{"id":"fd406468-9ce4-4158-a283-fee0e241906f","arxiv_id":"2508.12329","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":4.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper claims that reduction type, Tamagawa number, the BSD fudge factor, and the Galois representation of a curve are locally constant under p-adically small perturbations of the curve's defining equations.","lead":"Jakab Schrettner's paper claims that for curves over a p-adic field, sufficiently small perturbations of the coefficients of the defining equations leave the reduction type, the Tamagawa number, the Birch and Swinnerton-Dyer fudge factor, and the Galois representation unchanged.","discovery_kind":"unclear","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Local constancy of the full Galois representation is suspect: p-adically convergent coefficient perturbations with identical reduction can give non-isomorphic p-adic Tate modules, so the abstract's claim is ambiguous and, on the natural reading, false.","rationale":"The reader's weakest assumption concerned the existence of a uniform proper model for a 'family' defined by coefficient perturbation. That is a real under-specification. My stress-test identifies a sharper, more specific issue: even granting a model and fixing the reduction type, the Galois representation cannot be locally constant in the full sense. Tate's isogeny theorem shows that isomorphic p-adic Tate modules imply isogenous curves, and p-adically small perturbations can escape the isogeny class while preserving the reduction mod p. The explicit family E_7^n is a natural place to detect this. If the full text defines 'Galois representation' as the semisimplification or the residual representation, the abstract's wording should be corrected before acceptance; if it means the full representation, the central claim is false. Because the supplied material is only the abstract, I cannot fully confirm the intended definition, but the concern is load-bearing and warrants a conditional verdict: the paper should be accepted only after the terminology is clarified and the corresponding statement is proved. The reader's abstract-only UNVERDICTED status is reasonable, but the concrete red flag justifies moving to CONDITIONAL rather than leaving the mathematical claim unexamined.","tokens_in":825,"tokens_out":12413,"duration_ms":148881,"concrete_test":"Compute the 7-adic Galois representations V_7(E_n) for E_n: y^2 = x^3 + 7^n x + 1 over Q_7, n = 0,...,8, using the filtered φ-module (D = H^1_cris of the common reduction y^2 = x^3 + 1, same Frobenius, Hodge filtration line generated by the invariant differential) and compare them up to isomorphism. If the representations for n = 1 and n = 2 are non-isomorphic, local constancy of the full Galois representation fails in the ball |a|_7 < 1. Also inspect the full paper's definition of 'Galois representation'; if it is the semisimplified or residual representation, the same computation is consistent and the concern reduces to terminology.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The abstract asserts local constancy for 'the Galois representation' in the valuation topology on the coefficient space. Under the standard meaning (the p-adic or ℓ-adic étale cohomology representation of Gal(\\bar K/K)), this is not true: the representation determines the Q_p-isogeny class by Tate's isogeny theorem, and p-adically close coefficient values can belong to distinct isogeny classes. For example, E_n: y^2 = x^3 + 7^n x + 1 over Q_7 all have the same smooth ordinary reduction mod 7, but their j-invariants are distinct and their filtered (φ,N)-modules—equivalently, the isomorphism classes of V_7(E_n)—vary with n. Since 7^n → 0 in the 7-adic topology, every neighbourhood of the coefficient point 0 contains infinitely many non-isomorphic V_7's. The claim can therefore hold only for a cruder invariant such as the residual mod-7 representation or the semisimplification. The paper needs to state and prove which object is meant; as written, the headline assertion is ambiguous and, on the natural reading, incorrect.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript, as submitted, consists of an abstract alone. It claims that for a family of curves over a discretely valued field obtained by perturbing the coefficients of defining equations, the reduction type is locally constant in the valuation topology, and that analogous local constancy holds for the Tamagawa number, the Birch and Swinnerton-Dyer fudge factor, and the Galois representation. The document contains no definitions, hypotheses, theorem statements, or proofs.","tokens_in":966,"tokens_out":11944,"duration_ms":125246,"significance":"If the intended statement were made precise and proved, the result would be a useful complement to the theory of reduction in p-adic families, with potential applications to the arithmetic of curves in families. However, the submitted text provides no evidence for these claims, and the Galois representation assertion is ambiguous and, on its most natural reading, contradicted by standard examples. The contribution cannot be evaluated from the present text.","major_comments":[{"comment":"The assertion that 'the Galois representation' is locally constant is ambiguous and, under the usual meaning (the rational p-adic or ℓ-adic étale cohomology representation), false. For example, the elliptic curves E_n: y^2 = x^3 + 7^n x + 1 over Q_7 all have good ordinary reduction modulo 7 (they reduce to the same curve y^2 = x^3 + 1), yet their p-adic Tate modules V_7(E_n) are not generally isomorphic because their filtered (φ,N)-modules have extension data that depends on the j-invariant, which varies in every 7-adic neighbourhood of the family. A local constancy theorem cannot hold for the full representation; the manuscript must state precisely which invariant (e.g., the semisimplified mod-p representation or the Newton polygon) is claimed to be locally constant, and it must prove that weaker statement.","section":"Abstract"},{"comment":"The paper defines a 'family' only as 'a set of curves obtained by perturbing the coefficients of the defining equations'. This does not specify the parameter space, the base ring, the choice of model over the base, or the smoothness/properness hypotheses. In particular, the abstract does not say that the family avoids the discriminant locus or that a simultaneous regular (or semistable) model exists. Without such hypotheses, local constancy of the reduction type is not guaranteed. The full statement must include a precise base scheme and a proper flat model over it whose generic fiber is the family.","section":"Abstract"},{"comment":"The submission contains no body: there are no formal theorem statements, no proofs, no definitions of 'reduction type' or the 'fudge factor'. The abstract's promises ('we will show', 'we also derive') are not backed by verifiable mathematics. This is not a refereable paper in its current state; the authors need to submit a complete manuscript with precise statements and full arguments.","section":"Entire manuscript"}],"minor_comments":[{"comment":"The term 'Birch and Swinnerton-Dyer fudge factor' is informal and should be defined; presumably it denotes the leading coefficient of the L-function at s=1 (the product of period, Tamagawa number, regulator, and torsion factors), but the paper should say so explicitly.","section":"Abstract"},{"comment":"The phrase 'topology induced by the valuation' should be clarified: for each coefficient vector, the paper should indicate whether the neighborhood is a p-adic ball in the coefficient space and how the curves are parametrized by the coefficients.","section":"Abstract"},{"comment":"The abstract gives no references to prior results on constancy of reduction or Néron models in p-adic families; for instance, the constancy of the Euler characteristic and the existence of Néron models in proper flat families (e.g., Deligne's work) should be cited to frame the claimed result.","section":"References"}],"recommendation":"reject","confidential_remarks":"The submission appears to be an abstract-only placeholder rather than a complete paper. In addition to the lack of body, the headline claim about the Galois representation is likely false in its natural interpretation; if a full manuscript is ever submitted, that claim will need to be replaced by a statement about a suitable quotient or semisimplification, with a proof. I recommend rejecting the current submission and inviting the authors to submit a full, corrected paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know right away. First, this is an abstract-only submission; the full text in the material I have is empty, so every judgment below is about the abstract. Second, the abstract makes a claim about local constancy of the Galois representation that, under the standard arithmetic-geometric meaning, is false. Quick counterexample: over Q_7, the curves E_n: y^2 = x^3 + 7^n x + 1 all have the same smooth ordinary reduction mod 7 (the curve y^2 = x^3 + 1), but their p-adic Tate modules are not all isomorphic: the j-invariant varies with n, and the filtered (φ,N)-modules vary accordingly. Since 7^n → 0 in the 7-adic topology, every neighborhood of 0 contains non-isomorphic V_7's. So the assertion 'the Galois representation is locally constant' cannot be right as written.\n\nWhat might be right: local constancy of reduction type in p-adic families is a known phenomenon, and a proof for curves under explicit hypotheses on a simultaneous regular/minimal model would be a useful write-up. Adding Tamagawa numbers and the BSD fudge factor to the package is a nice organizing idea. If the full text does carry out that proof, the paper could be a helpful toolbox for deformation arguments and BSD computations.\n\nThe soft spots are not minor. The abstract does not define 'family' precisely: does the perturbed set admit one proper model over the base ring with flat special fiber? Without that stipulation, even local constancy of reduction type is not automatic and can fail. The novelty section is absent; the author should state explicitly what is new relative to known results for abelian varieties. And of course, the Galois-representation claim needs either a precise definition of which representation (mod p? semisimplification? a fixed residual representation?) or a withdrawal of that claim.\n\nI cannot check any proof step because no proof is before me. The paper might be salvageable, but as submitted the headline theorem is incorrect or meaningless under the standard reading.\n\nMy recommendation: do not send this to a referee yet. Ask the author to revise the abstract and supply the full text; then reconsider.","headline":"The abstract's Galois-representation claim is false on the standard reading; the rest of the paper is unverifiable from the abstract alone.","tokens_in":1550,"tokens_out":8633,"would_cite":false,"duration_ms":97749,"reading_group":"no","serious_thinker":"unclear","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14G20","14H25","11S15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Small p-adic changes leave a curve's reduction type fixed.","keywords":["reduction type","Tamagawa number","Birch-Swinnerton-Dyer fudge factor","Galois representation","p-adic families of curves","discretely valued field","local constancy","valuation topology"],"falsifier":"One could compute the reduction type and Tamagawa number for a sequence of hyperelliptic curves over $\\mathbb{Q}_p$ whose coefficient vectors converge $p$-adically to a limit; if any invariant changes infinitely often along the sequence, the local constancy claim is false.","tokens_in":535,"feed_emoji":"","tokens_out":6447,"duration_ms":61228,"temperature":0.7,"pith_summary":"This paper claims that when a curve over a discretely valued field is varied by perturbing the coefficients of its defining equations, the curve's reduction type is locally constant in the topology induced by the valuation. The same local constancy is claimed for the Tamagawa number, the Birch and Swinnerton-Dyer 'fudge factor', and the Galois representation attached to the curve. In plain terms, sufficiently small $p$-adic changes to the coefficients leave these arithmetic invariants unchanged on an open neighborhood of the coefficient space. A sympathetic reader would care because these invariants control the arithmetic of the curve, and local constancy makes them computable from a single point in a ball.","feed_headline":"Small p-adic changes leave a curve's reduction type fixed","feed_subtitle":"Sufficiently close p-adic perturbations change none of the curve's reduction invariants.","key_machinery":"The central object is a family of curves over a discretely valued field, defined as the set of curves obtained by perturbing the coefficients of the defining equations. The mechanism carrying the argument is the valuation topology on the coefficient space: the paper shows that under this topology the reduction type, and with it the Tamagawa number, the BSD fudge factor, and the Galois representation, are locally constant, meaning each invariant takes a single value on a sufficiently small open ball around any coefficient vector.","core_discovery":"The central discovery is that reduction type, Tamagawa number, the Birch and Swinnerton-Dyer fudge factor, and the Galois representation are all locally constant functions on the coefficient space of a family of curves defined over a discretely valued field, where the family is taken to be a set of curves obtained by perturbing the defining equations. The author states this as a theorem: within the topology induced by the valuation, every curve has a neighborhood on which these invariants do not change. The reduction type is the key invariant, and the other invariants are shown to share the same local constancy.","pith_inferences":["A testable extension would be to compute, for explicit families of hyperelliptic curves over $\\mathbb{Q}_p$, the radius of the largest ball on which the reduction type is constant; the paper does not give such radii.","If local constancy holds for curves, the same perturbation argument may apply to other arithmetic objects such as abelian varieties or higher-dimensional varieties, where reduction-type invariants might also be locally constant; this is an inference, not a claim of the paper.","The result suggests an algorithmic shortcut: instead of computing reduction invariants for a continuum of curves, one can compute them at a single representative coefficient vector in each sufficiently small ball."],"forward_implications":["If the result is correct, the reduction type of a curve over a discretely valued field is an invariant of its $p$-adic neighborhood, so any two curves whose defining equations are sufficiently close in the valuation topology share the same reduction type.","The Tamagawa number, which records the component structure of the special fiber, is locally constant, so arithmetic formulas that sum Tamagawa numbers over primes are stable under small coefficient changes.","The Birch and Swinnerton-Dyer fudge factor is locally constant, meaning the local contribution to the BSD conjecture remains unchanged under small $p$-adic perturbations of the curve.","The Galois representation attached to a curve is locally constant, so the action of the absolute Galois group on its étale cohomology is unchanged on a $p$-adic neighborhood of the coefficients."],"supporting_citations":[],"fun_headline_variants":["Local p-adic stability for curve reduction invariants","p-adic perturbations: reduction type fixed in neighborhoods","Curve invariants constant under sufficiently small p-adic changes","Reduction type locally constant across p-adic family","Nearby p-adic curves have identical reduction invariants"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The result rests on the assumption that for every sufficiently small perturbation, all the curves in the family admit one common proper model over the base ring, so that a single special fiber exists from which the reduction invariants can be read; if that common model does not exist, local constancy could fail.","fun_headline_variants_meta":{"raw":{"variants":["Local p-adic stability for curve reduction invariants","p-adic perturbations: reduction type fixed in neighborhoods","Curve invariants constant under sufficiently small p-adic changes","Reduction type locally constant across p-adic family","Nearby p-adic curves have identical reduction invariants"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000418,"raw_usage":{"total_tokens":2051,"prompt_tokens":742,"completion_tokens":1309,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":358,"completion_tokens_details":{"reasoning_tokens":1230}},"tokens_in":358,"tokens_out":1309,"duration_ms":12797,"temperature":1.0,"reasoning_tokens":1230,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:23:15.710179+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One could compute the reduction type and Tamagawa number for a sequence of hyperelliptic curves over $\\mathbb{Q}_p$ whose coefficient vectors converge $p$-adically to a limit; if any invariant changes infinitely often along the sequence, the local constancy claim is false.","supporting_citations":[],"review_version":2}