{"id":"dc733880-8a9b-4096-b56b-b7156dfff6d5","arxiv_id":"2602.19813","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Geometrically simple abelian surfaces over Q with small conductor are shown to have (Z/pZ)^2 in their Tate–Shafarevich groups for p = 5, 7, 11, 13, via modular-form congruences and visibility.","lead":"This paper constructs abelian surfaces over the rational numbers, with conductor below 10^8, whose Tate–Shafarevich groups contain a subgroup (Z/pZ)^2 for p = 5, 7, 11, and 13. It is the first time p = 11 and p = 13 torsion is exhibited for geometrically simple abelian surfaces.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The p=11 and p=13 examples hinge on the unpublished [CEH+a,b] curve-to-newform correspondence; an error in that database would invalidate the theorem's new cases.","rationale":"The reader's weakest_assumption identifies the same load-bearing dependency: the unpublished [CEH+a,b] database is needed to connect the explicit genus-2 curves to the LMFDB newforms, and a failure in that connection would destroy the p=11 and p=13 examples, which are the main novelties. My read of the paper supports this: the internal mathematics after the correspondence is largely standard (Kraus-Oesterle style congruence certification, Agashe-Stein-Fisher visibility with Tamagawa and rank checks), and the authors are transparent about dagger rows and unverified Tamagawa numbers, but the curve-to-newform bridge is not fully reproducible from the submitted text alone. The appropriate remedy is not rejection; it is to require a public, versioned, independent verification of the critical rows, exactly the conditionality already expressed. I see no reason to move the verdict, but I would emphasize that the p=11 and p=13 rows specifically should be the first targets of that verification.","tokens_in":19572,"tokens_out":27351,"duration_ms":248387,"concrete_test":"Independently certify the two critical rows without using [CEH+a,b]: for the Table B.3 curves labelled 9025.2.a.r (p=11) and 6776.2.a.r (p=13), compute conductor(J) via [DD19], compute End(J) via [CMSV19] to confirm the stated real multiplication, and compute trace/norm of Frobenius for all good primes ell up to the Sturm bound mu(N)/6; then check these match the LMFDB newform data for 9025.2.a.r and 6776.2.a.r. If these checks pass, the curve-to-newform identification is independently confirmed; if they fail, the corresponding Theorem 1.1 row is invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim for the genuinely new primes p=11 and p=13 rests on two rows of Table B.2: (11, 9025.2.a.r/9025.2.a.n) and (13, 6776.2.a.r/6776.2.a.b). For each such row, the proof that the explicit genus-2 curve in Table B.3 is the Jacobian attached to the stated LMFDB newform is not self-contained. Proposition 2.3 says the abelian surfaces are 'given explicitly' and that RM and conductor are verified, but the actual bridge to the LMFDB newform is described as relying 'crucially' on the in-preparation database [CEH+a,b], supplemented by the authors' checks. The published text does not provide the Hilbert modular surface point or the full matching computation, so a reader cannot certify the row-to-newform assignment from the arXiv version alone. A single misassignment, typo, or RM-field error in a critical row would remove that row's (Z/pZ)^2 subset of Sha(A/Q), and because the dagger-marked p=11 row is excluded from Theorem 1.1, the p=11 and p=13 novelty claims would collapse if the affected rows are wrong. This is not an internal inconsistency; the modular-forms congruence machinery and visibility argument are coherent conditional on the correspondence, but the correspondence is the least-secure link in the chain.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that certain explicitly listed geometrically simple abelian surfaces A/Q of conductor at most (10000)^2 have (Z/pZ)^2 contained in the Tate--Shafarevich group X(A/Q) for p = 5, 7, 11, 13. The method is to find congruences between weight-2 newforms using a Kraus--Oesterlé-style criterion, pass to RM abelian varieties, and then apply a visibility theorem of Agashe--Stein and Fisher (corrected in Appendix A). The paper also gives a conjectural example of 7-torsion in X(A/Q) not visible in any abelian threefold, and reports a computational enumeration of all such congruences in the LMFDB for coefficient fields of degree at most 4.","tokens_in":19839,"tokens_out":11821,"duration_ms":104275,"significance":"If the results are correct, the paper supplies the first geometrically simple abelian surfaces with nontrivial 11- and 13-torsion in X(A/Q). The computational framework is a natural extension of Cremona--Freitas, and the authors are admirably explicit about which rows depend on unpublished work, which Tamagawa numbers could not be computed, and which rows are excluded from the main theorem. The paper ships Magma/Sage code and uses a Sturm-bound certificate for the congruences and an independent visibility argument for the Sha elements, so the proof strategy is transparent and checkable in principle. The central weakness is that the genuinely new p = 11 and p = 13 rows rely on the unpublished database [CEH+a,b] for the curve-to-newform correspondence, and the published text does not contain enough data to certify that correspondence independently.","major_comments":[{"comment":"The new p=11 and p=13 examples in Theorem 1.1 rest on identifying the genus-2 curves in Table B.3 with the newforms 9025.2.a.r/9025.2.a.n and 6776.2.a.b. The proof of Proposition 2.3 says this is checked by a Hilbert modular surface point, conductor computation, and trace/norm matching, but none of the actual data is given; Table B.3 says it 'relies heavily' on the unpublished [CEH+a,b]. Since the other p=11 row (6962.2.a.q) is dagger-marked and excluded, an error in the 9025 row would collapse the p=11 novelty claim. Please include the full certificate: Hilbert point, conductor verification, and the trace/norm values for enough primes.","section":"Proposition 2.3 / Table B.2, rows (11, 9025.2.a.r) and (13, 6776.2.a.r)"},{"comment":"Theorem 1.4 is stated for 'any of the tuples in Table B.2', but Table B.2 contains rows marked * ('unproved, since we could not compute a Tamagawa number'), e.g. 6864.2.a.bf, 7632.2.a.v, and 9802.2.a.j. For those rows the Tamagawa-number hypothesis of Theorem 4.1(i) is not verified, so Theorem 1.4 is false as stated. The statement must be restricted to rows without * and without †, or the unproved/unpublished rows must be proved and the marks removed.","section":"Theorem 1.4 / Table B.2"}],"minor_comments":[{"comment":"The proof says 'there exists 1 ≤ i ≤ 6', but the sets are indexed L_0,...,L_5. This should be 0 ≤ i ≤ 5.","section":"Lemma 3.2, proof"},{"comment":"The statement 'we are left only with the examples given' is a finite computational completeness claim, but the paper does not report the number of hash collisions, the exact version of the code, or a log/certificate. Since Theorem 1.8 is not needed for Theorem 1.1 this is not a blocking issue, but reproducibility would be improved by a frozen repository snapshot or a checksummed output.","section":"Section 3, proof of Theorem 1.8"},{"comment":"Several typos: 'computing' should be 'computed', 'assocaited' should be 'associated', and 'yeilds' should be 'yields'. Please proofread the table notes.","section":"Table B.2 / Table B.3"},{"comment":"The 'direct calculation' that the local factors of f' and g' agree modulo m is not shown; a reference to the precise case in [KO92, Proposition 4] would help the reader.","section":"Section 2, Table 2.1"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is likely correct, but acceptance should wait until the [CEH+a,b] dependency is resolved or the relevant verification data is included. The inconsistency in Theorem 1.4 regarding asterisk/dagger rows is easily fixed but must be fixed. I also recommend asking the authors to make the Theorem 1.8 enumeration reproducible, even though it is not load-bearing for Theorem 1.1."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What should you know about this paper? It genuinely delivers something new: the first examples of geometrically simple abelian surfaces over Q with (Z/11Z)^2 and (Z/13Z)^2 in their Tate–Shafarevich groups, found by enumerating congruences between weight 2 newforms and then proving the Sha classes via visibility. The main theorem (1.1) is backed by a careful chain: Proposition 2.2 gives a Kraus–Oesterlé style congruence criterion with a Sturm bound, Proposition 2.3 promotes the newform congruences to congruences of Galois representations for explicit Jacobians, and Theorem 4.1 (with a corrected proof of Fisher's visibility criterion in Appendix A) turns the congruences into elements of Sha. The paper is unusually transparent: rows where the Tamagawa number couldn't be verified are marked with asterisks, rows relying on unpublished Dokchitser–Jakovac computations are daggered and excluded from the theorem, and the GitHub repository includes code and data.\n\nThe soft spots are real but addressable. The most loaded one is the bridge from the genus 2 curves in Table B.3 to the LMFDB newforms that carry the congruences. The authors state that the curves come from the in-preparation database [CEH+a,b], and while Proposition 2.3 says they verify real multiplication, conductor, and newform identification via trace/norm matching, the Hilbert modular surface point and the matching details are not in the text. So the p=11 and p=13 examples, which are the actual novelty, rest on an external unpublished database plus computations the paper does not fully expose. An error there would indeed knock out the new cases. That does not appear to be a live danger—the authors' checks are substantive rather than a black box—but a referee should ask for the missing details and a pinned commit of the [CEH+b] repository. The enumeration in Theorem 1.8 is also conditional on LMFDB completeness up to level 10000, which is a standard and reasonable assumption but worth stating.\n\nThe central argument holds up as far as I can tell. The math is careful, the criteria are provable, and the authors do not overclaim—they explicitly exclude unverified rows. The paper would benefit from a fuller audit trail, but this is not a reason to desk reject.\n\nWho should read it: anyone working on computational arithmetic geometry, Sha of abelian varieties, or modular congruences. It deserves a serious referee; I'd send it out and ask for a revised version with the external dependencies pinned down and the curve-to-newform verification made more explicit.","headline":"First examples of 11- and 13-torsion in Sha of simple abelian surfaces, carefully proven, with a reproducibility gap around unpublished database dependencies that a referee should close.","tokens_in":20399,"tokens_out":15773,"would_cite":true,"duration_ms":121400,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G10","11F33","11G40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that geometrically simple abelian surfaces over Q with conductor at most (10000)^2 exist whose Tate-Shafarevich groups contain a subgroup isomorphic to (Z/pZ)^2 for p = 5, 7, 11, 13, including the first known examples for","keywords":["Tate-Shafarevich group","abelian surface","modular forms","congruences","visibility","genus 2 Jacobian","real multiplication","conductor"],"falsifier":"Choose one row of Table B.3, recompute the conductor of the Jacobian of the given Weierstrass equation (it must equal the square of the newform level), verify real multiplication by the listed order, and check the mod-p Fourier-coefficient congruence up to the stated bound; any failure would disprove the corresponding claim of Theorem 1.1.","tokens_in":19413,"feed_emoji":"🧮","tokens_out":13305,"duration_ms":101290,"temperature":0.7,"pith_summary":"This paper proves that for each prime p = 5, 7, 11, 13 there is a geometrically simple abelian surface A/Q, with conductor at most (10000)^2, whose Tate-Shafarevich group X(A/Q) contains a subgroup isomorphic to (Z/pZ)^2. These are, according to the authors, the first known such examples for p = 11 and 13. The surfaces are given explicitly as Jacobians of genus-2 curves, and the proof works by detecting congruences between the associated weight-2 newforms and those of auxiliary abelian varieties, then using the visibility theorem to turn the congruence into a non-trivial element of Sha. The paper also gives a conjectural example of an order-7 class in Sha that is not visible in any abelian threefold, which would mean its visibility dimension is 4.","feed_headline":"First abelian surfaces with 11- and 13-torsion in Sha","feed_subtitle":"Explicit genus-2 Jacobians put (Z/pZ)^2 in the Tate-Shafarevich group for p = 5, 7, 11, and 13.","key_machinery":"The load-bearing identity is the (p,q)-congruence: an isomorphism A[p] ≅ B[q] of Galois modules for prime ideals p and q of the endomorphism rings of two abelian varieties. On the modular-forms side this is equivalent to congruence of Fourier coefficients modulo a prime above p, and the paper proves a finite criterion (generalising a classical one) so that only coefficients up to a computable bound need be checked. The visibility theorem then identifies a subgroup of X(A/Q) with a quotient of Mordell-Weil groups of an isogenous variety, provided the rank of A is 0, B has positive rank, and the Tamagawa numbers are coprime to p. This converts a congruence into an unconditional statement about","core_discovery":"On the paper's own terms, the discovery is a systematic way to produce unconditional examples of non-trivial Tate-Shafarevich groups. A finite enumeration of congruences between weight-2 newforms with coefficient fields of degree at most 4 and levels up to 10000 yields, after passing to the corresponding abelian varieties and applying a visibility criterion, abelian surfaces A/Q such that (Z/pZ)^2 embeds into X(A/Q). The key advance over previous constructions is that the congruences are proved rather than conjectured: a generalised finite-coefficient criterion checks only Fourier coefficients up to a prescribed bound and, once the mod-p representation is shown to be irreducible, upgrades to","pith_inferences":["The same sieve could be run on larger level bounds or higher-degree coefficient fields once the modular-forms database grows; the restriction to levels up to 10000 and degree up to 4 is a database artefact, not a limitation of the method.","The proof upgrades semisimple isomorphisms to true Galois-module isomorphisms using irreducibility of the mod-p representations, so a variant of the method might handle reducible cases only at the cost of an extra argument.","If Conjecture 1.5 is proved, it would supply the first explicit element of Sha of an abelian surface whose minimal visualizing abelian variety is of dimension 4, giving a sharp answer to a natural quantitative question about visibility.","One could test the method's reach by looking for congruences of the same type among newforms of conductor just above 10000; a first example there would suggest the phenomenon is not tied to small levels."],"forward_implications":["For each p = 5, 7, 11, 13 there is now an explicit, geometrically simple abelian surface over Q known unconditionally to have (Z/pZ)^2 in its Tate-Shafarevich group; the p = 11 and 13 cases are new.","Because the surfaces are Jacobians of explicit genus-2 curves, they can serve directly as test cases for the standard conjecture relating L-functions to ranks and Sha orders.","The paper's enumeration of congruences among level-at-most-10000 newforms with coefficient fields of degree at most 4 gives a complete list for this range, so future visibility constructions can be checked against it.","The conjectural p = 7 example (label 9603.2.a.o) provides evidence that the visibility dimension of a Sha element can be as large as 4, and reduces the conjecture to determining rational points on two explicit plane quartic curves."],"fun_headline_variants":["Abelian surfaces with (Z/pZ)^2 in Sha for p=5,7,11,13","Proven Sha elements on simple abelian surfaces of small conductor","Conductor-bounded abelian surfaces with nontrivial Tate-Shafarevich","Explicit Jacobians with 5,7,11,13-torsion in Sha","Unconditional Sha subgroups on small-conductor abelian surfaces"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The identification of each explicit genus-2 curve with the abelian variety attached to the stated newform depends on unpublished database work, and some Tamagawa-number computations are unpublished; if any of these data are wrong, the corresponding row of Theorem 1.1 would no longer be a proved example.","fun_headline_variants_meta":{"raw":{"variants":["Abelian surfaces with (Z/pZ)^2 in Sha for p=5,7,11,13","Proven Sha elements on simple abelian surfaces of small conductor","Conductor-bounded abelian surfaces with nontrivial Tate-Shafarevich","Explicit Jacobians with 5,7,11,13-torsion in Sha","Unconditional Sha subgroups on small-conductor abelian surfaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001181,"raw_usage":{"total_tokens":4737,"prompt_tokens":789,"completion_tokens":3948,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":533,"completion_tokens_details":{"reasoning_tokens":3843}},"tokens_in":533,"tokens_out":3948,"duration_ms":25696,"temperature":1.0,"reasoning_tokens":3843,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T21:30:46.004212+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Choose one row of Table B.3, recompute the conductor of the Jacobian of the given Weierstrass equation (it must equal the square of the newform level), verify real multiplication by the listed order, and check the mod-p Fourier-coefficient congruence up to the stated bound; any failure would disprove the corresponding claim of Theorem 1.1.","supporting_citations":[],"review_version":1}