REVIEW 4 major objections 10 minor
Iwasawa invariants of sharp/flat $2$-adic $L$-functions for quadratic twists of elliptic curves
T0 review · 4 major / 10 minor · reviewed 2026-07-07 · glm-5.2
Pith's one-line read Explicit formula tracks lambda-invariants under quadratic twists at p=2
desk verdict Kida-type formula for sharp/flat 2-adic lambda-invariants under quadratic twists, supersingular case at p=2 read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
Sprung's sharp/flat 2-adic L-functions; Mazur-Tate elements; the logarithmic matrix; finite-layer congruence relations; the Greenberg-Vatsal-type congruence; Selberg-Delange-type analytic number theory for distribution results
What would settle it
If the mu-invariant is nonzero for some curve satisfying the other hypotheses, the congruence in Proposition 3.13 would not imply equality of lambda-invariants, and the difference formula would fail. Computational verification against known examples (as in Section 5) could also falsify the formula if discrepancies arise.
Extended reading notes
Core claim
The central result is the explicit Kida-type difference formula (Theorem 3.16) for the sharp/flat 2-adic lambda-invariants under quadratic twists. The formula decomposes the change in lambda into local contributions from each prime dividing the twisting parameter D, where each local contribution is determined by whether the prime divides the conductor of E and by the parity of the point count of the reduction of E modulo that prime. The key mechanism is a Greenberg-Vatsal-type congruence (Proposition 3.13) between the sharp/flat L-function of the twist and an imprimitive sharp/flat L-function of the original curve, combined with an explicit computation of the lambda-invariants of the local h
Load-bearing premise
The entire formula depends on the hypothesis that the mu-invariant vanishes (mu = 0). This is a standard but unproved assumption in non-ordinary Iwasawa theory; if mu is nonzero, the key congruence between the twisted and imprimitive L-functions does not yield equality of lambda-invariants, and the formula breaks down.
Editorial extensions
If this is right
- The set of lambda-invariants arising from quadratic twists of a fixed supersingular elliptic curve at p=2 is unbounded, since one can choose twisting primes that each contribute a positive correction.
- The asymptotic lower bound X/(log X)^{2/3} for twists with prescribed lambda-invariant gives quantitative control over the distribution of analytic ranks in twist families, analogous to results in the ordinary case.
- The formula provides a computational tool: given E and D, one can predict the sharp/flat lambda-invariant of E_D without computing the full 2-adic L-function, using only local data (point counts and conductor divisibility).
- The supersingular 2-adic analogue is now on equal footing with the ordinary 2-adic case (Matsuno's formula), completing the picture for p=2 across both reduction types.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This paper studies the variation of analytic Iwasawa invariants of Sprung's sharp/flat 2-adic L-functions for elliptic curves over Q with good supersingular reduction at 2, under quadratic twists. Under the hypothesis that the mu-invariant vanishes, the author proves an explicit Kida-type difference formula for the sharp/flat lambda-invariants (Theorem 3.16), giving a supersingular analogue of Matsuno's formula in the ordinary case. As applications, the author shows that lambda-invariants can be made arbitrarily large via quadratic twists (Corollary 3.17) and obtains asymptotic lower bounds for the number of quadratic twists with prescribed lambda-invariant (Theorem 4.4), following the Hatley-Ray method. The proof proceeds by establishing period ratio results (Proposition 3.4, 3.6), a finite-layer congruence for Mazur-Tate elements (Proposition 3.8), a Greenberg-Vatsal-type congruence lifting to the inverse limit (Proposition 3.13), and an explicit computation of local lambda-contributions (Lemma 3.14).
Significance. The paper addresses a natural and timely problem in non-ordinary Iwasawa theory at p=2. The Kida-type formula (Theorem 3.16) is the central contribution and provides an explicit, computable formula for the variation of sharp/flat lambda-invariants under quadratic twists, filling a gap between the ordinary theory (Matsuno) and the odd supersingular theory (Pollack-Weston, Pratap-Ray). The inclusion of computational examples (Section 5) verifying the formula against Pollack's SageMath code adds value and falsifiability. The extension of Sprung's construction to general odd M and the explicit local factor computation (Lemma 3.14) are useful technical contributions. The asymptotic distribution result (Theorem 4.4) is a straightforward but welcome application.
major comments (4)
- §2.3, paragraph containing the statement 'Although Sprung formulates his construction only in the case M=1, the same argument applies verbatim to any odd integer M.' This is the most load-bearing unverified step in the paper. The factorization (2.2) in Proposition 2.9, which underpins Lemma 3.12 and hence the additive decomposition of lambda-invariants in Theorem 3.16, depends on Sprung's construction being valid for general odd M. The author's justification is that the Mazur-Tate elements with M still form a queue sequence and that the matrix C_j depends only on f. However, the kernel module M_n(f) and its inverse limit M(f) = {(0,0)} are stated to be 'independent of the auxiliary odd integer M' without proof. One needs to verify that the vanishing of the inverse-limit ambiguity for general M follows from the M=1 case, since the Mazur-Tate elements theta_{n,M} do depend on M. The author
- should either provide a more detailed justification (e.g., by explicitly tracing through Sprung's [34, Proposition 5.10] argument for general M) or at minimum explain precisely which steps of [34] go through unchanged and why the M-dependence of theta_{n,M} does not affect the kernel computation. As written, this is a gap in the proof of Proposition 2.9, which is load-bearing for Theorem 3.16.
- Proposition 3.13, proof: The passage from the finite-layer congruence (3.16) to the congruence (3.18) in the inverse limit uses the growth rate 2^n - q^{*/}_n -> infinity. The argument shows that if F* is nonzero mod 2, then for large n the T-adic order d + q*_n < 2^n, contradicting (3.16). This step appears correct, but the role of the hypothesis mu*_2(E, omega_i) = 0 is not explicitly invoked in the proof of Proposition 3.13 itself. The hypothesis is used only to conclude mu*_2(E_D, omega_i) = 0 via mu*_2,D(E, omega_i) = 0. The author should clarify where exactly mu=0 is needed: is it only for the final conclusion about mu(E_D)=0, or does the congruence (3.18) itself require mu=0? If the congruence (3.18) holds unconditionally, this should be stated; if it requires mu=0, the point in the proof where it enters should be identified.
- Proposition 3.8, proof: The congruence (3.9) states chi(a) = chi(2^{-(n+2)} c) = chi(2^{n+2})^{-1} chi(c) ≡ chi(c) mod 2. Since chi is a quadratic character and chi(2^{n+2}) = chi(2)^{n+2}, this requires chi(2)^{n+2} ≡ 1 mod 2, which is automatic since chi(2) = ±1. However, the step from chi(2^{-(n+2)} c) to chi(2^{n+2})^{-1} chi(c) implicitly assumes that c is coprime to 2D, which should be verified: under the change of variable c = bD + 2^{n+2} a with b coprime to 2^{n+2} and a ranging over Z/DZ, c is coprime to 2D only when gcd(a, D) = 1 and b is odd. The sum over a in Z/DZ includes non-units, so some terms c may not be coprime to D. The author should clarify how the non-coprime terms are handled (they should vanish since chi(a) = 0 for gcd(a, D) > 1).
minor comments (10)
- §1, Theorem 1.1: The notation '2 | #tilde_E_ell(F_ell)' in the summation condition is somewhat hard to parse. Consider writing '2 | #tilde_E_ell(F_ell)' with the vertical bar more clearly typeset, or rephrase as 'where #tilde_E_ell(F_ell) is even'.
- §2.1, line below Theorem 2.2: 'sgn(psi)' is used but not defined until the sentence following. Consider defining it before first use.
- §3.1, Proposition 3.3, proof: The reference '[22, Theorem 1.1 (2)]' for A(Q_2)[2] = {O} should be checked: the paper [22] is by Ozeki-Yoshida and is listed as a preprint. The result that A(Q_2)[2] = {O} for supersingular reduction at 2 is a known result, but the author should verify the reference is accurate.
- §3.1, Proposition 3.4, proof: The notation 'c_phi_A' for the dual isogeny is introduced but the letter 'c' is also used for the Manin constant c(A_0). Consider using a different notation such as 'phi_A^vee' to avoid confusion.
- §3.2, Lemma 3.7: The equality marked (*) uses the change of variable a -> -a and chi(-1) = 1. This is correct since D > 0 and D ≡ 1 mod 4, but the author should note that chi(-1) = 1 follows from D > 0 (not from D ≡ 1 mod 4).
- §3.3, Lemma 3.12: The proof is stated as 'We use the equality (2.2) repeatedly.' This is correct but a one-line elaboration (e.g., 'iterating (2.2) over the prime factors of D') would improve readability.
- §4, Theorem 4.4, proof: The set Q is defined as {ell ∤ 2N_E | ell ≡ 5 mod 8, ...} but the condition ell ≡ 5 mod 8 is not explicitly used in the proof. The author should clarify where this congruence condition enters (it appears to ensure n_ell = 0 via ord_2((ell^2-1)/8) = 0 for ell ≡ 5 mod 8, but this should be stated).
- §5, Remark 5.1: The statement 'these invariants coincide with the Iwasawa invariants of Sprung's sharp/flat 2-adic L-functions' via [34, Corollary 8.9] requires mu+ = mu- = 0. The author should note that this is verified computationally in each example, not assumed.
- References: [5] and [6] are listed as 'in preparation' and 'preprint' respectively. If the paper is accepted, the author should update these references with final publication details if available.
- §3.3, equation (3.19): The notation 'lambda_2(h_ell(E, omega_i, T)) := lambda_2(h_ell(f_E, omega_i, T))' uses ':=' which suggests a definition, but this is really an equality. Consider using '=' instead.
Simulated Author's Rebuttal
The referee raises three major comments concerning: (1) the extension of Sprung's construction from M=1 to general odd M, which underpins Proposition 2.9 and Theorem 3.16; (2) the role of the mu=0 hypothesis in the proof of Proposition 3.13; and (3) a technical point about coprimality in the proof of Proposition 3.8. We address each below and indicate revisions for all three points.
read point-by-point responses
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Referee: §2.3: The claim that Sprung's construction applies verbatim to general odd M is the most load-bearing unverified step. The kernel module M_n(f) and its inverse limit M(f) = {(0,0)} are stated to be independent of M without proof. The author should provide detailed justification or explain precisely which steps go through unchanged.
Authors: The referee is correct that this step is insufficiently justified in the current manuscript. We will revise §2.3 to provide a detailed argument. Specifically, the key observation is that the kernel module M_n(f) is defined as the kernel of the map ×C_1···C_nC^{-(n+3)}(-1,-1;β,α) on Λ_n^⊕2, where the matrices C_j depend only on f (via a_2(f) and ε_f(2)) and not on M. The Mazur–Tate elements θ_{n,M}(f,ω_i,T) do depend on M, but they enter Sprung's construction only through the queue sequence relation, which holds for all odd M by the same Hecke relation argument (cf. [34, (4.2)] and [21, §10]). The factorization in Proposition 2.6 expresses (θ_{n,M}, ν_{n-1/n}(θ_{n-1,M})) as a product of the vector L_{2,n,M}^{ω_i} with matrices depending only on f. The kernel M_n(f) is the set of vectors v ∈ Λ_n^⊕2 such that v·C_1···C_nC^{-(n+3)}(-1,-1;β,α) = 0, which is purely a statement about the matrix product and is therefore independent of M. The vanishing M(f) = lim M_n(f) = {(0,0)} then follows from [34, Proposition 5.10], whose proof depends only on properties of the matrix product C_1···C_nC^{-(n+3)}(-1,-1;β,α) and the supersingularity hypothesis ord_2(α) < 1, not on M. We will add this detailed explanation to the revised manuscript. revision: yes
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Referee: Proposition 3.13, proof: The role of the hypothesis mu*_2(E,ω_i)=0 is not explicitly invoked in the proof. The author should clarify where exactly mu=0 is needed: is it only for the conclusion about mu(E_D)=0, or does the congruence (3.18) itself require mu=0?
Authors: We thank the referee for this observation. The congruence (3.18) itself does not require the hypothesis μ=0; it is derived purely from the finite-layer congruence (3.16) and the growth rate 2^n - q*_n → ∞, which is a statement about the matrix C_1···C_n(f) mod 2. The hypothesis μ*_2(E,ω_i)=0 enters only in the final step: combining (3.18) with Lemma 3.12 to conclude μ*_2(E_D,ω_i)=0 and λ*_2(E_D,ω_i)=λ*_{2,D}(E,ω_i). Specifically, (3.18) gives L*_{2,1}(f⊗χ,ω_i,T) ≡ U_χ L*_{2,D}(f,ω_i,T) mod 2 with λ_2(U_χ)=0, so λ*_2(E_D,ω_i)=λ*_{2,D}(E,ω_i) holds whenever both sides have μ=0. The hypothesis μ*_2(E,ω_i)=0, combined with μ_2(h_ℓ)=0 for each ℓ|D (which follows from Lemma 3.14), gives μ*_{2,D}(E,ω_i)=0 via (3.13), and then (3.18) gives μ*_2(E_D,ω_i)=0. We will revise the proof of Proposition 3.13 to state explicitly that (3.18) holds unconditionally and that μ=0 is used only for the conclusions about μ(E_D)=0 and the equality of λ-invariants. revision: yes
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Referee: Proposition 3.8, proof: The step from chi(2^{-(n+2)}c) to chi(2^{n+2})^{-1}chi(c) implicitly assumes c is coprime to 2D. Under the change of variable c=bD+2^{n+2}a with b coprime to 2^{n+2} and a ranging over Z/DZ, c may not be coprime to D when gcd(a,D)>1. The author should clarify how non-coprime terms are handled.
Authors: The referee is correct that some terms in the sum over a ∈ Z/DZ may yield c not coprime to D, and this should be clarified. The key point is that χ is the quadratic Dirichlet character modulo D, so χ(c)=0 whenever gcd(c,D)>1. Under the change of variable c = bD + 2^{n+2}a, we have gcd(c,D) = gcd(2^{n+2}a, D) = gcd(a,D) since D is odd. Thus for terms with gcd(a,D)>1, we have χ(a) = 0 (since χ is a Dirichlet character modulo D and a is not coprime to D), and correspondingly χ(c) = 0. The congruence χ(a) ≡ χ(c) mod 2 then holds trivially (0 ≡ 0) for these terms. For terms with gcd(a,D)=1, c is coprime to D (since b is coprime to 2^{n+2} and hence odd, and gcd(a,D)=1), so χ(c) is well-defined and nonzero, and the congruence χ(2^{n+2})^{-1}χ(c) ≡ χ(c) mod 2 holds since χ(2^{n+2}) = χ(2)^{n+2} = ±1. We will add this explanation to the proof of Proposition 3.8 in the revised manuscript. revision: yes
Circularity Check
No circularity found; derivation chain rests on external citations (Sprung, Matsuno) with no load-bearing self-citations.
full rationale
The paper's central result (Theorem 3.16) is assembled from three independent components: (1) the factorization Proposition 2.9, whose proof cites Matsuno's calculations [19, Lemmas 2.2 and 3.3] and Sprung's Theorem 2.8 [34] — both external; (2) the Greenberg–Vatsal-type congruence Proposition 3.13, proved via finite-layer matrix computations using Sprung's kernel-vanishing result M(f) = {(0,0)} from [34, Proposition 5.10] — external; and (3) Lemma 3.14, a direct computation of λ_2(h_ℓ) from the definition of h_ℓ. No step reduces to its own inputs by construction. The author's only self-citation [2] (joint with Nomoto and Shii) appears solely in the acknowledgments and is not load-bearing for any theorem. The assertion in §2.3 that Sprung's construction extends from M=1 to general odd M is a correctness/gap concern (the author claims 'the same argument applies verbatim'), not a circularity: it does not cite the author's own prior work to justify itself, and the downstream factorization (2.2) is verified against Sprung's external Theorem 2.8. The examples in §5 check the formula against Pollack's independent SageMath code [25], providing external benchmarking. Score 1 reflects the minor unverified extension claim, which is a correctness risk rather than circularity.
Assumptions & free parameters
assumptions (5)
- domain assumption Vanishing of the mu-invariant: mu*_2(E, omega_i) = 0 for the base curve E.
- domain assumption Square-free conductor: N_E is square-free.
- standard math The Manin constant c(A_0) is not divisible by 2 (Theorem 3.2, Abbès-Ullmo).
- domain assumption Sprung's construction of sharp/flat L-functions extends verbatim from M=1 to general odd M.
- domain assumption D > 0 and D ≡ 1 mod 4.
Cite this review
Pith. "Pith review of Iwasawa invariants of sharp/flat $2$-adic $L$-functions for quadratic twists of elliptic curves." pith.science (2026). https://pith.science/paper/IYRBP64I
@misc{pith2026260705305,
author = {Pith},
title = {Pith review of: Iwasawa invariants of sharp/flat $2$-adic $L$-functions for quadratic twists of elliptic curves},
year = {2026},
howpublished = {\url{https://pith.science/paper/IYRBP64I}},
note = {Machine review of arXiv:2607.05305}
}
abstract
The aim of this paper is to study the variation under quadratic twists of the analytic Iwasawa invariants of Sprung's sharp/flat 2-adic $L$-functions for elliptic curves over $\mathbb{Q}$ with good supersingular reduction at $2$. Under the hypothesis that the $\mu$-invariant vanishes, we obtain an explicit formula for the sharp/flat $\lambda$-invariants. This formula gives a supersingular analogue of Matsuno's formula in the good ordinary case. As an application, we show that the sharp/flat $\lambda$-invariants can be made arbitrarily large even among quadratic twists by single primes. Moreover, using the method of Hatley-Ray, we obtain an asymptotic lower bound for the number of quadratic twists with a prescribed sharp/flat 2-adic Iwasawa $\lambda$-invariant.
Reviewed July 7, 2026 · model on record in the stance chip above.
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