{"id":"1480bf46-05eb-4ff2-ac3a-a58ba3a7fc01","arxiv_id":"2607.05305","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"An explicit Kida-type difference formula for sharp/flat 2-adic lambda-invariants of supersingular elliptic curves under quadratic twists is proved, yielding asymptotic lower bounds for twists with prescribed invariants.","lead":"This paper proves an explicit formula for how certain 2-adic L-function invariants (lambda-invariants) change when an elliptic curve with good supersingular reduction at 2 is quadratically twisted. It matters because it extends Kida-type formulas to the supersingular setting at p=2, enabling distribution results for these invariants across twist families.","discovery_kind":"extension","skeptic_critique":{"model":"glm-5.2","headline":"The extension of Sprung's construction from M=1 to general odd M is asserted without proof and underpins the key factorization (2.2); this is the most load-bearing unverified step.","rationale":"The reader correctly identifies mu=0 as a necessary hypothesis, but this is a standard and clearly stated assumption in the field — it is not a gap in the proof so much as a condition on the theorem's applicability. The more pressing concern is the unverified extension of Sprung's construction to general odd M, which is needed for Proposition 2.9 (the factorization by h_ℓ) and hence for Lemma 3.12 (additivity of lambda over primes dividing D). The paper asserts this extension is verbatim from the M=1 case, but the dependence of the Mazur–Tate elements and the kernel module on M makes this nontrivial. The computational examples in §5 provide partial verification (the formula checks out for specific curves and twists), which is reassuring but does not constitute a proof of the general M case. The proof of Proposition 3.13 (the Greenberg–Vatsal congruence) is also somewhat compressed, particularly the T-adic order argument, but the logic appears sound: if F* ≠ 0 mod 2, the T-adic order d is finite, and for large n the product T^{q*_n} · F*_n is nonzero mod (2, T^{2^n}), contradicting (3.16). The examples cover curves with a_2 ∈ {0, 2, -2}, testing different supersingular cases. The condition D ≡ 1 mod 4 is used crucially in Proposition 3.6 (period ratio) and Proposition 3.8 (congruence of Mazur–Tate elements). Overall, the paper makes a solid contribution, but the result should remain conditional pending verification of the general-M extension of Sprung's construction. The CONDITIONAL verdict is appropriate.","tokens_in":22098,"tokens_out":1058,"duration_ms":837255,"concrete_test":"Verify Proposition 2.9 directly for a specific example: take E = 101a1 (Example 5.1) and M = 5, compute L^{♯/♭}_{2,5}(f,ω_0,T) and L^{♯/♭}_{2,1}(f,ω_0,T) using Sprung's algorithm at finite levels, and check whether L^{♯/♭}_{2,5} = h_5 · L^{♯/♭}_{2,1} holds in Λ. If it fails, the factorization (2.2) and hence Theorem 3.16 are in jeopardy.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central formula (Theorem 3.16) depends on Proposition 2.9, which gives the factorization L^{♯/♭}_{2,M}(f, ω_i, T) = h_ℓ(f, ω_i, T) · L^{♯/♭}_{2,M/ℓ}(f, ω_i, T) for ℓ | M. This factorization is the key input to Lemma 3.12, which decomposes the lambda-invariant additively over primes dividing D. Proposition 2.9's proof relies on the identity (L_{2,M}(f,α,ω_i,T), L_{2,M}(f,β,ω_i,T)) = h_ℓ · (L_{2,M/ℓ}(f,α,ω_i,T), L_{2,M/ℓ}(f,β,ω_i,T)), which in turn requires that Sprung's construction of sharp/flat L-functions is valid for general odd M, not just M=1. The paper states in §2.3: 'Although Sprung formulates his construction only in the case M=1, the same argument applies verbatim to any odd integer M.' This is a nontrivial claim: Sprung's construction involves a limiting process over Mazur–Tate elements θ_{n,M} and the matrix product C_1···C_n·C^{-(n+3)}, and one must verify that the queue sequence relation, the module M_n(f) = ker(...), and the vanishing of the inverse-limit ambiguity M(f) = {(0,0)} all remain valid for general M. While the matrix C_j depends only on f (not M), the Mazur–Tate elements θ_{n,M}(f,ω_i,T) do depend on M, and the identification of the kernel module and its inverse limit for general M is not a purely formal consequence of the M=1 case. If the factorization (2.2) fails for some M, then Lemma 3.12 fails, and the additive decomposition of λ-invariants in Theorem 3.16 is unverified. The reader's identified concern (mu=0) is a standard hypothesis that is clearly stated; the extension to general M is a structural gap in the proof.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","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).","tokens_in":22383,"tokens_out":2293,"duration_ms":616615,"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":[{"comment":"§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","section":null},{"comment":"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.","section":null},{"comment":"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.","section":null},{"comment":"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).","section":null}],"minor_comments":[{"comment":"§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'.","section":null},{"comment":"§2.1, line below Theorem 2.2: 'sgn(psi)' is used but not defined until the sentence following. Consider defining it before first use.","section":null},{"comment":"§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.","section":null},{"comment":"§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.","section":null},{"comment":"§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).","section":null},{"comment":"§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.","section":null},{"comment":"§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).","section":null},{"comment":"§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.","section":null},{"comment":"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.","section":null},{"comment":"§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.","section":null}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about the extension of Sprung's construction to general odd M is well-founded and is the primary reason for the major_revision recommendation. The author's claim that 'the same argument applies verbatim' is plausible but insufficiently justified for a load-bearing step. I note that the reader's concern about the mu=0 hypothesis is standard in this area and is clearly stated as an assumption in all results; it does not constitute a gap but rather a limitation of the method, which is acceptable. The paper is otherwise well-written and the core computations (especially Lemma 3.14 and the examples) are valuable. If the author can provide a convincing justification for the general-M case (or cite a source where this is verified), the paper should be suitable for publication after minor revisions."},"author_rebuttal":{"model":"glm-5.2","summary":"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.","responses":[{"response":"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_made":"yes","referee_comment":"§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."},{"response":"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_made":"yes","referee_comment":"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?"},{"response":"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_made":"yes","referee_comment":"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."}],"tokens_in":22384,"tokens_out":1450,"duration_ms":70888,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"The paper proves a Kida-type difference formula for sharp/flat 2-adic lambda-invariants under quadratic twists (Theorem 3.16), extending Matsuno's ordinary-case result to the supersingular setting at p=2. This is a genuine new result — no such formula existed for Sprung's sharp/flat L-functions. The distribution result (Theorem 4.4) giving asymptotic lower bounds is a clean application following Hatley-Ray's framework. Computational examples using Pollack's Sage code check out and provide concrete verification. The proof structure is well-organized: Proposition 3.4 (period ratios are 2-adic units, via an isogeny argument), Proposition 3.8 (finite-layer congruence for Mazur-Tate elements), and Proposition 3.13 (Greenberg-Vatsal-type lifting) are the key steps, and they are carried out carefully. The local factor computation in Lemma 3.14 is explicit and clean. The mu=0 hypothesis is standard and clearly stated; it is a real limitation but not a defect. Now, the stress-test concern about extending Sprung's construction from M=1 to general odd M. On reading the paper, I think this concern is somewhat overstated but not groundless. The author's argument in Section 2.3 is that the queue sequence relation still holds for general M (citing Mazur-Tate-Tateitelbaum) and that the matrix C_j depends only on f, not M. The kernel module M_n(f) is also stated to be independent of M. This is plausible — the M-dependence enters through the Mazur-Tate elements, not the logarithmic matrix — but the verification is asserted rather than fully written out. Someone familiar with Sprung's construction can probably fill the gap, but a referee should ask for either a more detailed justification or a reference where the general-M case is treated. The T-adic order argument in Proposition 3.13 is compressed but I followed it; the key point that 2^n - q_n grows without bound while the T-adic order of the difference stabilizes is sound. This paper is for Iwasawa theorists working at p=2 and researchers studying arithmetic statistics of elliptic curves in twist families. It deserves a serious referee who can verify the Sprung extension claim and check the congruence-lifting argument in detail.","headline":"Kida-type formula for sharp/flat 2-adic lambda-invariants under quadratic twists, supersingular case at p=2","tokens_in":23195,"tokens_out":565,"would_cite":true,"duration_ms":33555,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Explicit formula tracks lambda-invariants under quadratic twists at p=2","keywords":[],"falsifier":"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.","tokens_in":22252,"feed_emoji":"🜨","tokens_out":981,"duration_ms":45716,"temperature":0.7,"pith_summary":"For an elliptic curve E over Q with good supersingular reduction at 2, the sharp/flat 2-adic L-functions decompose the classical p-adic L-function into two components (denoted sharp and flat) that each live in the Iwasawa algebra. The paper proves an explicit difference formula: when you pass to the quadratic twist E_D (for D > 0 square-free, D = 1 mod 4), the lambda-invariant of the twisted curve equals the original lambda-invariant plus a correction term that depends only on the prime factorization of D. For each prime ell dividing D, the correction is 2*n_ell if ell divides the conductor of E, and 2*n_ell + 1 if ell does not divide the conductor but the reduction of E mod ell has an even number of rational points. Here n_ell is the 2-adic valuation of (ell^2 - 1)/8. If ell does not divide the conductor and the point count is odd, there is no correction from that prime. This is a Kida-type formula: it mirrors classical formulas for the change of genus under field extensions, but now for analytic Iwasawa invariants at the supersingular prime 2. The proof proceeds by establishing a congruence between the sharp/flat L-function of the twist and an imprimitive version of the original L-function, then computing the Iwasawa invariants of the difference factor. As an application, the formula yields asymptotic lower bounds: the number of quadratic twists D up to X with a prescribed lambda-invariant grows at least on the order of X / (log X)^{2/3}.","feed_headline":"Explicit formula tracks lambda-invariants under quadratic twists at p=2","feed_subtitle":"Supersingular analogue of Matsuno's formula gives local correction terms for sharp/flat 2-adic L-functions of twisted elliptic curves","key_machinery":"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","core_discovery":"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","pith_inferences":[],"forward_implications":["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."],"fun_headline_variants":["Kida-type formula for lambda-invariants of 2-adic twists of elliptic curves","Local corrections govern lambda-shifts in sharp/flat 2-adic L-functions under twists","Greenberg-Vatsal congruence yields lambda-invariant formula for supersingular curves at 2","Quadratic twists of elliptic curves: explicit lambda-invariant difference formula at p=2","Supersingular analogue of Matsuno tracks sharp/flat lambda-invariants under twisting"],"cache_read_input_tokens":0,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Kida-type formula for lambda-invariants of 2-adic twists of elliptic curves","Local corrections govern lambda-shifts in sharp/flat 2-adic L-functions under twists","Greenberg-Vatsal congruence yields lambda-invariant formula for supersingular curves at 2","Quadratic twists of elliptic curves: explicit lambda-invariant difference formula at p=2","Supersingular analogue of Matsuno tracks sharp/flat lambda-invariants under twisting"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":628,"prompt_tokens":513,"completion_tokens":115,"prompt_tokens_details":null},"tokens_in":513,"tokens_out":115,"duration_ms":8475,"temperature":1.0,"reasoning_tokens":null,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-07T18:29:00.875431+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"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.","supporting_citations":[],"review_version":1}