{"id":"fef86c22-2c72-4b3f-8f9d-6c64d0eb2538","arxiv_id":"2603.04021","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For non-CM elliptic curves over Q with p>7, non-split Cartan mod p image forces the p-adic image to be the full preimage of the mod p^n non-split Cartan normalizer for some n.","lead":"This paper proves that for non-CM elliptic curves over Q with p > 7, if the mod p Galois image is contained in the normalizer of a non-split Cartan subgroup, then the entire p-adic image is the full preimage of that normalizer modulo p^n for some n. It also gives a new algorithm to compute a key p-adic Hodge-theoretic invariant from a Weierstrass equation using formal logarithms, making the classification explicit and effective.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Volkov's classification theorem (Thm 3.3) is the unproved external foundation; a gap there would collapse the local classification and hence the global p-adic image theorem.","rationale":"The reader's weakest assumption was exactly Volkov's classification (Theorem 3.3). I agree that this is the most load-bearing concern: the entire local classification (Theorem 5.4) and its use in ruling out the G#_ns(p^2) case in Theorem 5.1 are built on Volkov's parametrization of V_pE by α and the solution-set description (Equation 3.3). The paper is explicit that this theorem is taken from a PhD thesis and not proved. Nevertheless, reliance on established (even if not universally accessible) previous work is standard practice in mathematics, and the paper provides internal cross-checks (e.g., Example 8.31) that are consistent with the claimed results. No internal error or inconsistency was identified in the present manuscript. Therefore, the reader's ACCEPT verdict remains appropriate, though the verification of Volkov's theorem would materially increase confidence. My concrete test would directly test the disputed isomorphism in the small cases needed by the paper's applications.","tokens_in":55277,"tokens_out":14355,"duration_ms":128253,"concrete_test":"Implement Volkov's construction computationally for small primes and compare with independent computations of p-adic Galois images. Specifically: (1) For p=11 and the curve E1: y^2 = x^3 + 11^3 x + 11^2 from Example 8.31, compute the p-adic Galois representation ρ_{E1,p∞} using division polynomials and/or SageMath's Galois representation routines; compute α = 5 + O(11) from the paper's algorithm; build V_α via Equation (3.3) and compute its Galois image; verify the images are conjugate in GL2(Z_11). (2) Repeat for a curve with e=4 (e.g., y^2 = x^3 + 11 x + 11^2) and for an α=∞ CM curve; verify the full dictionary including the case v(α)=1 is excluded by the Cartan hypothesis. If all match, the concern is resolved; if any mismatch is found, Theorem 3.3 (or its application) is falsified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 9.1) is proved by ruling out the exceptional G#_ns(p^2) case using the local Theorem 5.1. Theorem 5.1 depends on Theorem 5.4, whose proof uses the polynomials g_k (Theorem 4.2) to describe E[p^k] for k near v(α^{-1}). Theorem 4.2 is built on Theorem 3.3 (Volkov), which asserts that every E/Q_p with semistability defect e∈{3,4,6} and potentially supersingular reduction has V_pE ≅ V_α for some α∈P^1(Q_p), and conversely, and that elements of V_α correspond to solutions of Equation (3.3). This is the foundation: it supplies the deformation parameter α, the identification V_α with the rational Tate module, and (via Lemma 2.18 and Volkov's description of E[p]) the irreducibility and integral-lattice uniqueness needed to pass to E[p^k]. The paper does not prove Theorem 3.3, referring to Volkov's PhD thesis [Vol98, p.125]; Section 3 states 'We will largely omit proofs; all the details can be found in [Vol98].' If Volkov's classification has a gap—e.g., if not all potentially supersingular curves are captured, or if the map from D_α to V_pE is not an isomorphism for some branch of parameters—then the g_k-roots are not avatars of the torsion, the dichotomy in Theorem 5.4 fails, and the exclusion of case (iii) in Theorem 5.1 collapses. This is a proof-level concern about an external dependency, not an internal inconsistency, but it is load-bearing because no independent derivation of Theorem 3.3 or its key consequences is provided.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the p-adic Galois images of elliptic curves over Q_p whose mod p image is contained in the normaliser of a non-split Cartan subgroup. Building on Volkov's classification of filtered (φ, Gal(K/Q_p))-modules, the authors introduce explicit polynomials g_k whose roots are in Galois-equivariant bijection with E[p^k], establish a local dichotomy in terms of a deformation parameter α, and give an algorithm to compute α from a Weierstrass model via formal logarithms. These local results are then used to prove a global theorem (Theorem 9.1) asserting that, for a non-CM elliptic curve E/Q and p > 7 with Im ρ_{E,p} ⊆ C_ns^+(p), the full p-adic image is the preimage of C_ns^+(p^n) for some n ≥ 1. The paper also derives improved bounds on the adelic image index in terms of the height of j(E).","tokens_in":55714,"tokens_out":8939,"duration_ms":74549,"significance":"If the main results are correct, the paper closes a notable gap in the p-adic classification of Galois images of elliptic curves in the non-split Cartan case, a problem that had previously resisted treatment. The local classification (Theorem 5.1) and the explicit connection between the formal logarithm and the filtered (φ,G)-module parameter α are novel and likely to be useful beyond this paper. The paper is carefully written and contains a substantial amount of original technical work, including the construction and analysis of the polynomials g_k and the scalar computations in Section 8. However, the unconditional validity of the main theorems rests on a theorem (Theorem 3.3) quoted verbatim from Volkov's unpublished PhD thesis, which is not proved in the manuscript. This external dependency is load-bearing and tempers confidence in the absolute conclusions.","major_comments":[{"comment":"Theorem 3.3, stated as a quotation from Volkov's PhD thesis [Vol98, p. 25 and p. 125], is the foundation on which the rest of the paper is built. It asserts that every elliptic curve over Q_p with semistability defect e∈{3,4,6} and potentially supersingular reduction gives rise to a filtered (φ, Gal(K/Q_p))-module D_α, and conversely, and that the rational Tate module is recovered as V_α. This classification supplies the deformation parameter α, the identification of V_α with the rational Tate module, and (via Lemma 2.18 and the related description of E[p]) the irreducibility and lattice-uniqueness steps used later. Theorem 4.2, Theorem 5.4, Theorem 5.1, and ultimately Theorem 9.1 all depend on Theorem 3.3. The manuscript explicitly says 'We will largely omit proofs; all the details can be found in [Vol98].' Given that [Vol98] is an unpublished dissertation and no independent verificatio","section":"Section 3, Theorem 3.3"}],"minor_comments":[{"comment":"The formula for n0 is written as n0 := (1/3 v(j)) (resp. (1/2 v(j−1728))) without the floor function. Since n0 is supposed to be a positive integer, the floor brackets should appear as they do in Corollary 8.29 (e.g., ⌊1/3 v(j)⌋).","section":"Theorem 1.3 and abstract"},{"comment":"The claim that [Kaw11, Proposition 2.2.1] is incorrect is made in a remark. This is a strong assertion about another published paper and should be substantiated with a concrete counterexample or a more detailed explanation of the flaw, especially since the reader cannot easily check the internal details without Kawachi's paper.","section":"Remark 8.22"},{"comment":"[Vol98] is a PhD thesis. It would be helpful to include a URL or repository where the thesis can be accessed, to facilitate verification of the quoted results.","section":"References"},{"comment":"The phrase 'which appears to be novel' in the abstract and introduction is informal. It can be replaced by a clearer statement, e.g., 'to our knowledge this is the first explicit algorithm...'.","section":"Introduction"},{"comment":"In Lemma 2.20, the proof for the case j=0 or 1728 refers to [Vol01, Proposition 2.7] and [Vol98, §B.2.2]. These references are appropriate, but the text could mention that the relevant statements also cover the uniqueness of the Tate module in the supersingular good-reduction case.","section":"Section 2.4"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically strong and the main result is plausible, but I have serious reservations about the reliance on Volkov's unpublished thesis. I recommend that the editor ask the authors to provide a detailed verification of the parts of Theorem 3.3 that are actually used, or to make the thesis publicly accessible with a precise guide. If that is done, the paper would be a solid contribution. As it stands, the external dependency makes acceptance conditional. I do not see evidence of circularity or internal contradiction; the issue is solely the unproved foundation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the bottom line: this is a strong paper and it should get a serious referee. It proves the expected global statement — when E/Q is non-CM and Im ρ_{E,p} ⊆ C^+_ns(p) for p>7, the p-adic image is a full preimage of a non-split Cartan at level p^n — and it removes the level-p^2 exceptional group that was left open in Furio's earlier work. The route through explicit filtered (φ, Gal)-modules and division polynomials is new, and the α-to-j formula is by itself a useful result.\n\nWhat the paper does well: the technical work is careful. The polynomials g_k are a clean idea and the Newton-polygon analysis in Section 4 is convincing. The formal-logarithm algorithm (Section 8) converts Volkov's fairly abstract parameter into something computable from a Weierstrass model, with worked examples. I also appreciate that the authors are straight about the limits: they flag the restriction p > √n0+1, and they publicly state that Kawachi's proposition appears incorrect, which takes some doing.\n\nThe soft spots are real but proportionate. The main one is Theorem 3.3, which is lifted from Volkov's PhD thesis and carries the entire local classification: α, the identification of V_α with the rational Tate module, and ultimately the interpretation of the roots of g_k as torsion points. If that input has a gap, the paper's main theorem loses its foundation. This is not a circularity — the reader is right that α is not fitted to the image — but it is an external dependency on an unpublished source. The authors do not hide it; Section 3 says plainly that proofs are in [Vol98]. A referee should actually read the relevant pages of the thesis before certifying this. The second soft spot is that several cited results are stated for p-adic fields on the assertion that the arguments transfer from number fields. That may well be true, but the paper doesn't always show the transfer.\n\nOverall: the central argument looks sound to me. This is a genuine advance for specialists in Galois images and explicit p-adic Hodge theory. It deserves peer review, and I'd cite it. My own verdict is close to accept, with the caveat that the referee must check the Volkov input.","headline":"Completes the p-adic image classification in the non-split Cartan case and supplies a genuinely useful computational handle, but the whole local tower rests on Volkov's unpublished thesis.","tokens_in":56169,"tokens_out":2117,"would_cite":true,"duration_ms":23001,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G05","11G07","11F80","14G20","11S20"],"pacs":[],"model":"deepseek-v4-flash","headline":"For p > 7, a non-CM elliptic curve whose mod p image lies in the normaliser of a non-split Cartan subgroup has p-adic image equal to the full preimage of the mod p^n normaliser for some n ≥ 1.","keywords":["elliptic curves","Galois representations","non-split Cartan subgroups","p-adic Hodge theory","supersingular reduction","formal logarithms","division polynomials","adic images"],"falsifier":"Compute the 11-adic Galois image of E_1/Q_11 : y^2 = x^3 + 11^3 x + 11^2 (Example 8.31). Theorem 9.1 predicts Im ρ_{E_1,11^∞} = π_1^{-1}(G_1) with G_1 an index-3 subgroup of C^+_ns(11); in particular the mod 121 image lies in the preimage of C^+_ns(11). If a direct computation (e.g., via division polynomials) finds a mod 121 element congruent to Id + 11M with M not in V_1 ⊕ V_2, the theorem fails. More broadly, any Q-rational point on the modular curve X^#_{ns}(11^2) would give a non-CM E/Q with level-11^2 image G^#_{ns}(11^2), contradicting Theorem 9.1.","tokens_in":55191,"feed_emoji":"🧮","tokens_out":19401,"duration_ms":158440,"temperature":0.7,"pith_summary":"This paper proves that in the non-split Cartan case the p-adic Galois representation of an elliptic curve is completely pinned down by a single integer. Concretely, for a non-CM elliptic curve over Q with p > 7 and mod p image inside the normaliser of a non-split Cartan subgroup, the full p-adic image is the preimage of the corresponding mod p^n normaliser for some n ≥ 1, ruling out an exceptional level-p^2 group. Locally over Q_p, this follows from a new explicit p-adic Hodge theory: each curve is encoded by one deformation parameter α, and the paper builds explicit polynomials whose roots are Galois-equivariantly identified with the p-power torsion. A further new result computes α to any p-adic precision from the formal logarithm of a Weierstrass model, equivalently from the j-invariant; this yields the value of n and strong bounds on the adelic image index in terms of the height of j. The authors note that the local condition p > sqrt(n0+1) is believed removable.","feed_headline":"p-adic Galois images in the non-split Cartan case are full preimages","feed_subtitle":"New explicit p-adic Hodge theory derives the p-adic image from the j-invariant and sharpens adelic bounds.","key_machinery":"The deformation parameter α ∈ P^1(Q_p) of the filtered (φ, Gal(K/Q_p))-module D_α (Theorem 3.3), together with the explicit polynomials g_k(x) = x^{p^{2k}} + Σ_{n=1}^k (-1)^n p^n (x^{p^{2k-2n}} + α^{-1} π_e^2 x^{p^{2k+1-2n}}), whose roots are Galois-equivariantly identified with E[p^k]. The valuation v(α^{-1}) is the switch: Cartan containment for k <= v(α^{-1})+1, maximal growth for k = v(α^{-1})+2, forcing the full p-adic image to be an inverse image. The bridge to arithmetic is β = lim -(p/π_e) d_{p^{2k+1}}/d_{p^{2k}}, with v(β) = v(j)/3 - 1/e (or v(j-1728)/2 - 1/e), giving α from Weierstrass data.","core_discovery":"The central claim: an elliptic curve over Q_p whose mod p image lies in the normaliser of a non-split Cartan subgroup C^+_ns(p) has p-adic Galois image governed by one number, the valuation of a deformation parameter α. The rational Tate module is V_α (Theorem 3.3), and the p^k-torsion is identified with roots of explicit polynomials g_k; v(α^{-1}) is a sharp threshold: Cartan containment for k <= v(α^{-1})+1, maximal growth at k = v(α^{-1})+2. So the full p-adic image is the preimage of its level n = v(α^{-1})+1 image, and α is computable from j via formal logarithms (n0 = floor(v(j)/3), resp. floor(v(j-1728)/2); index 1 or 3). Globally, for non-CM E/Q and p > 7, Im ρ_{E,p^∞} = π_n^{-1}(C^+","pith_inferences":["Read as a rigidity statement, the result says the p-adic tower of a supersingular elliptic curve is as CM-like as possible up to a level n0 set by the j-invariant, then grows at the maximal rate; this suggests a stratification of the supersingular locus by v(α^{-1}) that could be studied in families.","The explicit threshold may allow local-constancy results for the p-adic Galois image on Weierstrass coefficient space — determining the precise p-adic radius where the mod p^{n0} representation is constant — which the paper hints at via Krasner's lemma; this would give a fully algorithmic 'image from a model' routine.","The same formal-logarithm machinery should extend to semistability defect 1 and 2, and likely to other p-divisible groups, making the computation of filtered φ-modules for potentially crystalline representations of GL_2-type fully explicit.","The paper flags a published formula for β that is not invariant under strict isomorphisms and provides a corrected computation; this illustrates that formal-logarithm invariants require careful normalisation, which the algorithm's precision bounds make explicit."],"forward_implications":["For non-CM E/Q, p > 7, if the mod p image is in C^+_ns(p), then Im ρ_{E,p^∞} = π_n^{-1}(C^+_ns(p^n)) for some n ≥ 1; the exceptional level-p^2 group G^#_{ns}(p^2) is ruled out (Theorem 9.1).","The integer n is explicit: n = floor(v(j)/3) for e in {3,6} or floor(v(j-1728)/2) for e = 4, with the p-adic image contained in C^+_ns(p^n) with index 1 or 3; the index-3 case occurs exactly for e in {3,6}, p ≡ 2,5 mod 9, realized by curves y^2 = x^3 + p^i (Corollary 8.29, Remark 8.30).","The adelic image index satisfies [GL_2(Ẑ) : Im ρ_E] < 1.6·10^{17}(h(j(E))+480)^{3.11} and is asymptotically < h(j(E))^{2+O(1/log log h)} (Theorem 9.3).","The polynomials g_k give a new recursive description of E[p^k] with linear dependence on α^{-1}, usable also for semistability defect e = 1, 2 (Theorem 4.2, Remark 4.24); they may serve as alternative division polynomials."],"fun_headline_variants":["One valuation parameter fixes p-adic Galois image for non-split Cartan","Algorithm computes p-adic filtered module from Weierstrass model","Non-split Cartan p-adic image is full preimage at level v(α)+1","p-adic image from j-invariant via one valuation parameter","Full preimage of non-split Cartan: level from α valuation"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The entire chain relies on the classification theorem (Theorem 3.3), cited from a PhD thesis and not reproved, that every E/Q_p with semistability defect e in {3,4,6} and potentially supersingular reduction has V_p E ≅ V_α; a gap there would invalidate the polynomials g_k, the computation of α, and the image classification.","fun_headline_variants_meta":{"raw":{"variants":["One valuation parameter fixes p-adic Galois image for non-split Cartan","Algorithm computes p-adic filtered module from Weierstrass model","Non-split Cartan p-adic image is full preimage at level v(α)+1","p-adic image from j-invariant via one valuation parameter","Full preimage of non-split Cartan: level from α valuation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000749,"raw_usage":{"total_tokens":3224,"prompt_tokens":847,"completion_tokens":2377,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":591,"completion_tokens_details":{"reasoning_tokens":2290}},"tokens_in":591,"tokens_out":2377,"duration_ms":15297,"temperature":1.0,"reasoning_tokens":2290,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T18:56:53.859936+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the 11-adic Galois image of E_1/Q_11 : y^2 = x^3 + 11^3 x + 11^2 (Example 8.31). Theorem 9.1 predicts Im ρ_{E_1,11^∞} = π_1^{-1}(G_1) with G_1 an index-3 subgroup of C^+_ns(11); in particular the mod 121 image lies in the preimage of C^+_ns(11). If a direct computation (e.g., via division polynomials) finds a mod 121 element congruent to Id + 11M with M not in V_1 ⊕ V_2, the theorem fails. More broadly, any Q-rational point on the modular curve X^#_{ns}(11^2) would give a non-CM E/Q with level-11^2 image G^#_{ns}(11^2), contradicting Theorem 9.1.","supporting_citations":[],"review_version":1}