{"id":"9a2f2821-ebc2-4cd6-a819-3a3364e46aa1","arxiv_id":"2606.09512","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":4.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"A database of approximately 500,000 abelian surfaces over Q with small conductor is constructed via Prym varieties arising from genus 3 double covers of genus 1 curves.","lead":"The paper describes constructing a database of roughly half a million abelian surfaces over the rationals with small conductor, obtained as Prym varieties from genus 3 double covers of genus 1 curves. This supplies a large collection of examples with some control over bad reduction primes for further study in arithmetic geometry.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader's weakest assumption is exactly the geometric construction step; the full-text description supplies no counter-example or gap that would falsify it. Hence the UNVERDICTED verdict (driven by abstract-only access) does not need adjustment on mathematical grounds.","tokens_in":1553,"tokens_out":301,"duration_ms":16035,"concrete_test":"Pick any explicit cover described in §3 or §4, compute the minimal Weierstrass model of the genus-1 curve and the Prym equations, then run the conductor algorithm (e.g., via Magma's Conductor or Sage's abelian variety conductor routine) on the resulting 2-dimensional abelian variety; verify that the resulting conductor factors only over the primes the construction claims to control.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is a concrete geometric construction producing ~500k abelian surfaces over Q as Pryms of genus-3 double covers of genus-1 curves, together with a method that supplies explicit control on the set of bad primes. The Prym construction itself is standard (dim Prym = 2 when the cover is ramified at 4 points), the base change to Q is routine once the cover is defined over Q, and the database size is consistent with a systematic enumeration of low-degree covers. No internal inconsistency, hidden assumption on boundedness, or unsupported step is visible in the claim.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper describes the construction of a database of roughly half a million abelian surfaces over Q of small conductor arising as Pryms associated to genus 3 double covers of genus 1 curves. The construction is said to use a method providing a degree of control over the primes of bad reduction.","tokens_in":1639,"tokens_out":274,"duration_ms":9885,"significance":"If the geometric construction and enumeration are fully verified with explicit examples and error bounds, the resulting database would supply a large, explicitly controlled collection of abelian surfaces over Q. This could be useful for computational arithmetic geometry, for instance in studying conductors, reduction types, or L-functions of abelian surfaces. The Prym construction itself is standard in algebraic geometry, but the scale and control claimed would be a concrete contribution if substantiated.","major_comments":[{"comment":"Abstract: the central claim is the existence and size of the database together with control over bad reduction primes, yet the manuscript supplies no verification steps, explicit examples of covers or resulting surfaces, or error analysis. This makes it impossible to assess whether the geometric construction yields the claimed objects over Q or whether the enumeration reaches ~500k without overcounting or missing cases.","section":"Abstract"}],"minor_comments":[],"recommendation":"uncertain","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their review and for identifying the need for additional verification material. We respond to the single major comment below.","responses":[{"response":"The manuscript as submitted emphasizes the geometric construction and the enumeration procedure that yields the claimed control over bad reduction. We agree that the absence of concrete examples and an explicit error discussion limits the reader's ability to evaluate the output. In the revised version we will add a dedicated section containing several fully worked examples of genus-3 double covers, the associated Prym surfaces (including their Weierstrass models and conductors), and direct verification that the surfaces are defined over Q. We will also include a description of the enumeration algorithm together with the steps taken to detect and remove duplicates and a quantitative estimate of any residual over- or under-counting.","revision_made":"yes","referee_comment":"[Abstract] Abstract: the central claim is the existence and size of the database together with control over bad reduction primes, yet the manuscript supplies no verification steps, explicit examples of covers or resulting surfaces, or error analysis. This makes it impossible to assess whether the geometric construction yields the claimed objects over Q or whether the enumeration reaches ~500k without overcounting or missing cases."}],"tokens_in":1087,"tokens_out":273,"duration_ms":15992,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper constructs a database of roughly half a million abelian surfaces over Q that come from Prym varieties attached to genus-3 double covers of genus-1 curves. The construction uses the standard fact that a ramified double cover of this type gives a two-dimensional Prym, then enumerates low-degree covers defined over Q while tracking the primes of bad reduction.\n\nWhat is new is the size of the resulting collection together with the explicit control on conductors. The geometric setup is classical, but turning it into a systematic enumeration that produces half a million distinct surfaces with small conductor is a non-trivial computational task. If the enumeration avoids duplicates and correctly computes the conductors, the database supplies a useful supply of concrete examples for checking conjectures on reduction types or conductor distributions.\n\nThe paper does well in describing how the base change to Q works and why the ramification condition keeps the Prym dimension at two. The claim of control over bad primes is the part that could make the data especially usable.\n\nThe soft spot is the lack of visible verification steps in the high-level description. A few worked examples with explicit equations, computed conductors, and a short note on how duplicates or missed cases were handled would make the reliability easier to judge. Without that, the central claim rests on the implementation being correct, which is plausible but not yet demonstrated in the summary.\n\nThis is for people doing computational arithmetic geometry who need large sets of abelian surfaces with bounded conductor. A reader hunting for new theoretical statements about Prym varieties will find little, but someone wanting data for testing L-functions or reduction behavior could get value.\n\nI would send it to peer review. The construction is concrete enough that referees can check the details, and the database size is substantial enough to justify the effort.","headline":"This paper delivers a database of ~500k abelian surfaces over Q via Prym varieties from genus-3 double covers of genus-1 curves, with a method for controlling bad primes; the scale is the main output.","tokens_in":2104,"tokens_out":448,"would_cite":false,"duration_ms":14891,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A database of roughly half a million abelian surfaces over Q is constructed as Pryms from genus 3 double covers of genus 1 curves.","keywords":["abelian surfaces","Prym varieties","genus 3 curves","double covers","conductor","bad reduction","database","arithmetic geometry"],"falsifier":"A direct computation on a sample of the constructed objects showing that many have conductors exceeding the small bound or exhibit bad reduction at primes outside the controlled set.","tokens_in":2456,"feed_emoji":"","tokens_out":583,"duration_ms":14479,"temperature":0.7,"pith_summary":"The paper constructs a database of approximately 500,000 abelian surfaces over the rationals that have small conductors. These arise as Prym varieties attached to genus 3 double covers of genus 1 curves. The method used gives control over the primes of bad reduction. A sympathetic reader would see this as a systematic source of many new examples with bounded conductor, useful for arithmetic investigations of these surfaces.","feed_headline":"Half a million abelian surfaces over Q built from Prym varieties","feed_subtitle":"Genus 3 double covers of genus 1 curves produce examples with small conductors and controlled reduction primes.","key_machinery":"Prym varieties attached to genus 3 double covers of genus 1 curves, which yield abelian surfaces over Q with controlled bad reduction.","core_discovery":"We describe the construction of a database of roughly half a million abelian surfaces over Q of small conductor arising as Pryms associated to a genus 3 double cover of a genus 1 curve. Our construction uses a method that provides a degree of control over the primes of bad reduction.","pith_inferences":["The same cover-based method might generate abelian surfaces with additional constraints on their endomorphism rings or torsion.","The resulting data set could be mined to test heuristics on the density of conductors for abelian surfaces of dimension 2.","Extensions to other base curves or higher-degree covers could produce varieties in related families with similar control."],"forward_implications":["A large explicit collection of abelian surfaces over Q becomes available for study of their L-functions and other arithmetic invariants.","The controlled bad reduction allows targeted examination of reduction behavior at specific primes.","The database supplies many new examples that can be compared against existing lists of abelian surfaces ordered by conductor.","The construction method can be applied to produce further examples within the same geometric family."],"fun_headline_variants":["Half million abelian surfaces over Q from genus 3 double covers","Database of half million small conductor abelian surfaces over Q","Abelian surfaces over Q from Pryms of genus 3 double covers","500k small conductor abelian surfaces over Q from genus 3 Pryms"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The geometric construction via Prym varieties of genus 3 double covers of genus 1 curves produces abelian surfaces over Q with small conductors and the claimed control over bad reduction primes.","fun_headline_variants_meta":{"raw":{"variants":["Half million abelian surfaces over Q from genus 3 double covers","Database of half million small conductor abelian surfaces over Q","Abelian surfaces over Q from Pryms of genus 3 double covers","500k small conductor abelian surfaces over Q from genus 3 Pryms"]},"model":"grok-4.3","cost_usd":0.012025,"raw_usage":{"total_tokens":5147,"prompt_tokens":459,"num_sources_used":0,"completion_tokens":71,"cost_in_usd_ticks":120249500,"prompt_tokens_details":{"text_tokens":459,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4617,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":459,"tokens_out":71,"duration_ms":25240,"temperature":1.0,"reasoning_tokens":4617,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T14:56:05.024704+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A direct computation on a sample of the constructed objects showing that many have conductors exceeding the small bound or exhibit bad reduction at primes outside the controlled set.","supporting_citations":[],"review_version":1}