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REVIEW 2 major objections 5 minor 38 references

Non-split Cartan curves and a characterization of fake elliptic curves

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Pith's one-line read An abelian surface over an imaginary quadratic field has quaternionic multiplication precisely when its residual Galois image lies in a non-split Cartan subgroup modulo at least two primes (assuming Conjecture 5.6).

desk verdict The computational core is solid, but the main characterization has a genuine twist-vs-isogeny gap that makes the theorem false as stated. read the letter →

arxiv 2608.03934 v1 pith:BS6WFHOJ submitted 2026-08-04 math.NT math.AG

Enric Florit This is my paper · ORCID
classification math.NTmath.AG MSC 11G1814K1511F80
keywords fakeellipticcurvesquaternionicmultiplicationnon-splitCartanmodularquadraticpointsabeliansurfacesimaginaryfieldsGaloisrepresentationssymmetricChabauty
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

Quaternionic-multiplication (QM) abelian surfaces, often called fake elliptic curves, produce Galois representations that look like those of elliptic curves, so no criterion was known to tell them apart. This paper gives such a criterion for surfaces over imaginary quadratic fields: under a stated conjecture, a surface satisfying three natural conditions has QM exactly when its mod-p Galois image is contained in a non-split Cartan subgroup for at least two primes. The direction QM implies two such primes is unconditional, since an indefinite division quaternion algebra ramifies in at least two primes. The reverse direction is reduced to a statement about quadratic points on non-split Cartan modular curves; the paper proves that statement for levels 6, 7, 10, 11, 13 and 15, making the characterization unconditional for every indefinite quaternion discriminant up to 33. A reader should care because it turns a representation-theoretic ambiguity into a finite, checkable condition: count the non-split Cartan primes.

What carries the argument

The load-bearing objects are the non-split Cartan modular curves X_ns(N), which parametrize elliptic curves whose mod-N Galois image lies in C_ns(N), and their double quotients X_ns^+(N). A quadratic point on X_ns(N) is non-exceptional when it is the pullback of a rational point on one of the degree-2 quotients; such points have rational j-invariant. For N=6,7,10,11,13,15 the paper shows every quadratic point is non-exceptional, using Jacobian computations, elliptic-curve Chabauty for X_ns^+(15), and symmetric Chabauty with a sieve for X_ns(15). The characterization itself is carried by the set C_A: quaternion ramification puts residual images in C_ns(p), and an elliptic curve with two such

What would settle it

Find an imaginary quadratic field K and an elliptic curve E/K with j(E) not in Q whose mod-p and mod-q representations both land in non-split Cartan subgroups for two distinct primes p and q. By Theorem 5.1 its j-value must lie in the finite set S_{p,q}, so the search can be made exhaustive: for p,q with p at least 17, compute S_{p,q}, or equivalently the exceptional quadratic points on X_ns(pq), and check whether any element lies in K. A positive hit would disprove Conjecture 5.6 and the if direction of Theorem 6.2; the pair 17 and 19 is the first natural test.

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Extended reading notes

Core claim

The central claim is Theorem 6.2. Let K be an imaginary quadratic field over which Conjecture 5.6 holds, and let A/K be an abelian surface that is not the twist of a base change of a surface defined over Q, whose endomorphism algebra does not grow under base change, and whose endomorphism algebra is either Mat2(Q) or an indefinite division quaternion algebra with centre Q. Let C_A be the set of primes p for which the semisimplification of A[p] is two copies of a representation whose image is contained in the non-split Cartan subgroup C_ns(p). Then A is simple with quaternionic multiplication if and only if #C_A is at least 2; otherwise A is isogenous to the square of an elliptic curve. The n

Load-bearing premise

The load-bearing premise is Conjecture 5.6, that for every imaginary quadratic field K and every pair of distinct primes p and q every K-point of the non-split Cartan curve X_ns(pq) is non-exceptional, equivalently that the finite exceptional set S_{p,q} is disjoint from K; if one such point had non-rational j-invariant, an elliptic curve square over K could have two non-split Cartan primes and the converse would collapse.

Editorial extensions

If this is right

  • If #C_A is at least 2, no further p-adic information is needed: the surface must be QM, not the square of an elliptic curve.
  • An elliptic curve over an imaginary quadratic field cannot have non-split Cartan image at two primes unless its j-invariant is rational or belongs to a finite exceptional set; under Conjecture 5.6, no exceptions remain.
  • For indefinite quaternion discriminants up to 33 the characterization is unconditional, because every possible pair of ramified primes is covered by the levels 6, 7, 10, 11, 13 and 15.
  • The paper computes the 14 rational points on X_ns^+(15), showing that the two non-CM points are explained by an involution, and proves that every quadratic point on X_ns(15) is non-exceptional.
  • Over an imaginary quadratic field of class number greater than one, there are only finitely many primes p for which any elliptic curve can have non-split Cartan image mod p; for p beyond a field-dependent bound, X_ns(p)(K) is empty.

Reading between the lines

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

  • Inference: the criterion suggests a practical algorithm for the modularity conjecture stated in the introduction—checking whether a candidate abelian surface has two non-split Cartan primes would decide between an elliptic curve and a QM surface without computing its endomorphism algebra.
  • Inference: if Conjecture 5.6 holds, the finite exceptional sets S_{p1,p2} for p at least 17 are empty; computing even one such set, for example for the pair 17 and 19, would either confirm the conjecture for that pair or produce a concrete counterexample with non-rational j-invariant.
  • Inference: the methods used for X_ns(15) should transfer to other composite non-split Cartan levels whose plus-quotients have manageable genus, with the Mordell-Weil rank of the Jacobian, rather than the structure of the argument, being the practical bottleneck.
  • Inference: a similar two-prime Cartan criterion may hold for other abelian varieties attached to division algebras whose ramification at two places forces the residual representation to be a sum of conjugate copies, since nothing in the proof appears to depend on dimension two except the curve computations.
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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

2 major / 5 minor

Summary. The paper studies abelian surfaces A over imaginary quadratic fields K with End(A)⊗Q equal to either Mat₂(Q) or an indefinite division quaternion algebra, and it aims to characterize the quaternionic-multiplication (QM) case by the presence of at least two primes for which the residual Galois representation is contained in a non-split Cartan subgroup. The main theorem (Theorem 6.2) states this characterization under a conjecture (Conjecture 5.6) on K-points of the non-split Cartan modular curves X_ns(pq). The unconditional claim for indefinite quaternion discriminants up to 33 is also made in the abstract. The paper contains substantial new computations: rational points on X_ns^+(15) via elliptic curve Chabauty, and proofs that all quadratic points on X_ns(6), X_ns(10), and X_ns(15) are non-exceptional, together with a Mordell–Weil sieve using symmetric Chabauty. The computational sections are detailed and accompanied by Magma code.

Significance. If the main theorem were correct, it would give a clean, computationally checkable criterion distinguishing QM abelian surfaces from elliptic-square isogeny classes over imaginary quadratic fields, advancing the program initiated by Ohta, Serre, and Michaud-Jacobs. The new rational-point computations on X_ns^+(15) and the quadratic-point results for X_ns(6), X_ns(10), and X_ns(15) are valuable in their own right and appear reproducible from the supplied code. However, the central characterization has a genuine proof gap involving the distinction between isogeny and twist, so the headline result is not yet established as stated.

major comments (2)
  1. [§6, proof of Theorem 6.2] The contradiction step asserting that j(E)∈Q contradicts assumption (1) is invalid. Condition (1) says A is not an isomorphism twist of a base change of a Q-surface; it says nothing about the isogeny class of A. From A∼E^2 and j(E)∈Q, one may only conclude that the isogenous surface E^2 is a twist of a base change. A surface in the same K-isogeny class need not be a twist: if E_1/K has j(E_1)∈Q and E_2/K is K-isogenous to E_1 with j(E_2)∉Q, then A=E_1×E_2 satisfies (2) and (3), is not a twist of a base change (the Galois orbit of j(E_2) prevents a Q-model), and has #C_A≥2 whenever E_1 has non-split Cartan image at two primes. Thus the proof needs a new argument excluding this configuration, or the statement must be modified, e.g. by replacing 'twist' with 'isogenous to a twist' in condition (1) or by adding an extra hypothesis. This is load-bearing for the main characterization.
  2. [§4.2, Lemma 4.8(3)] The proof of the index bound is not correct as written. For a double cover ϱ, the norm identity is ϱ_*ϱ^* = 2 on divisors, not ϱ^*ϱ_* = 2. Moreover, from ϱ_*(2X−ϱ^*Y)=0 one cannot deduce that 2X−ϱ^*Y is torsion, because the kernel of ϱ_* on J(Q) contains the rational points of the Prym variety, which is positive-dimensional. The integer I with I | 2^5·3^4 is used in the Mordell–Weil sieve in the proof of Theorem 4.9, so its validity is load-bearing. Please replace this argument by a direct computation of the saturation index in Magma (which the supplied code should allow) or give a correct proof.
minor comments (5)
  1. [§4.1, Theorem 4.1] The theorem states that X_ns^+(15) has exactly 14 rational points, but the displayed list contains 15 entries. The affine point (1,−1) does not satisfy the given equation (for x=1 the right-hand side is 2 while y^2+y=0), and a comma is missing before (3,−37). The correct count is 12 affine points plus the two points at infinity.
  2. [Notation section] The base-change notation defines X_K := X ×_{Spec K} Spec L; the left-hand side should be X_L.
  3. [§2] The sentence 'C_ns^+(p) is generated by C_ns^+(p) and the matrix ...' should read '... generated by C_ns(p) and the matrix ...'.
  4. [Abstract / end of §5] The claim that the characterization is unconditional for all indefinite quaternion discriminants up to 33 is not stated or proved in the body. It follows implicitly from Theorem 5.1 and Proposition 6.1, but an explicit corollary or remark should be added, especially since the abstract advertises it.
  5. [§5, proof of Theorem 5.1] For the case p_1∈{7,11,13}, the text says the point on X_ns(p_1)(K) is non-exceptional and hence j(E)∈Q. If the cited result [MJ22] in fact gives only rational j-invariant, that is already sufficient for the conclusion; please clarify the wording so that it matches Definition 2.2.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof chain is self-contained or cites independent external results; the main theorem is conditional on an open conjecture but does not reduce to its own inputs.

full rationale

The paper's derivation chain is not circular. Theorem 6.2 is conditional on Conjecture 5.6, which is explicitly an open conjecture about quadratic points on non-split Cartan modular curves; it is an external assumption, not a restatement of the target theorem. The forward direction uses Proposition 6.1 and the external theorem of Ohta [Oht74]. The converse uses Theorem 5.1, whose proof rests on the paper's own computations for X_ns(6), X_ns(10), X_ns(15), on the independently proven prime-level results of Michaud-Jacobs [MJ22] for 7, 11, 13, and on standard finiteness results [HS91]. These computations (Pic^0 triviality, elliptic curve Chabauty, symmetric Chabauty, Mordell-Weil sieve) do not fit any parameter to the desired characterization; they are self-contained and reproducible from the given equations and code. There are no load-bearing self-citations: the only self-citation is the author's GitHub repository [Flo26], which is for code availability and not used as mathematical input. The use of Conjecture 5.6 is flagged by the author as a conjecture and is not equivalent to the theorem. A potential gap in the converse—the inference from j(E) in Q to a contradiction with condition (1) when A is only isogenous to E^2, since isogeny need not imply being isomorphic to a twist of a base change—is a correctness risk rather than a circularity; it does not make the derivation equivalent to its input. Therefore no circular step is identified.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted numerical parameters and no new entities. The central conditional claim rests on Conjecture 5.6, a genuinely unproven input, together with standard theorems in the field and Magma computations. The ledger is therefore small: one ad hoc conjecture, several established theorems, and no invented objects.

assumptions (6)
  • ad hoc to paper Conjecture 5.6: every K-point on X_ns(pq) is non-exceptional for every imaginary quadratic field K and every pair of distinct primes p, q.
    Assumed verbatim in Theorem 6.2. Unproven for pairs outside the sets {2,3,5} and {7,11,13} that are handled unconditionally.
  • standard math Ohta's theorem [Oht74] on the Galois image in (O tensor Z_p)^x for QM abelian surfaces and its surjectivity for all but finitely many p.
    Used in Section 6 to construct the residual representations and to show that the set C_A is finite.
  • standard math Serre's theorem [Ser72] on Galois properties of torsion points, including potentially multiplicative reduction and open image results.
    Used in Proposition 5.3 and Theorem 6.2 to bound exceptional primes and to exclude twists.
  • standard math Kolyvagin-Logachev [KL89] finiteness of Shafarevich-Tate group and rank determination for modular abelian varieties.
    Used in Lemma 4.8 to compute the rank of J(Q) from analytic ranks.
  • standard math Harris-Silverman [HS91]: a curve that is neither hyperelliptic nor bielliptic has finitely many quadratic points.
    Used in Theorem 5.1 to define the finite sets S_{p1,p2} for primes p >= 17.
  • standard math Larson-Vaintrob [LV14a, Theorem 6.4] bounding exceptional primes for Galois representations attached to abelian varieties.
    Used in Proposition 5.3 to obtain a constant C_K unless the inertial character equality psi^12 = chi_p^6 holds.

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Pith. "Pith review of Non-split Cartan curves and a characterization of fake elliptic curves." pith.science (2026). https://pith.science/paper/BS6WFHOJ

@misc{pith2026260803934,
  author       = {Pith},
  title        = {Pith review of: Non-split Cartan curves and a characterization of fake elliptic curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BS6WFHOJ}},
  note         = {Machine review of arXiv:2608.03934}
}
abstract

The $\ell$-adic Tate module of an abelian surface with quaternionic multiplication (QM) decomposes as two copies of a two-dimensional $\mathbb{Q}$-rational representation. To this day, there is no criterion to distinguish these representations from those attached to elliptic curves. For this reason, QM abelian surfaces are usually called fake elliptic curves. In this paper we characterize QM abelian surfaces defined over an imaginary quadratic field $K$. Namely, we consider an abelian surface $A/K$ without potential CM and whose $L$-function is a square. Under a reasonable conjecture, we show that $A$ has QM if and only if it has residual image contained in a non-split Cartan group modulo at least two primes. The characterization is unconditional for all indefinite quaternion discriminants up to 33. The proof is based on previous work of Siksek and Michaud-Jacobs on quadratic points on non-split Cartan modular curves. In particular, we compute the rational points on the curve $X_{ns}^+(15)$, and we prove that all quadratic points on $X_{ns}(6)$, $X_{ns}(10)$ and $X_{ns}(15)$ are non-exceptional.

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