REVIEW 1 major objections 5 minor 11 references
Explicit p-adic Hodge theory for elliptic curves and non-split Cartan images
T0 review · 1 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash
Pith's one-line read 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.
desk verdict 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non-split Cartan case and supplies a genuinely useful computational handle, but the whole local tower rests on Volkov's unpublished thesis. 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
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.
What would settle it
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.
Extended reading notes
Core claim
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^+
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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).
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 (1)
- [Section 3, Theorem 3.3] 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
minor comments (5)
- [Theorem 1.3 and abstract] 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)⌋).
- [Remark 8.22] 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.
- [References] [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.
- [Introduction] 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 2.4] 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.
Circularity Check
No circularity: the p-adic image theorem is derived from independent p-adic Hodge theory inputs and external classifications, not from its own conclusion.
full rationale
The claimed derivation chain is not circular. The deformation parameter α is introduced through Volkov's classification (Theorem 3.3), which is an external result from [Vol98] and does not assume the non-split Cartan image conclusion. The paper then computes v(α) from the j-invariant and Weierstrass coefficients via formal logarithms in Section 8, which is independent of the Galois image statement. Theorem 5.4 translates v(α) into a prediction about Im ρ_{E,p^k} using the auxiliary polynomials g_k derived from Equation (3.3); the exclusion of the exceptional case G^#_ns(p^2) in Theorem 5.1 follows from the dichotomy v(α^{-1}) = 0 versus v(α^{-1}) ≥ 1, not from assuming the target result. Self-citations to [Fur26] and [FL23] are prior independent classifications that do not contain Theorem 9.1; in fact [Fur26] leaves open the exact case that this paper rules out. The only notable dependency is Volkov's Theorem 3.3, which the paper explicitly does not prove ('We will largely omit proofs; all the details can be found in [Vol98]'); this is an external correctness/verification risk, not circularity. No fitted parameter is renamed as a prediction, and no definition presupposes the theorem being proved.
Assumptions & free parameters
assumptions (5)
- domain assumption Volkov's classification theorem (Theorem 3.3): 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; V_α = Hom_{φ,G_K}(D_α, BW(R)).
- domain assumption The description of V_α as roots of Equation (3.3) and the identification of E[p^k] with A_c/p^k A_c (Lemmas 3.7, 4.11) depend on the filtered-module functor D_cris,K/Q_p of Fontaine/Volkov.
- domain assumption Group-theoretic classification of non-split N-Cartan lifts (Theorem 2.14) from [Fur26, Theorem 3.14].
- domain assumption The no-canonical-subgroup results of [Smi23, Theorem 4.6] and [FL23, Theorem 3.10] hold over p-adic fields as claimed in Theorem 2.9.
- domain assumption Kawachi's Dieudonné module / formal logarithm isomorphism (Theorem 8.17) and Lemma 8.8 (Yasuda's explicit formal logarithm formula).
Cite this review
Pith. "Pith review of Explicit p-adic Hodge theory for elliptic curves and non-split Cartan images." pith.science (2026). https://pith.science/paper/T3RFGANN
@misc{pith2026260304021,
author = {Pith},
title = {Pith review of: Explicit p-adic Hodge theory for elliptic curves and non-split Cartan images},
year = {2026},
howpublished = {\url{https://pith.science/paper/T3RFGANN}},
note = {Machine review of arXiv:2603.04021}
}
abstract
Let $E/\mathbb{Q}_p$ be an elliptic curve whose mod $p$ Galois image is contained in the normaliser of a non-split Cartan. We classify the possible $p$-adic images of $E$ using tools from $p$-adic Hodge theory via a careful analysis of the local Galois structure of the $p$-power torsion of $E$. We pay special attention to the case where $E$ has potentially supersingular reduction, where we give an algorithm to determine the corresponding filtered $(\varphi,\operatorname{Gal}(K/\mathbb{Q}_p))$-module from a Weierstrass model (which appears to be novel), and introduce alternative division polynomials that may be of independent interest. We deduce global consequences for elliptic curves $E/\mathbb{Q}$: when the mod $p$ representation of $E$ has non-split Cartan image and $E$ doesn't have CM, the $p$-adic image must be the full preimage of the normaliser of a mod $p^n$ non-split Cartan for some $n \geq 1$. As an application, we sharpen existing bounds on the adelic image in terms of the Weil height of the $j$-invariant.
Reference graph
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Reviewed August 2, 2026 · model on record in the stance chip above.
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