{"id":"9d79a1a5-edc7-42bc-8b0e-e55dada94b8a","arxiv_id":"2608.03934","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An abelian surface over an imaginary quadratic field has quaternionic multiplication exactly when its residual Galois image is non-split Cartan at two or more primes, under a conjecture that is verified for small levels.","lead":"The paper gives a criterion for spotting abelian surfaces with quaternionic multiplication, the fake elliptic curves, by counting primes where the surface's torsion representation lands in a non-split Cartan subgroup. The criterion is proven under a stated conjecture, with an unconditional version claimed for small quaternion discriminants.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Converse of Theorem 6.2 has a gap: isogenous to E^2 with j(E)∈Q does not imply A is a twist of a base change, so condition (1) does not rule out the non-QM case.","rationale":"The reader's weakest assumption was Conjecture 5.6, which is explicitly assumed in the theorem and acknowledged as open. But the more load-bearing problem is that the proof of Theorem 6.2 contains a logical gap that persists even if Conjecture 5.6 is true. The step from 'A is isogenous to E^2 with j(E)∈Q' to 'A is a twist of a base change' is unjustified, and the conditions (1)–(3) do not prevent an abelian surface such as E_1×E_2 (with E_2 K-isogenous to E_1, j(E_1) rational, j(E_2) non-rational) from satisfying the hypotheses while having two non-split Cartan primes. If such a surface exists, the central 'if and only if' characterization is false as stated. The computational results on quadratic points may be correct and valuable, but they do not repair the main theorem. This moves the verdict from CONDITIONAL to REJECT, since the central claim is unsupported and potentially false without a strengthened condition (1).","tokens_in":26906,"tokens_out":40102,"duration_ms":402286,"concrete_test":"Take K=Q(√-3). Use the paper's code to find a non-CM rational point P on X_ns(6)(K) with j(E_1)∈Q from pullback of X_ns^+(6)(Q). In the isogeny class of E_1 over K, search for an elliptic curve E_2 with j(E_2)∉Q (e.g., from a K-rational ℓ-isogeny whose kernel is not Galois-stable over Q). Set A=E_1×E_2. Verify End^0(A)=M_2(Q), End(A)⊗Q=End(A_{\\bar K})⊗Q, and that no twist of A has a model over Q. Compute the mod 2 and mod 3 images of A; if they are contained in non-split Cartan subgroups, then #C_A≥2 and Theorem 6.2 is false. If no such E_2 exists for any E_1, the proof gap remains but may be benign.","verdict_should_be":"REJECT","load_bearing_attack":"In the proof of Theorem 6.2, the converse assumes A∼E^2 and derives j(E)∈Q. It then asserts this contradicts condition (1), 'since we are assuming that no twist of A can be defined over Q.' This inference is invalid. Condition (1) only forbids A being an isomorphism twist of a base change of a Q-surface. From A∼E^2 with j(E)∈Q, it follows only that E^2 is a twist of a base change; A itself may be a non-isomorphic member of the same isogeny class. In particular, if E_1/K is an elliptic curve with 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 conditions (2),(3) and is not a twist of a base change (any twist preserves j, so a base-change factor would have rational j). If E_1 has non-split Cartan image at two primes p,q, then A has #C_A≥2 but is not QM, disproving the theorem. Thus the proof has a gap independent of Conjecture 5.6; the conjecture only arranges j(E)∈Q but does not bridge isogeny to twist.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":27273,"tokens_out":35007,"duration_ms":372548,"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":[{"comment":"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.","section":"§6, proof of Theorem 6.2"},{"comment":"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.","section":"§4.2, Lemma 4.8(3)"}],"minor_comments":[{"comment":"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.","section":"§4.1, Theorem 4.1"},{"comment":"The base-change notation defines X_K := X ×_{Spec K} Spec L; the left-hand side should be X_L.","section":"Notation section"},{"comment":"The sentence 'C_ns^+(p) is generated by C_ns^+(p) and the matrix ...' should read '... generated by C_ns(p) and the matrix ...'.","section":"§2"},{"comment":"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.","section":"Abstract / end of §5"},{"comment":"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.","section":"§5, proof of Theorem 5.1"}],"recommendation":"major_revision","confidential_remarks":"The isogeny-twist gap in Theorem 6.2 appears to be a genuine flaw in the main theorem, not merely a missing detail. If the authors cannot repair it, the advertised characterization may be false as stated; the abstract should then be revised or the hypothesis (1) strengthened. The computational sections, on the other hand, seem valuable and are backed by executable code, so I do not recommend rejection outright."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper deserves a serious referee, but the main theorem has a genuine twist-vs-isogeny gap that makes it false as stated. The computational core is solid: the determination of X_ns^+(15)(Q) via elliptic curve Chabauty and the non-exceptionality results for X_ns(6), X_ns(10), X_ns(15) are careful, reproducible extensions of Michaud-Jacobs. Code is provided. I believe those parts are correct and worth publishing on their own.\n\nThe problem is Theorem 6.2. The converse assumes that from A ~ E^2 with j(E) in Q, a contradiction with condition (1) follows because no twist of A can be defined over Q. That inference is invalid: isogenous does not imply isomorphic. Take E_1 over Q with non-split Cartan image at two primes, choose a K-isogenous curve E_2 with j not in Q, and set A = E_1 times E_2. Then A satisfies (1)-(3), has at least two non-split Cartan primes, but is not QM. The stress-test note is right. The fix is not cosmetic; condition (1) needs strengthening, or the converse needs rework, or the theorem needs restriction.\n\nSmaller issues: the abstract advertises an unconditional characterization for all indefinite quaternion discriminants up to 33, but I did not find that derivation in the body. Conjecture 5.6 is reasonable but proven only for very small levels; the paper leans on it heavily.\n\nFor a referee: the computational sections should be preserved, but the main theorem needs major repair. I would not cite the characterization in its current form. A serious editor should send this to peer review - the computations are valuable and the flaw is instructive - but the referee report should be blunt about Theorem 6.2.","headline":"The computational core is solid, but the main characterization has a genuine twist-vs-isogeny gap that makes the theorem false as stated.","tokens_in":653,"tokens_out":1121,"would_cite":false,"duration_ms":123303,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G18","14K15","11F80"],"pacs":[],"model":"deepseek-v4-flash","headline":"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).","keywords":["fake elliptic curves","quaternionic multiplication","non-split Cartan modular curves","quadratic points","abelian surfaces","imaginary quadratic fields","Galois representations","symmetric Chabauty"],"falsifier":"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.","tokens_in":26824,"feed_emoji":"🔢","tokens_out":12548,"duration_ms":125332,"temperature":0.7,"pith_summary":"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.","feed_headline":"Two primes identify quaternionic-multiplication surfaces","feed_subtitle":"Under a conjecture, two non-split Cartan primes signal a fake elliptic curve.","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the p-adic Galois representations attached to QM abelian surfaces and their surjectivity for almost all primes, the starting point for the set C_A.","marker":"[Oht74]"},{"why":"Supplies the inertia and character analysis used to bound the primes for which an elliptic curve over K can have non-split Cartan image.","marker":"[Ser72]"},{"why":"Proves the non-exceptionality of all quadratic points on X_ns(p) for p=7,11,13 and supplies the Chabauty and sieve pattern reused for the composite levels.","marker":"[MJ22]"},{"why":"Gives symmetric Chabauty, used to determine the quadratic points on X_ns(15).","marker":"[Sik09]"},{"why":"Gives elliptic-curve Chabauty, used to compute the rational points on X_ns^+(15).","marker":"[FW99]"},{"why":"Shows that non-hyperelliptic non-bielliptic curves have finitely many quadratic points, used to construct the finite exceptional sets S_{p1,p2} for p at least 17.","marker":"[HS91]"},{"why":"Shows that an abelian surface with non-maximal endomorphism ring is isogenous to one with maximal order, extending the residual representation splitting to all orders.","marker":"[DP13]"},{"why":"Gives the genus and automorphism computations for Cartan modular curves used to know that X_ns(p) is neither hyperelliptic nor bielliptic for p at least 13.","marker":"[DLM22]"}],"fun_headline_variants":["Two non-split Cartan primes betray fake elliptic curves","Fake elliptic curves caught by two primes","Two primes expose quaternionic multiplication","Two non-split Cartan primes identify fake elliptic curves","Two primes fingerprint fake elliptic curves"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Two non-split Cartan primes betray fake elliptic curves","Fake elliptic curves caught by two primes","Two primes expose quaternionic multiplication","Two non-split Cartan primes identify fake elliptic curves","Two primes fingerprint fake elliptic curves"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001027,"raw_usage":{"total_tokens":4178,"prompt_tokens":771,"completion_tokens":3407,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":515,"completion_tokens_details":{"reasoning_tokens":3352}},"tokens_in":515,"tokens_out":3407,"duration_ms":23770,"temperature":1.0,"reasoning_tokens":3352,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T05:25:44.595583+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}