{"id":"d2fcd0d6-5bb7-4814-baa1-3329679116c4","arxiv_id":"1908.00421","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper explicitly realizes a modular abelian surface over Q(√61) with everywhere good reduction and no principal polarization as the Prym of a genus 3 curve.","lead":"Mathematicians constructed an explicit equation for a curve whose associated abelian surface has good reduction at every prime over the quadratic field Q(√61). The example completes the last open case of a classification and is the first unconditional surface of its kind without a principal polarization.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Faltings–Serre step in Theorem 5.7 depends on unverified computational class field theory (uniqueness of the (2)-ramified extension and generation by primes above 3, 5, 61, 97); a wrong generation statement would invalidate the good-reduction proof.","rationale":"The reader's weakest assumption correctly identifies the load-bearing point. The central new claim is the everywhere good reduction of the Prym A, and the paper itself states that direct verification of good reduction is not algorithmically available. The modularity route through Faltings–Serre is standard, but it is only as sound as the finite-generation certificate and the uniqueness statement. These are explicitly computational assertions in Section 5, and no code or data files are shipped, so independent verification is not currently possible. If the generation statement were false, checking only primes above 3, 5, 61, 97 would not suffice, and the proof of part (c) would fail. The same applies to the uniqueness of the degree-3-or-6 extension, which is used to identify the residual representation. That said, the rest of the paper has independent support: exact Gröbner computations for smoothness and bad reduction, an exact twist matching Euler factors, a rigorous endomorphism certificate, and a clean norm obstruction for the principal-polarization claim. These make the theorem highly plausible but do not remove the need for the computational certificate. Since the reader already assigned CONDITIONAL, my concern reinforces that verdict without moving it.","tokens_in":19222,"tokens_out":7798,"duration_ms":84216,"concrete_test":"Independently recompute in Magma or PARI/GP: compute the splitting field L of p0(t) over K, verify (5.10) and (5.11); then compute Z_L[1/2]^×/squares and test whether the prime ideals above 3, 5, 61, 97 generate it. Enumerate all extensions of K of degree 3 or 6 unramified outside (2) by class field theory and check uniqueness. If generation fails, determine whether enlarging the prime set (all primes of L of norm up to 200, or further) yields a certificate; if no certificate is found, the Faltings–Serre conclusion in Theorem 5.7(c) is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 5.7(c) gives A everywhere good reduction by proving ρA,l ≃ ρfK,l via Faltings–Serre. This requires two computational class field theory statements in §5: (i) there is a unique extension of K of degree 3 or 6 ramified only at (2), namely the normal closure of p0(t); and (ii) Z_L[1/2]^×/Z_L[1/2]^×2 ≅ (Z/2Z)^9 and the primes of L above 3, 5, 61, 97 generate the corresponding elementary 2-abelian extension. These are asserted as computed facts, but no scripts, Magma/PARI code, or certificate files are provided. The trace/equality checks at only those primes are justified by (ii); if (ii) fails, they do not bound all possible primes, so ρA,l ≃ ρfK,l is not established. Similarly, (i) is used to identify the residual representations; if the extension is not unique, the residual comparison via 'uniqueness' is unsupported. Since the authors explicitly note they cannot verify good reduction of the Prym directly, this computational certificate is the sole bridge to part (c).","agreement_with_reader":"agree"},"referee_report":null,"author_rebuttal":null,"desk_editor":null,"rs_alignment":null,"lean_confirmation":null,"pith_extraction":null,"created_at":"2026-08-14T15:57:49.504752+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":null,"supporting_citations":[],"review_version":1}