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Effective bounds for adelic Galois representations attached to elliptic curves over the rationals

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Pith's one-line read For a non-CM elliptic curve over $\mathbb{Q}$, the adelic Galois image index is less than $10^{21}(h_{\mathcal{F}}(E)+40)^{4.42}$.

desk verdict Strong, mostly sound effective bounds for adelic Galois images; Lemma 5.9 has a repairable numerical gap in the printed proof, not a fatal flaw. read the letter →

arxiv 2412.10340 v4 pith:6WFEQRRH submitted 2024-12-13 math.NT math.AG

Lorenzo Furio This is my paper · ORCID
classification math.NTmath.AG MSC 11G0511F8011G1811R32
keywords ellipticcurvesGaloisrepresentationsadelicimageopentheoremFaltingsheightnon-splitCartansubgroupseffectiveboundstorsionfields
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

The open image theorem for elliptic curves guarantees that a non-CM elliptic curve over $\mathbb{Q}$ has adelic Galois image of finite index in $\operatorname{GL}_2(\widehat{\mathbb{Z}})$, but it does not reveal how large that index can be. This paper answers the quantitative version for rational curves: the index $[\operatorname{GL}_2(\widehat{\mathbb{Z}}):\operatorname{Im}\rho_E]$ is less than $10^{21}(h_{\mathcal{F}}(E)+40)^{4.42}$, where $h_{\mathcal{F}}(E)$ is the stable Faltings height, a logarithmic measure of the curve's arithmetic size, and for large heights the bound improves to $h_{\mathcal{F}}(E)^{3+o(1)}$. To prove this, the paper classifies the possible $p^n$-level images when the mod $p$ image sits in the normaliser of a non-split Cartan subgroup, and shows that the product of the corresponding prime powers grows at most like a power of the height. A companion bound in terms of the conductor makes the estimate effective in practice. If the main theorem is right, the missing portion of the Galois image is controlled by a computable function of the curve's size rather than by an unknown curve-dependent constant.

What carries the argument

The main working object is the N-Cartan lift: a closed subgroup $G<\operatorname{GL}_2(\mathbb{Z}_p)$ whose reduction modulo $p$ is contained in the normaliser of a Cartan subgroup, is not contained in the Cartan, has surjective determinant, and contains a non-scalar element from the Cartan. For such a group the graded pieces $g_n=(G_n/G_{n+1})\hookrightarrow\mathfrak{gl}_2(\mathbb{F}_p)$ obey a forced decomposition $\mathfrak{gl}_2(\mathbb{F}_p)=V_1\oplus V_2\oplus V_3$ (Lemma 3.7), and the dimension dynamics of these pieces force the level-$p^n$ image to be either the full normaliser $C^+_{ns}(p^n)$ or one of a few exceptional groups (Propositions 3.13 and Theorem 3.14). The second machinery is an effective surjectivity theorem (Theorems 5.1 and 5.17): from the Cartan subgroups one builds a quotient of $E\times E$ whose degree records the prime powers contributing, and the period-theoretic isogeny theorem used in Section 5 bounds that degree by the Faltings height, giving $\Lambda<21000(h_{\mathcal{F}}(E)+40)^{1.308}$ for rational curves. The third machinery is entanglement control: in the non-split Cartan case the prime $p$ is almost totally ramified in $\mathbb{Q}(E[p^n])$, while at primes not dividing the conductor and $p$ the ramification is small, so the overlap of division fields at different primes is small; Lemma 7.16 converts the product of the $p$-adic indices into the adelic index with an additional factor at most $1536\cdot 6^\alpha$.

What would settle it

Run the one-variable check behind Lemma 5.9 across $x\in[-0.75,0]$: compare $\frac{3}{\pi}\log\left(\frac{12}{\pi}x+5.52+4e^2\right)+\frac{6}{\pi}x+2.76$ with $2.29x+6.21$. If the first expression exceeds the second anywhere in that interval, the displayed proof does not establish the lemma's constant, and the constants depending on it (1454, 1266.4, 21000, 2488320) need revision; otherwise the chain is consistent.

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Extended reading notes

Core claim

The central claim is that for a non-CM elliptic curve over $\mathbb{Q}$ the size of the missing part of the full adelic Galois image is polynomially controlled by the stable Faltings height. Theorem 1.8 states $[\operatorname{GL}_2(\widehat{\mathbb{Z}}):\operatorname{Im}\rho_E]<10^{21}(h_{\mathcal{F}}(E)+40)^{4.42}$, and as $h_{\mathcal{F}}(E)\to\infty$, $[\operatorname{GL}_2(\widehat{\mathbb{Z}}):\operatorname{Im}\rho_E]<h_{\mathcal{F}}(E)^{3+o(1)}$; an explicit conductor version bounds the same index by $2488320\,(51N(1+\log\log N)^{1/2})^{3\omega(N)}$. The route is to show that the only primes that can contribute seriously are those for which $\operatorname{Im}\rho_{E,p}$ lies in the normaliser of a non-split Cartan subgroup, and that at level $p^n$ the image is almost always the full normaliser $C^+_{ns}(p^n)$. The product of these prime powers is then bounded by an effective surjectivity theorem in terms of $h_{\mathcal{F}}(E)$, and the entanglement between division fields at different primes is shown to cost only a small multiplicative factor. The paper also classifies the possible (conjecturally non-existent) images of $\rho_{E,p^n}$ whenever $\operatorname{Im}\rho_{E,p}$ is contained in a non-split Cartan normaliser.

Load-bearing premise

The explicit constants in the main theorems depend on Lemma 5.9, a numerical inequality comparing an average of inverse squared periods to 2.29 times the Faltings height plus 6.21; if that inequality is not valid as stated, the displayed constants must be revised even though the polynomial form is likely to survive.

Editorial extensions

If this is right

  • For every non-CM elliptic curve over $\mathbb{Q}$, the adelic Galois image index is bounded by a fixed polynomial in the stable Faltings height with explicit constants.
  • For sufficiently large heights the bound becomes $h_{\mathcal{F}}(E)^{3+o(1)}$; in particular, for every $\varepsilon>0$, the index is eventually smaller than $h_{\mathcal{F}}(E)^{3+\varepsilon}$.
  • The conductor bound $[\operatorname{GL}_2(\widehat{\mathbb{Z}}):\operatorname{Im}\rho_E]<2488320(51N(1+\log\log N)^{1/2})^{3\omega(N)}$ gives an effective, lower-exponent replacement for the previous conductor bound.
  • When $\operatorname{Im}\rho_{E,p}$ is contained in the normaliser of a non-split Cartan subgroup, the possible images of $\rho_{E,p^n}$ form a short classified list: the full normaliser, listed $p=3$ and $p=5$ exceptional groups, or one specific level-$p^2$ semidirect-product group.
  • If that specific level-$p^2$ group can be excluded for all curves, the asymptotic height exponent improves from $3+o(1)$ to $2+o(1)$.

Reading between the lines

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

  • Going beyond the paper: the same three-step decomposition—residual-image classification, a height-based product bound, and entanglement control—should produce analogous effective index bounds for other compatible families of 2-dimensional Galois representations such as modular forms, whenever their residual images are classified.
  • Going beyond the paper: the explicit constants invite a finite numerical test: compute the actual adelic index for all non-CM curves up to a moderate height threshold and compare it with $10^{21}(h_{\mathcal{F}}(E)+40)^{4.42}$; the paper contains no such data, and the comparison would indicate how far the constant is from the truth.
  • Going beyond the paper: the asymptotic statement $h_{\mathcal{F}}(E)^{3+o(1)}$ is more robust than the displayed constant $10^{21}$, because it does not depend on the precise constant in Lemma 5.9; optimizing the constant would therefore focus on Lemma 5.9 and on the $1536\cdot 6^\alpha$ entanglement factor.
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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

1 major / 4 minor

Summary. The paper proves fully explicit, height-dependent upper bounds for the index of the adelic Galois image of a non-CM elliptic curve over Q. The main theorem (Theorem 1.8) gives [GL2(Ẑ):Im ρ_E] < 10^21 (h_F(E)+40)^4.42, an asymptotic bound h_F(E)^{3+o(1)}, and a conductor bound, improving previous work of Zywina and Lombardo. The proof combines a structural classification of p-adic images in the non-split Cartan case (Sections 3 and 6), an effective surjectivity theorem via the Gaudron–Rémond period/isogeny estimates (Section 5), and new entanglement estimates among division fields (Sections 7.1–7.2). The paper also classifies possible images of ρ_{E,p^n} when Im ρ_{E,p} lies in the normaliser of a non-split Cartan subgroup (Theorem 1.9).

Significance. If the proof is completed as intended, this is a substantial contribution: it supplies the first fully explicit polynomial bound in the Faltings height for the adelic index over Q, with the polynomial exponent 4.42 and the asymptotic exponent 3+o(1) going well beyond Lombardo's effective but enormous bounds. The use of prior external results (Gaudron–Rémond, Mazur, RSZB, Le Fourn–Lemos, and the author's earlier work with Lombardo) is honest and non-circular, and the classification in the non-split Cartan case is a useful complement to the RSZB database. The paper is carefully structured and many of the estimates are explicit rather than merely existential.

major comments (1)
  1. [Section 5, Lemma 5.9] The proof of Lemma 5.9 does not establish the displayed inequality. After applying [57, Lemma B.1], the proof claims (3/π) log((12/π)h_F + 5.52 + 4e^2) + (6/π)h_F + 2.76 < 2.29 h_F + 6.21, citing x > 32.2 and log x / x ≤ log(32.2)/32.2 for x ≥ 32.2. At h_F = -0.75 the left side evaluates to about 4.643, whereas 2.29(-0.75)+6.21 = 4.4925, so the asserted inequality is false; the cited monotonicity bound does not close this gap. This lemma is load-bearing: it is used in (5.10) and (5.12) to bound the average inverse period and hence feeds into the constants 1454 and 1266.4 in Theorem 5.1, the constants 21000 and 14400 in Theorem 5.17, and ultimately the 10^21 (h_F+40)^4.42 bound in Theorem 1.8(1). The lemma may be salvageable by solving the implicit inequality πT ≤ 3 log T + 6h_F + 8.66 more sharply than the printed chain does, but the proof as written must be repaired and all resulting constants re-verified before the main theorem can be accepted as stated.
minor comments (4)
  1. [Throughout] Several phrases are repeated or have minor typos, e.g. 'attache d' in the title line, 'the same p roblems' in the introduction, and 'we also give an improved and effective version' missing the object 'of Zywina's bound'; these do not affect the mathematics but should be cleaned up.
  2. [Lemma 2.8 and Proposition 7.10] The paper relies on the MAGMA function FindOpenImage and on MAGMA computations for finitely many cases (e.g. the four j-invariants in (2.1), the list (7.2), and the RSZB labels). It would help reproducibility if the relevant scripts or explicit outputs were included in an ancillary file; as written the reader cannot independently verify these finite computations without reimplementing them.
  3. [Section 5.2, Theorem 5.14] In the proof of Theorem 5.14, the sentence 'using d h(j)> 2 > e/(1.6n)' is not immediately clear: the last inequality seems to require n > 1, but the case n = 1 is also covered by the surrounding argument; please clarify the range of n in this step.
  4. [Theorem 7.1, statement] The second assertion of Theorem 7.1 is stated with a function δ(x) whose denominator can be small for x near -0.75; the later proof restricts to h_F > 4·10^15 for that branch, but the statement as written may suggest the bound holds uniformly. Please state the ranges in the displayed asymptotic claim more precisely.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main index bound is derived from external effective period estimates and group-theoretic classification, not from fitting or self-referential inputs.

full rationale

The paper's central claim, Theorem 1.8, is obtained by combining an effective surjectivity bound (Section 5) with p-adic index computations (Section 6) and entanglement bounds (Section 7). The constants 1454, 1266.4, 21000, 2488320 and 10^21 are produced by explicit analytic inequalities, not by fitting the adelic index to itself. The apparent boundary issue in Lemma 5.9 is a numerical-estimate concern, not circularity: the lemma invokes external results [24, Proposition 3.2] and [57, Lemma B.1] as inputs and does not presuppose the desired index bound. The author's prior work [22] is cited as an input theorem, for example in Theorem 1.5 and Theorem 2.4, and the author is a coauthor of [22]; however, the present argument does not reduce the main theorem to [22] by construction, and [22] is combined with independent results of Zywina, Balakrishnan et al., and Rouse-Sutherland-Zureick-Brown. No parameter is fitted to the quantity being bounded, and no known empirical pattern is merely renamed as a prediction. The manuscript candidly discloses a flaw in Claim 2.6 (Remark 2.7) and a corrected reference in Remark 6.4; these are limitations, not circular steps.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted parameters or new entities. All constants are derived from explicit estimates. The central claim depends on deep external theorems and computational classifications, which are listed as axioms.

assumptions (5)
  • domain assumption Gaudron-Remond effective period and isogeny theorem (Theoreme des periodes)
    Invoked in the proof of Theorem 5.1 to bound Lambda in terms of the Faltings height. If this deep estimate has an error, the explicit bound collapses.
  • domain assumption Completeness of the RSZB classification of l-adic images (Rouse-Sutherland-Zureick-Brown [51])
    Used in Theorem 6.5, Lemma 7.17 and Section 5 to list possible p-adic images for small primes and to rule out exceptional curves. The full listing relies on extensive computation and online data.
  • domain assumption Zywina's classification of mod p images and FindOpenImage algorithm outputs [63,65]
    Used in Propositions 7.10 and Lemma 2.8 to compute exact adelic indices for exceptional j-invariants. Correctness depends on algorithm implementation.
  • standard math Serre's open image theorem and Mazur's theorem on rational isogenies
    Background facts used throughout to reduce to non-surjective primes p in a finite set.
  • domain assumption Effective Mertens-type bound [59, Proposition 4] and Robin's bound on omega(n) [50]
    Used in Lemmas 2.9 and Section 5 to get explicit constants; these are published results but not proved in the paper.

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Pith. "Pith review of Effective bounds for adelic Galois representations attached to elliptic curves over the rationals." pith.science (2026). https://pith.science/paper/6WFEQRRH

@misc{pith2026241210340,
  author       = {Pith},
  title        = {Pith review of: Effective bounds for adelic Galois representations attached to elliptic curves over the rationals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6WFEQRRH}},
  note         = {Machine review of arXiv:2412.10340}
}
abstract

Given an elliptic curve $E$ defined over $\mathbb{Q}$ without complex multiplication, we provide an explicit sharp bound on the index of the image of the adelic representation $\rho_E$. In particular, if $\operatorname{h}_{\mathcal{F}}(E)$ is the stable Faltings height of $E$, we show that $[\operatorname{GL}_2(\widehat{\mathbb{Z}}) : \operatorname{Im}\rho_E]$ is bounded above by $10^{21} (\operatorname{h}_{\mathcal{F}}(E)+40)^{4.42}$, and, for $\operatorname{h}_{\mathcal{F}}(E)$ tending to infinity, by $\operatorname{h}_{\mathcal{F}}(E)^{3+o(1)}$. We also classify the possible (conjecturally non-existent) images of the representations $\rho_{E,p^n}$ whenever $\operatorname{Im}\rho_{E,p}$ is contained in the normaliser of a non-split Cartan. This result improves previous work of Zywina and Lombardo.

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Cited by 1 Pith paper

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  1. Explicit p-adic Hodge theory for elliptic curves and non-split Cartan images

    math.NT 2026-03 accept novelty 7.0 of 10

    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.

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Reviewed August 11, 2026 · model on record in the stance chip above.