{"id":"b9c3dc99-b772-4604-9b22-194aff2e0ca3","arxiv_id":"2411.12404","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under finiteness of the Tate-Shafarevich group and local Assumption 7.2, the normalized leading term at s=1 of the Hasse-Weil-Artin L-function of (A, chi) equals a product of a volume term, a ramification correction, a twisted regulator, and a characteristic ideal.","lead":"This paper proves an explicit Birch and Swinnerton-Dyer type formula for the leading term of Hasse-Weil-Artin L-functions attached to abelian varieties over global function fields, conditional on finiteness of Tate-Shafarevich groups and a set of local hypotheses. It is the function field analogue of a recent number field result of Burns and Macias Castillo, and the p-primary part requires a new equivariant Riemann-Roch theorem for weakly ramified covers.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Load-bearing dependency on unpublished [BKK]: the extension to maximal local subgroups is not checkable; if it fails, the p-part formula lacks support.","rationale":"I read the paper in good faith and found no internal mathematical error in the written argument as far as it can be checked without full formal verification. The reader's weakest-assumption analysis of Assumption 7.2(3) is accurate: the explicit p-part formula is only established when the chosen subbundle L = Lie(A_F)(-D_F) satisfies the local freeness condition (3.2a), and the paper itself explains in Remarks 6.7 and 7.17 that outside that setting one would need an inexplicit stable subbundle. That is a genuine scope limitation, but it is explicitly disclosed and does not by itself constitute a flaw in the theorem as stated. The more load-bearing risk is the reliance on the unpublished preprint [BKK] for Theorem 6.14, which is the bridge between the equivariant BSD theorem and the two Euler-characteristic terms used in Corollary 6.15. Since the proof of Theorem 6.14 contains a 'verbatim' extension claim that cannot be audited from the paper alone, this is the point at which the central claim is least secure. I nevertheless recommend no change to the reader's CONDITIONAL verdict: the paper is transparent about its hypotheses and dependencies, the internal computation from Corollary 6.15 onward is coherent, and no counterexample or concrete failure was identified.","tokens_in":1027,"tokens_out":1429,"duration_ms":157987,"concrete_test":"Obtain the [BKK] preprint and verify Proposition 3.7(i) and Theorem 4.9 for the choice U_v = A_F^0(p_v) under Assumption 7.2(3). Specifically, check that the Selmer complex is perfect and that the associated coherent bundle is literally L = Lie(A_F)(-D_F) rather than a proper subbundle; then trace the resulting formula through Corollary 6.15 and Proposition 7.10 to confirm that the BSD term is v_lambda(Reg^chi_lambda / |G|^{r_alg(chi)}) + length(X^vee_chi,lambda(A/F)). A mismatch in either step would invalidate Theorem 7.12.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is internally coherent once Theorem 6.14 is granted, but Theorem 6.14 is quoted from the unpublished preprint [BKK] (Theorem 4.9 and Proposition 5.6). In the proof of Theorem 6.14 the authors assert that [BKK]'s construction, originally made for carefully chosen subgroups U_v inside A_F^0(p_v), extends 'verbatim' to the maximal choice U_v = A_F^0(p_v) and yields the coherent term chi_p^G(Lie(A_F)(-D_F)). This extension is exactly what converts the equivariant BSD theorem into the two-term Euler-characteristic identity of Corollary 6.15, which is then used in Proposition 7.10 and Theorem 7.12. The paper does not reproduce the proof, so a reader cannot independently verify that the maximal choice of U_v is cohomologically trivial under Assumption 7.2(3), nor that no hidden boundary or torsion term appears when comparing the Selmer complexes. If [BKK]'s theorem is correct, the local computations in Sections 4 and 5 appear consistent; if the 'verbatim' extension fails, every p-primary formula in Section 7 would need correction. This is a load-bearing external dependency rather than an internal inconsistency, and the authors are transparent about it, but it is the single most serious risk to the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves an explicit BSD-type leading-term formula for Hasse-Weil-Artin L-functions attached to an abelian variety A over a global function field K of characteristic p>0 and an irreducible complex Galois representation chi, assuming finiteness of the relevant Tate-Shafarevich group and certain local hypotheses. The main result, Theorem 7.12 (with the simplified version Theorem 1.2), expresses the normalized leading term L_S(A,chi) at s=1 as a product of a volume term, a local ramification correction loc_{D/F}(A,chi), a chi-twisted regulator divided by |G|^{r_alg(chi)}, and a characteristic ideal of the chi-isotypic Tate-Shafarevich module. The proof combines an equivariant Riemann-Roch theorem for weakly ramified covers (Theorem 3.2), a detailed analysis of Lie algebras of Neron models under tame and weakly ramified base change (Section 5), and the equivariant BSD theorem of Burns-Kakde-Kim [BKK] quoted in Theorem 6.14. Section 8 gives examples, including Ulmer elliptic curves and an Artin-Schreier wild extension, where Assumption 7.2 holds for l=p.","tokens_in":69670,"tokens_out":7300,"duration_ms":79539,"significance":"If the result is correct, it is a substantial contribution: it provides the function-field analogue of Burns-Macias Castillo, extends the explicit leading-term formula to the p-primary part under natural hypotheses, and introduces a new local correction term loc_{D/F}(A,chi) that is explicit enough to compute in examples. The equivariant Riemann-Roch theorem of Section 3 generalizing Nakajima, Kock, and Fischbacher-Weitz-Kock is of independent interest. The paper is also honest about its hypotheses: it explicitly states that finiteness of X(A/F) remains open, that the explicit p-part relies on Assumption 7.2(3), and that without weak ramification only an inexplicit stable subbundle is available (Remarks 6.7 and 7.17). No parameters are fitted: the regulator, characteristic ideal, volume, and local terms are independently defined arithmetic objects. The principal weakness is the heavy reliance on the unpublished preprint [BKK], including an asserted 'verbatim' extension whose proof is not reproduced.","major_comments":[{"comment":"The proof of Theorem 6.14(2) states that a construction from the unpublished preprint [BKK], originally made for carefully chosen subgroups U_v inside A_F^0(p_v), extends 'verbatim' to the maximal choice U_v = A_F^0(p_v) and yields the coherent term chi_p^G(Lie(A_F)(-D_F)). This extension is load-bearing: it is exactly what converts the equivariant BSD theorem into Corollary 6.15, which is then used in Proposition 7.10 and Theorem 7.12. The manuscript does not reproduce the argument that the maximal choice is cohomologically trivial under Assumption 7.2(3), nor does it show that no hidden boundary or torsion term appears when comparing the Selmer complexes. Since [BKK] is an unpublished preprint, a reader cannot independently verify this step. If the 'verbatim' extension fails, every p-primary formula in Section 7 would need correction. I recommend that the authors either include a complete proof of this extension as a lemma or appendix, or explicitly reformulate the main theorem as conditional on a precisely stated theorem from [BKK] and make the relevant part of [BKK] available.","section":"Section 6, Theorem 6.14 and its proof"},{"comment":"The p-primary part of Theorem 7.12 inherits the unresolved dependency described above, because Corollary 6.15 for l=p is stated under the hypotheses of Proposition 6.6(2), whose proof relies on the same 'verbatim' extension and on [BKK, Proposition 3.7(i)]. The paper does provide a detailed proof that the explicit line bundle Lie(A_F)(-D_F) satisfies condition (3.2a) under Assumption 7.2(3) via Corollary 5.8(3), but the Selmer-complex side is quoted rather than proved. This means the central p-primary claim is currently not checkable from the manuscript alone, even assuming the rest of the paper is correct.","section":"Section 7, Theorem 7.12 and Corollary 6.15"},{"comment":"The examples are useful and do verify Assumption 7.2 in nontrivial settings, but they do not compute the new terms loc_{D/F}(A,chi) or the characteristic ideal in any concrete case, so no numerical or even explicit finite-field check of the main formula is presented. In particular, Example 8.6 contains a wild Artin-Schreier extension where the paper cannot verify finiteness of X(A/F); the authors state this, but as a consequence the example does not yield an unconditional application. A worked computation of loc in a tame cyclic or Artin-Schreier example would substantially increase confidence in the explicit local formula of Section 4.","section":"Section 8, Examples"}],"minor_comments":[{"comment":"The text refers to a 'Hochschield-Serre-type spectral sequence'; the standard spelling is Hochschild-Serre.","section":"Section 4, proof of Proposition 7.10"},{"comment":"There is a typo in the phrase 'a rational function field of characteristic of characterist ic p'; it should read 'of characteristic p'.","section":"Section 8, Example 8.6"},{"comment":"In equation (7.12a), the expression |Lie(A_F)(k_{\\tilde v})^{G_{\\tilde v}}| denotes the cardinality of a finite set; using absolute-value notation for cardinality is potentially confusing and should be explicitly defined.","section":"Section 7, Theorem 7.12"},{"comment":"The notation D_1, D_2, D and D_F is introduced only in the proof of Theorem 7.12; defining this decomposition earlier, near Proposition 5.5, would improve readability.","section":"Section 5, Proposition 5.5 and Corollary 5.8"},{"comment":"The paper relies on the unpublished preprint [BKK] for several central inputs, including Theorem 6.14 and Proposition 6.6(2). It would be helpful to include the precise statement number of the relevant results from [BKK] in each citation rather than only 'cf. [BKK, Theorem 4.9]' and 'cf. [BKK, Proposition 3.7(i)]', so that a reader can locate the exact assertions being used.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The main risk is the unpublished dependency: the central p-primary theorem rests on an assertion that a construction in [BKK] extends 'verbatim' to the maximal local subgroup, and this assertion is not checkable from the manuscript. I would not reject the paper on this basis alone, because the argument is otherwise coherent and the authors are transparent about the dependency. However, for a serious journal, either the proof of this extension should be included, or the main theorem should be stated as conditional on a precisely specified theorem from [BKK]. I also suggest asking the authors to make the relevant parts of [BKK] publicly available or to add a short appendix containing the extension proof. The paper fits the journal's scope and, if the dependency is resolved, would be a strong contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this is a serious conditional theorem, not a survey or a conjecture. The genuinely new pieces are the equivariant Riemann–Roch theorem for weakly ramified covers (Theorem 3.2) and the explicit local correction term loc_{D/F}(A,chi). The formula itself is the function-field analogue of Burns–Macias Castillo, and it upgrades the [BKK] equivariant BSD result from an equality up to torsion to an explicit leading-term identity, under Assumption 7.2. That upgrade is real work: the proof of Theorem 3.2, reducing to Brauer characters on cyclic subgroups, is a cleaner argument than Köck's original, and the local Néron-model analysis in Section 5 that makes Lie(A_F)(-D_F) satisfy (3.2a) is careful and convincing.\n\nThe soft spots are mostly where the paper is transparent. The single biggest risk is the dependency on [BKK, Theorem 4.9] (Theorem 6.14 here). The authors assert that the construction in [BKK], made for carefully chosen proper subgroups U_v inside A_F^0(p_v), extends 'verbatim' to the maximal choice U_v = A_F^0(p_v), and that this gives the coherent term chi_p^G(Lie(A_F)(-D_F)). That extension is exactly what converts the up-to-torsion theorem into the two-term Euler characteristic identity used in Proposition 7.10 and Theorem 7.12. The paper does not reproduce the proof, so a reader cannot independently verify the maximal choice is cohomologically trivial under Assumption 7.2(3), nor that no boundary term appears. If the 'verbatim' claim fails, the p-primary formulas in Section 7 would need correction. This is an external dependency, not an internal contradiction; the authors flag it in Remarks 6.7 and 7.17. But it is the first thing a referee should check.\n\nThe other softness is the usual one: finiteness of X(A/F). The paper is honest about it, and Section 8 gives examples where it is known (constant abelian varieties, Ulmer curves) and one wild-ramification example (Artin–Schreier, Example 8.6) where finiteness is not verified. That is a limitation, not a flaw.\n\nOverall: the math is coherent, the local computations seem consistent, and the examples are non-trivial. This deserves a serious referee. I would send it to a strong journal and ask the referee to scrutinize the [BKK] dependency; ideally the authors should include a precise statement of the needed [BKK] theorem, or at least make the preprint available for verification. Recommend: engage, conditional on the [BKK] input checking out.","headline":"A serious conditional p-adic BSD formula for Hasse–Weil–Artin L-functions over function fields; the new equivariant Riemann–Roch and explicit local terms are real, but the p-part rests on an unpublished [BKK] input whose 'verbatim' extension needs scrutiny.","tokens_in":70198,"tokens_out":2779,"would_cite":true,"duration_ms":27588,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G40","14G10","11J95","14C40"],"pacs":[],"model":"deepseek-v4-flash","headline":"For abelian varieties over characteristic-p function fields, the leading term of the Hasse–Weil–Artin L-function is computed by an explicit BSD-type formula, conditional on finiteness of the Tate–Shafarevich group and a local ramification…","keywords":["Birch and Swinnerton-Dyer conjecture","Hasse-Weil-Artin L-functions","global function fields","equivariant Riemann-Roch theorem","weakly ramified covers","Tate-Shafarevich group","Néron models","characteristic p"],"falsifier":"Compute both sides of Theorem 7.12 for a constant supersingular abelian variety over $\\mathbb{F}_q(t)$ with an Artin–Schreier $p$-extension $F/K$: the left side is obtained from the Grothendieck–Lefschetz trace formula for the $L$-function, and the right side from the Néron-model fibre data and the regulator; any mismatch in the $p$-adic valuation at the place above $p$ would disprove the formula, and agreement for many such pairs would confirm the $p$-part computation.","tokens_in":69235,"feed_emoji":"🧮","tokens_out":11512,"duration_ms":110864,"temperature":0.7,"pith_summary":"The paper establishes a Birch–Swinnerton-Dyer-type formula for the leading term at $s=1$ of the Hasse–Weil–Artin $L$-function attached to an abelian variety $A$ over a global function field $K$ of characteristic $p>0$ and an irreducible complex Galois representation $\\chi$. The formula expresses the $\\lambda$-adic valuation of the normalized leading term as a sum of explicit arithmetic terms: a volume factor, a local ramification correction, the twisted regulator, and the length of a twisted Tate–Shafarevich module, with the $p$-part included when a local weak-ramification hypothesis holds. A sympathetic reader should care because this makes the equivariant refinement of BSD over function fields explicit enough to compute individual leading terms, not merely classes up to torsion, and it covers the troublesome $p$-part even for weakly wildly ramified characters. The main new engine is an equivariant Riemann–Roch theorem for weakly ramified covers of curves over finite fields.","feed_headline":"Explicit BSD formula pins down Hasse-Weil-Artin L-values","feed_subtitle":"Under finiteness of Tate-Shafarevich, leading terms become products of volumes, regulators, and characteristic ideals.","key_machinery":"The central engine is an equivariant Riemann–Roch theorem for weakly ramified covers of curves over a perfect field of characteristic $p$ (Theorem 3.2). It computes the class $[\\mathrm{R}\\Gamma(C_F,E)] \\in K_0(k[G])$ of the cohomology of a $G$-equivariant vector bundle $E$ on $C_F$, under the local condition (3.2a) that at wildly ramified points the completed stalk is a sum of fractional ideals $\\mathfrak{p}_v^{-b_{v,i}}$ with $b_{v,i} \\equiv -1 \\bmod |P_v|$; the formula is written in terms of induced modules $\\operatorname{Ind}_{G_v}^G(P_v(j))$ attached to the inertia action. The other load-bearing piece is Köck's local integral normal basis theorem (Theorem 2.11), which guarantees freeness of those fractional ideals under weak ramification. Applied to $E=\\operatorname{Lie}(A_F)(-D_F)$, the theorem converts the incoherent-cohomology correction term of the equivariant BSD conjecture into an explicit local ramification sum, yielding the loc factor.","core_discovery":"On the paper's own terms, the central discovery is Theorem 7.12: for an abelian variety $A$ over a global function field $K$ of characteristic $p>0$, a finite Galois extension $F/K$ with group $G$, and an irreducible complex character $\\chi$ of $G$ defined over a number field $E$, if the Tate–Shafarevich group $\\Sha(A/F)$ is finite and the local Assumption 7.2 holds, then for any place $\\lambda$ of $E$ above a prime $\\ell$ the normalized leading term $\\mathcal{L}_S(A,\\chi)$ satisfies the equality of fractional ideals $\\mathcal{L}_S(A,\\chi) O_{E,\\lambda} = (\\operatorname{vol}_{D_1}(A/K) \\prod_{v \\in D_2} |\\operatorname{Lie}(A_F)(k_{\\tilde v})^{G_{\\tilde v}}|)^{\\deg \\chi} \\cdot \\operatorname{loc}_{D/F}(A,\\chi) \\cdot \\frac{\\operatorname{Reg}^\\chi_\\lambda}{|G|^{r_{\\mathrm{alg}}(\\chi)}} \\cdot \\operatorname{Char}_\\lambda(X^\\vee_{\\chi,\\lambda}(A/F))$. The paper's specific contribution is to make this explicit at $\\ell=p$ by showing that the equivariant vector bundle $L=\\operatorname{Lie}(A_F)(-D_F)$ satisfies the local freeness condition (3.2a) whenever $F/K$ is weakly ramified everywhere and tame at places of non-semistable reduction, so that the new equivariant Riemann–Roch theorem computes the coherent-cohomology factor directly.","pith_inferences":["The paper's Assumption 7.2(3) is probably not optimal: if a wild place of $F/K$ meets a non-semistable fibre of $A$, the stable-subbundle construction mentioned in Remarks 6.7 and 7.17 suggests the correct $p$-part correction should be a local term attached to that subbundle; finding an explicit formula for it would extend the theorem verbatim.","Theorem 3.2 is a standalone equivariant Riemann–Roch theorem: any equivariant vector bundle satisfying the stalk condition (3.2a) has its Euler characteristic in $K_0(\\mathbb{F}_p[G])$ computed by the displayed ramification formula, so the theorem should apply to other arithmetic bundles (e.g., sheaves of differentials or structure sheaves twisted by ramified divisors), not just Lie algebras of Né","Because Assumption 7.2 holds for all but finitely many primes, the theorem gives a complete proof of the leading-term BSD formula for every irreducible character away from a finite set; the only obstruction to a full theorem is the $p$-part, so the paper reduces the problem to understanding one local phenomenon.","The explicit $\\operatorname{loc}$ term could be tested numerically: for cyclic tame extensions it is a sum over ramified places weighted by the character's inertia type, so one can compare it with the ratio between the $L$-value and the regulator in explicit Ulmer-curve examples."],"forward_implications":["For every prime $\\ell$ coprime to $|G|$, the $\\lambda$-adic valuation of the normalized leading term is given by the explicit formula (7.13b), involving only volumes, the local correction, the twisted regulator, and characteristic ideals of the twisted Tate–Shafarevich group, with no torsion ambiguity.","When the character is trivial and $F=K$, the formula specializes to the $\\ell$-part of the classical BSD formula over function fields, so the result is compatible with the known full BSD theorem.","The $p$-part of the leading term is computed for weakly wildly ramified characters, provided $A$ has semistable reduction everywhere in $F$, $F/K$ is weakly ramified, and $F/K$ is tame at the places where $A$ does not have semistable reduction.","The local ramification correction $\\operatorname{loc}_{D/F}(A,\\chi)$ is $1$ for $p$-extensions and unramified covers, and has a simple logarithmic expression in tame cyclic cases, giving a new explicit description even in the cyclic tame setting.","The theorem applies unconditionally (with $\\Sha(A/F)$ known finite) to constant supersingular abelian varieties, to Ulmer elliptic curves with $p\\equiv 2 \\bmod 3$, and to Artin–Schreier extensions made from them; in these cases the $p$-part formula is fully explicit."],"supporting_citations":[{"why":"Supplies the equivariant refinement of BSD over function fields and the torsion statement (Theorem 6.14) whose Selmer and coherent terms the paper computes explicitly.","marker":"[BKK]"},{"why":"Number-field analogue whose proof strategy (twisted regulators and Selmer-complex computation) the paper adapts to function fields.","marker":"[BMC24]"},{"why":"Local integral normal basis theorem for weakly ramified extensions, the key input for the explicit subbundle and for Theorem 2.11.","marker":"[Köc04]"},{"why":"Tame equivariant Riemann–Roch theorem that the paper generalizes to weakly ramified covers; provides the Brauer-character base case.","marker":"[Nak86]"},{"why":"Rank-1 equivariant Riemann–Roch theorem for perfect fields, generalized here to arbitrary rank and an explicit $K_0$-formula.","marker":"[FWK09]"},{"why":"Classical BSD theorem in characteristic $p$ that supplies the volume normalization, Selmer-complex construction, and the compatibility check with the trivial-character case.","marker":"[KT03]"},{"why":"Chinburg's local Galois-structure results used in Section 2 to characterize freeness and cohomological triviality of semilinear lattices.","marker":"[Chi94]"},{"why":"Néron-model base-change facts used in Section 5 to relate $\\operatorname{Lie}(A)$ and $\\operatorname{Lie}(A_F)$ and to define $D_2$.","marker":"[BLR90]"},{"why":"Néron models under tame ramification; used to identify $A$ with the invariants of the Weil restriction in Proposition 5.3.","marker":"[Edi92]"},{"why":"Ulmer elliptic curves provide examples with finite Tate–Shafarevich and large rank where the $p$-part formula applies unconditionally.","marker":"[Ulm02]"}],"fun_headline_variants":["Function-field BSD formula for Hasse-Weil-Artin L-functions","Explicit p-part of BSD leading term in char p","Equivariant Riemann-Roch powers new BSD proof","Tate-Shafarevich finite? Then L-value is explicit","Hasse-Weil-Artin L-values pinned down in char p"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the Tate–Shafarevich group $\\Sha(A/F)$ is finite and that, at the characteristic prime, the extension $F/K$ is weakly ramified everywhere, semistable at places where wild ramification occurs, and tame at places where the abelian variety has bad reduction; without these local conditions the explicit $p$-part formula is not proved.","fun_headline_variants_meta":{"raw":{"variants":["Function-field BSD formula for Hasse-Weil-Artin L-functions","Explicit p-part of BSD leading term in char p","Equivariant Riemann-Roch powers new BSD proof","Tate-Shafarevich finite? Then L-value is explicit","Hasse-Weil-Artin L-values pinned down in char p"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000207,"raw_usage":{"total_tokens":1491,"prompt_tokens":1129,"completion_tokens":362,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":745,"completion_tokens_details":{"reasoning_tokens":273}},"tokens_in":745,"tokens_out":362,"duration_ms":4167,"temperature":1.0,"reasoning_tokens":273,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T17:34:15.350066+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute both sides of Theorem 7.12 for a constant supersingular abelian variety over $\\mathbb{F}_q(t)$ with an Artin–Schreier $p$-extension $F/K$: the left side is obtained from the Grothendieck–Lefschetz trace formula for the $L$-function, and the right side from the Néron-model fibre data and the regulator; any mismatch in the $p$-adic valuation at the place above $p$ would disprove the formula, and agreement for many such pairs would confirm the $p$-part computation.","supporting_citations":[],"review_version":1}