used both for the sum over weights and for the algebra of cusp forms; in characteristic p this sum is not direct, so statements such as S(N,B)ℓ-new ⊆ S(N,B)Uℓ−new in Theorem 5.10 should explicitly state whether they are made in each graded piece or in the non-direct sum. The proof argues in fixed weight, but the theorem statements would benefit from an explicit convention.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central char-p equality is proven by lifting and external theorems, not by definitional identity.

full rationale

The paper's main claimed new result, Theorem 6.18, equates two algebraically defined subspaces, S^chi(N,B)^{U^chi_ell-new} and S^chi(N,B)^{Tr^chi_ell-new}. These are defined independently as kernels of the operators D^chi_ell and Tr^chi_ell (with Tr^chi_ell W_ell), and the equality is not a built-in consequence of those definitions. The proof proceeds by reducing to the old subspace via a 'slight modification of Proposition 5.9' and then by a p-adic lifting argument that uses the characteristic-zero equality, whose proof rests on independent external results such as the Weil bound and Li's newform theory. The constant chi'(ell) entering D^chi_ell is taken from [Li75, Theorem 3], not fitted to the target equality. The thesis's reliance on Deo-Medvedovsky is explicit and external rather than self-citational: the author of the thesis is not an author of [DM19]. The unproved integrality property of the Fricke operator cited to [Loe] affects Definition 6.20 and Theorem 6.21, not the stated central claim Theorem 6.18, and is a support gap rather than a circularity. Similarly, the possible missing character-twisted old-subspace decomposition in the proof sketch of Theorem 6.18 is a correctness concern, not an instance of an output being equivalent to its input by construction. No fitted parameter is renamed as a prediction, and no load-bearing claim reduces to a self-citation chain. The derivations are therefore not circular.

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

The thesis introduces no physical or mathematical entities beyond new operators and definitions (U^chi_ell-new, Tr^chi_ell-new'). Its central claims rest on standard external theorems (Weil bound, Goren, Li) and on one non-peer-reviewed citation for Fricke integrality. No fitted parameters appear.

assumptions (4)

standard math Deligne's Weil bound for Hecke eigenvalues of modular forms (with character)
Used in Lemma 3.10 and Theorem 3.9 to exclude ell-old forms from ker(D_ell), and cited to [Del74]. The thesis does not prove it.
domain assumption Integrality of Atkin-Lehner operator over Z[1/ell] (Conrad) and of Fricke operator over Z[1/N, zeta_d, chi] (MathOverflow [Loe])
Needed to define W_ell and H_N on modular forms over rings B. The Fricke integrality is not proven in the thesis and is cited to a non-peer-reviewed source, Section 6.5 property (iv).
domain assumption Goren's theorem on the kernel of the q-expansion map in characteristic p
Proposition 5.3 relies on [Gor02, Theorem 5.4] to decompose M(N,B) by weight mod p-1. This is a deep external result.
standard math Characteristic-zero theory of newforms with character (Li, Weisinger), including [Li75, Theorem 4] and [Wei77, Proposition 16/19]
Used in Theorem 6.21 and Section 6.5 to identify ell-new spaces with kernels of trace/Fricke operators over C.

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Pith. "Pith review of Defining newforms in characteristic $p$." pith.science (2026). https://pith.science/paper/X4DZCXTT

@misc{pith2026241220606,
  author       = {Pith},
  title        = {Pith review of: Defining newforms in characteristic $p$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X4DZCXTT}},
  note         = {Machine review of arXiv:2412.20606}
}

abstract

The theory of newforms, due to Atkin and Lehner, provides a powerful method for decomposing spaces of modular forms. However, many problems occur when trying to generalise this theory to characteristic $p$. Recently, Deo and Medvedovsky have suggested a way around these problems by using purely algebraic notions to define newforms. In this thesis, we describe the methods of Deo and Medvedovsky in detail and generalise their results where possible.

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Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Hecke operators on Γ 0(m)

[AL70] L Atkin AO and J Lehner. “Hecke operators on Γ 0(m)”. In: Math. Ann 185 (1970), pp. 134–160. [AM69] M Atiyah and IG MacDonald. Introduction to Commutative Algebra . Addison- Wesley,

work page 1970

[2] [2]

Formes modulaires et fonction s zêta p-adiques

Warszawa. 1983, pp. 503–514. [Ser73] Jean-Pierre Serre. “Formes modulaires et fonction s zêta p-adiques”. In: Modular functions of one variable III . Springer, 1973, pp. 191–268. 77 [Ste07] William A Stein. Modular forms, a computational approach . American Math. Soc.,

work page 1983

[3] [10]

Ring-theoretic pr operties of certain Hecke algebras

URL : https://math.mit.edu/classes/ [TW95] Richard Taylor and Andrew Wiles. “Ring-theoretic pr operties of certain Hecke algebras”. In: Annals of Mathematics 141.3 (1995), pp. 553–572. [Wei77] James Roger Weisinger. “Some results on classical E isenstein series and modular forms over function ﬁelds”. PhD thesis. Harvard University ,

work page 1995

[4] [1969]

On the modularity of elli ptic curves over Q: wild 3- adic exercises

[Bre+01] Christophe Breuil et al. “On the modularity of elli ptic curves over Q: wild 3- adic exercises”. In: Journal of the American Mathematical Society 14.4 (2001), pp. 843–939. [Car99] David Carlton. On a Result of Atkin and Lehner

work page 2001

[5] [1976]

Newforms and functional equa tions

[Li75] Wen-Ch’ing Winnie Li. “Newforms and functional equa tions”. In: Mathematis- che Annalen 212.4 (1975), pp. 285–315. [Loe] David Loefﬂer. Integrality of Atkin-Lehner operator for Γ 1(N). Math Overﬂow. URL : http://mathoverflow.net/q/353743 (visited on 04/26/2020). [MO03] William J McGraw and Ken Ono. “Modular form congruenc es and Selmer groups”. In:...

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[6] [1977]

Modular elliptic curves and Fermat’ s last theorem

[Wil95] Andrew Wiles. “Modular elliptic curves and Fermat’ s last theorem”. In: Annals of mathematics 141.3 (1995), pp. 443–551. 78

work page 1995

[7] [1999]

On a Result of Atkin and Lehner

arXiv: math/9903131 [math.NT]. [CS17] Henri Cohen and Fredrik Strömberg. Modular forms: A Classical Approach. V ol

work page Pith review arXiv

[8] [2002]

Congruences between systems of eigenvalues of modular forms

76 [Joc82] Naomi Jochnowitz. “Congruences between systems of eigenvalues of modular forms”. In: Transactions of the American Mathematical Society 270.1 (1982), pp. 269–285. [Kat73] N Katz. “p-Adic properties of modular forms and modu lar schemes”. In: Modular functions of one variable, III (Proc. Internat. Summer Scho ol, Univ. Antwerp, Antwerp, 1972), LN...

work page 1982