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Towards the $p$-adic derived Hecke algebra for weight one forms

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Pith's one-line read A p-adic conjecture ties weight-one Hecke actions to Stark-unit logarithms

desk verdict A promising p-adic complement to Harris–Venkatesh, with a solid theorem on old forms but a load-bearing gap in the flat cohomology construction that leaves Conjecture 2 undefined. read the letter →

arxiv 2506.09139 v1 pith:BUSAZAWY submitted 2025-06-10 math.NT

Robin Zhang This is my paper
classification math.NT MSC 11F1111F3311F8511G1811R42
keywords p-adicderivedHeckealgebraweightonemodularformsShimuraclassesStarkunitscompletedcohomologyflatregulatorHarris–Venkateshconjecture
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 tries to move the Harris–Venkatesh derived Hecke algebra story from modulo-$p$ coefficients to the $p$-adic side for weight-one modular forms. It outlines $p$-adic Shimura classes on completed cohomology of modular curves and a $p$-adic derived Hecke operator, then proposes Conjecture 2: for each newform $f$ of weight one, the resulting $p$-adic norm $||f||^2_{\mathbb{Z}_p}$ equals the $p$-adic regulator of a unit in the Stark unit group attached to the adjoint representation of $f$. If true, the $\mathbb{Z}_p$-component of the Shimura class completes the mod-$p$ Harris–Venkatesh conjecture and links derived Hecke actions to $p$-adic logarithms of Stark units. Much of the construction, especially the flat-cohomology passage in Section 5.1, is sketched rather than fully proved, so the conjecture is proposed conditionally on that step.

What carries the argument

The engine is the $p$-adic Shimura class $S_{\mathbb{Z}_p,{\rm fl}}$, obtained as the $\mathbb{Z}_p$-factor of a tower of finite flat covers $X_1(p^n) \to X_0(p^n)$. Although these covers are only \'etale at $n=1$, the paper argues they still define classes in flat cohomology $H^1_{\rm fl}$, which after pushforward and Serre duality become elements of $\operatorname{Hom}_{\mathbb{Z}_p}(\widehat{H}^0(X_0(p^\infty), \Omega), \mathbb{Z}_p)$. The other load-bearing object is the $p$-adic system $\widehat{f^*}$ of the dual weight-one form: a one-dimensional projective limit in the ordinary case, generated by $\widehat{f^*}_n = \alpha^{1-n} f^*_n - \alpha^{-n} \omega(p) f^*_{n-1}$, where $\alpha$ is the unique unit root of $X^2 - a_p X + \omega(p)p$. Pairing the Shimura class with $f \cdot \widehat{f^*}$ defines the $p$-adic norm, and the same construction with a Hecke lifting produces the operator $T_{\mathbb{Z}_p,N}$ on the ordinary subspace of coherent cohomology.

What would settle it

Write the cochain on $X_1(p^n) \times_{X_0(p^n)} X_1(p^n)$ for $n > 1$ explicitly and verify it is a flat $1$-cocycle; if it fails, the construction of $S_{\mathbb{Z}_p,{\rm fl}}$ collapses. Independently, compute $||f||^2_{\mathbb{Z}_p}$ via the displayed limit for a concrete dihedral form and compare it with $\operatorname{Reg}_{\mathbb{Z}_p}(u)$ for a range of ordinary primes; any mismatch disproves Conjecture 2, while a single matching prime would be genuine evidence.

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Extended reading notes

Core claim

On the paper's own terms, the central proposed discovery is a conjectural equality: there exists an element $u \in U(\operatorname{Ad}(\rho_f)) \otimes \mathbb{Q}$ and a prime $p_0$ such that for all primes $p \geq p_0$, $$||f||^2_{\mathbb{Z}_p} = \operatorname{Reg}_{\mathbb{Z}_p}(u),$$ equivalently $\langle f^*, T_{\mathbb{Z}_p,N}(f) \rangle_{\rm SD} = \operatorname{Reg}_{\mathbb{Z}_p}(u)$. The norm is defined as a $p$-adic period pairing the $p$-adic Shimura class $S_{\mathbb{Z}_p}$ against $f \cdot \widehat{f^*}$, where $\widehat{f^*}$ is built from the $p$-adic system of the dual form by the limit $\widehat{f^*}_n = \alpha^{1-n} f^*_n - \alpha^{-n} \omega(p) f^*_{n-1}$. This $\mathbb{Z}_p$-component is complementary to the $(\mathbb{Z}/p\mathbb{Z})^\times$ component of the Harris–Venkatesh conjecture, and together they give a single equality over $\mathbb{Z}_p^\times$.

Load-bearing premise

Everything rests on the claim, sketched in Section 5.1 rather than proved, that the non-\'etale finite flat covers $X_1(p^n) \to X_0(p^n)$ for $n > 1$ genuinely produce flat cohomology classes that survive passage to Zariski cohomology and Serre duality; without that, neither the $p$-adic norm nor $T_{\mathbb{Z}_p,N}$ is defined.

Editorial extensions

If this is right

  • For every ordinary prime $v$ over $p$, Theorem 4 gives a one-dimensional projective limit $\widehat{f^*}$ in the completed cohomology, so the $p$-adic norm is a specific element of the completed coefficient ring $\mathcal{O}_v$ rather than an abstract class.
  • If Conjecture 2 holds, the ordinary subspace carries a $p$-adic derived Hecke operator $T_{\mathbb{Z}_p,N}$ whose Serre-duality pairing on $(f^*, f)$ is exactly the $p$-adic regulator of a Stark-unit-group element.
  • Together with the mod-$p$ Harris–Venkatesh conjecture, the $p$-adic statement yields a single equality $||f||^2_{\mathbb{Z}_p^\times} = \operatorname{Reg}_{\mathbb{Z}_p^\times}(u)$, combining the torsion and $\mathbb{Z}_p$ components for all sufficiently large primes.
  • The construction of $\widehat{f^*}$ via the unit root $\alpha$ predicts that only ordinary primes contribute nonzero $p$-adic periods, matching the known arithmetic distinction between ordinary and non-ordinary weight-one forms.

Reading between the lines

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

  • Beyond the paper, the conjecture can be tested numerically for dihedral forms: compute $\operatorname{Reg}_{\mathbb{Z}_p}(u)$ from the splitting of $p$ in the quadratic field and compare it with the explicit limit formula for $||f||^2_{\mathbb{Z}_p}$; existing computations for the mod-$p$ case make this a finite check.
  • Should the flat-cohomology step fail for $n > 1$, a fallback exists: the limit formula in Section 5.3 is written explicitly through $\mathbb{Z}/p^n\mathbb{Z}$-cohomology, so the norm could be defined by that limit directly without the non-\'etale flat classes; the paper does not discuss this possibility.
  • The author's hope of relating the conjecture to $p$-adic $L$-functions suggests a stronger identity: for dihedral $f$, $\operatorname{Reg}_{\mathbb{Z}_p}(u)$ should equal the derivative of a $p$-adic Artin $L$-function at $s=0$, giving the derived Hecke operator a $p$-adic $L$-value interpretation that the paper leaves open.
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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 / 5 minor

Summary. The paper proposes a p-adic analogue of the Harris–Venkatesh derived Hecke algebra for weight-one modular forms. It reviews the mod-p constructions, proves a non-vanishing/uniqueness theorem for a p-adic family of dual forms in completed cohomology under ordinariness (Theorem 4), sketches constructions of p-adic Shimura classes via flat cohomology, and formulates Conjecture 2 asserting that the resulting p-adic norm equals the p-adic regulator of a Stark unit.

Significance. Should Conjecture 2 hold, it would provide a new p-adic complement to the Harris–Venkatesh conjecture, linking derived Hecke actions to p-adic logarithms of Stark units and potentially to p-adic L-functions. The paper also contains a complete, transparent proof of Theorem 4, and it openly distinguishes established results from speculation. However, the central construction of the p-adic Shimura class is not established, and the p-adic regulator is not defined in sufficient detail; these gaps currently prevent Conjecture 2 from being a well-defined mathematical statement.

major comments (3)
  1. [§5.1] Section 5.1 claims that the finite flat covers π_n: X_1(p^n) → X_0(p^n) define flat cohomology classes S_{(Z/p^nZ)^×,fl} ∈ H^1_fl(X_0(p^n), ((Z/p^n Z)^×)_a). This is not justified. In fppf cohomology, H^1_fl(Y,G) classifies G-torsors. A group action on a finite flat cover yields a torsor only if the action is free, i.e., X×_Y X ≅ X × G. For n>1, the action of (Z/p^nZ)^× on X_1(p^n) has nontrivial stabilizers at cusps and at supersingular points in characteristic p. The assertion that the fiber product 'is a group scheme over X' and 'gives a cochain' does not produce a torsor cocycle. Hence the existence of S_{Z_p,fl} is unproved, and since ||f||²_{Z_p} (Section 5.3) and T_{Z_p,N} (Section 5.4) are defined through this class, Conjecture 2 is not well-posed as written. This point must be fixed—either by a rigorous construction (for example, via a stack-theoretic or syntomic form of the class) or by explicitly stating the existence as an additional hypothesis.
  2. [§2] Section 2 defines the p-adic regulators Reg_{Z_p} and Reg_{Z^×_p} in one sentence by 'omitting the reduction modulo w step' and applying the p-adic logarithm. No proof is given that the result is independent of the choice of place w over p, independent of the branch of the p-adic logarithm, or even that the evaluation at x_{Frob_w} lands in a domain where the p-adic logarithm is defined. Since Conjecture 2 asserts an equality with Reg_{Z_p}(u), this regulator must be a canonically defined map. The manuscript should provide a precise definition and prove its well-definedness, or state the independence as an assumption.
  3. [§5.3] The sentence in Section 5.3, 'Concretely, what we have done in Section 5 is the construction of the class S_{Z/p^nZ,fl} ∈ H^1(X_{0,1}(p^{n+1},N), Z/p^nZ) using the étale cover X_1(p^n) → X_0(p^n),' contradicts Section 5.1, which states that these covers are not étale for n>1. This internal inconsistency underlines the unresolved status of the flat cohomology construction and must be resolved.
minor comments (5)
  1. [§3.1] The diagram for the mod-p Shimura class is hard to follow because the steps (a)–(d) are listed after the diagram containing arrows labeled '(a)', etc.; please reorder or add explicit references to the maps. Also, the notation 'Tr Np p' in §3.2 appears garbled; the trace should be written with clear indices.
  2. [§5.1] The quotient sheaf ((Z/p^nZ)^×)_a is introduced without definition; please define its embedding into G_a and the topology used.
  3. [§5.2] The isomorphism ∆ × Z_p ≅ Z_p^× and the decomposition S_{Z_p^×,fl} = S_{∆,fl} × S_{Z_p,fl} are stated without proof; at least indicate how this splitting is compatible with the flat cohomology classes.
  4. [Conjecture 2] Conjecture 2 and the surrounding text use the notation ⟨f^*, T_{Z_p,N}(f)⟩ without specifying the pairing used; the pairing identity in §5.4 defines it for ordinary forms, but the conjecture should restate this.
  5. [Remark 3] The paper would benefit from a precise statement of the relation between u and the Stark unit u_f; Remark 3 acknowledges this as a hope, but the unclarity weakens the conjecture's testability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Conjecture 2 equates a geometrically defined p-adic norm with an independently defined p-adic regulator; neither side is fitted to the other.

full rationale

The central new claim, Conjecture 2, asserts the existence of a global u in U(Ad(ρ_f))⊗Q and a prime p0 such that ||f||^2_Zp = Reg_Zp(u) for all p ≥ p0. The left-hand side is defined in Section 5.3 as S_Zp(f·bf^*), i.e. the pairing of the explicitly constructed element f·bf^* with the proposed p-adic Shimura class. The right-hand side is the p-adic regulator on the Stark unit group defined in Section 2 by evaluating at a distinguished Frobenius element, embedding into a p-adic field, and applying the p-adic logarithm. These two quantities are constructed independently: the norm depends on the modular curve coverings and the projective-limit element bf^*, while the regulator depends only on the global Galois representation Ad(ρ_f) and a chosen place over p. No parameter is fitted to force the equality, and no equation in the paper defines one side in terms of the other. The self-citations to [Zha23b] and [Zha23c] supply background, the mod-p framework, and the speculative Remark 3 about a unique u_f; they are not used to force u = u_f or to define Reg_Zp. The genuine weakness flagged in Section 5.1, namely whether the non-étale covers X_1(p^n)→X_0(p^n) actually produce well-defined flat cohomology classes, is a potential gap in the construction that would make Conjecture 2 ill-posed, but it is not a circular reduction of the conjecture to its own inputs. Therefore the appropriate circularity finding is no significant circularity.

Assumptions & free parameters 2 free parameters · 5 assumptions · 3 invented entities

The paper introduces no numerically fitted constants; its underdetermination comes from the existential unit u and threshold p0 in Conjecture 2. It depends on standard theorems (Deligne-Serre, Serre duality, Stark background) and on two sketched constructions: the flat cohomology Shimura class and the p-adic regulator. The invented entities are mathematical objects proposed for the conjecture; none has independent falsifiable evidence outside the paper.

free parameters (2)
  • existential unit u in Conjecture 2 = unspecified
    The conjecture asserts existence but gives no recipe for u; Remark 3 admits the relation to u_f from [Zha23b] is unknown.
  • prime threshold p0 = unspecified
    Both Conjecture 1 and Conjecture 2 require equality for all sufficiently large primes; no effective p0 is given.
assumptions (5)
  • standard math Every weight-one newform f has an associated two-dimensional Artin representation ρ_f (Deligne-Serre).
    Used throughout Sections 1 and 2 to define Ad(ρ_f), the Stark unit group, and the adjoint representation.
  • domain assumption The Stark conjectures predict a Stark unit u_Stark in U(Ad(ρ_f)) ⊗ Q whose regulator gives the leading term of L(Ad(ρ_f), s) at s = 0.
    Section 2 invokes Stark's conjectures to name the unit group and motivate Conjecture 2; the truth of the Stark conjecture is not needed to state Conjecture 2, but it underpins the interpretation.
  • ad hoc to paper The finite flat covers π_n: X_1(p^n) -> X_0(p^n) define flat cohomology classes S_{(Z/p^nZ)^×,fl} as stated, including for non-etale n > 1.
    Section 5.1 sketches this; it is load-bearing for S_{Z_p} and Conjecture 2.
  • ad hoc to paper The p-adic regulator Reg_{Z_p} is well-defined and independent of choices of place w_v over p and branch of the p-adic logarithm.
    Defined in one sentence in Section 2; Conjecture 2 equates a norm to this regulator.
  • domain assumption For weight-one newforms, ordinarity holds automatically for p >= 5 with a_p != 0; only p = 2, 3 need an explicit assumption.
    Section 4.2 argues this via roots of unity and gives dihedral details; the general statement is only sketched.
invented entities (3)
  • p-adic Shimura class S_{Z_p}
    purpose: Element of Hom_{Z_p}(bH^0(X_0(p^∞), Ω), Z_p) used to define the p-adic norm ||f||^2_{Z_p}.
    Introduced in Section 5.2 through flat cohomology and Serre duality; no external falsifiable prediction is attached.
  • p-adic derived Hecke operator T_{Z_p,N}
    purpose: Degree-raising operator on ordinary coherent cohomology whose Serre-duality pairing with f* is conjectured to equal Reg_{Z_p}(u).
    Defined in Section 5.4 via a lifting and cup product with S_{Z_p}; its conjectural meaning is exactly Conjecture 2.
  • bf* in completed cohomology
    purpose: Generator of the projective limit bV_f* of old-form spaces; f·bf* is paired with S_{Z_p}.
    Constructed and proved in Theorem 4 under ordinariness; no external handle beyond the paper.

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Pith. "Pith review of Towards the $p$-adic derived Hecke algebra for weight one forms." pith.science (2026). https://pith.science/paper/BUSAZAWY

@misc{pith2026250609139,
  author       = {Pith},
  title        = {Pith review of: Towards the $p$-adic derived Hecke algebra for weight one forms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BUSAZAWY}},
  note         = {Machine review of arXiv:2506.09139}
}
abstract

This note outlines an approach to defining $p$-adic Shimura classes and $p$-adic derived Hecke operators on the completed cohomology of modular curves from upcoming work by the author. After reviewing the modulo-$p$ constructions of Harris and Venkatesh, we formulate a conjecture relating the action of $p$-adic derived Hecke operators on cusp forms of weight $1$ and level $\Gamma_1(N)$ to the $p$-adic logarithm of the Stark unit for the corresponding adjoint Deligne-Serre representation. This new $p$-adic conjecture can be viewed as complementary to the Harris-Venkatesh conjecture.

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