{"id":"ed7fdef2-e833-4bd7-9177-6dbd1a4c4943","arxiv_id":"2603.10576","paper_version":3,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A p-adic L-function interpolating twisted Hasse-Weil values is defined for ordinary elliptic curves over function fields and shown to equal the dual Selmer characteristic ideal in multiple Iwasawa main conjecture cases.","lead":"The paper constructs a p-adic L-function for ordinary elliptic curves over global function fields that interpolates twisted Hasse-Weil special values. It proves this matches the characteristic ideal of the dual Selmer group (Iwasawa main conjecture) in several cases and reduces higher-rank cases to rank-2 via Grassmannians.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The paper's central claim is the construction of an analytic p-adic L-function that interpolates twisted Hasse-Weil values and matches the characteristic ideal of the dual Selmer group in several cases, together with a clean Grassmannian reduction for higher-rank extensions. The ordinary-reduction hypothesis is indispensable for the local Euler factors and the local cohomology modules that produce the correction terms; it is stated from the first paragraph and used uniformly. The proofs of interpolation (Lemma 4.1.1), functional equation (Prop. 4.2.3) and specialization (Prop. 4.3.1) are direct once the local factors are granted, and the main-conjecture cases cite established results. The group-ring arguments of §6 are independent of the arithmetic and appear correct. The single concrete check proposed above simply reconfirms the most technical local calculation; a mismatch would be a genuine gap, but the text already supplies a detailed proof. No adjustment to the reader's ACCEPT verdict is warranted.","tokens_in":43780,"tokens_out":592,"duration_ms":4930,"concrete_test":"Independently recompute the local factor ϑ_{L/L',v} for a non-split multiplicative place with Ψ≃Z_p and Γ'_v=0 (the case treated in Lemma 2.2.4) by direct calculation of H^1(Ψ_w,A(L_w)) via the Tate curve twist; verify that CH_Λ'(W^1_v)=(ϑ_{L/L',v}) matches Definition 2.1.4(d) and that W^2_v=0. If the characteristic ideal differs, the specialization formula (10) fails for that place.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest assumption (ordinary reduction at all ramified places) is the natural and necessary hypothesis of the whole subject; the local cohomology calculations of §2.2 and the definitions of ϑ_{L/L'} (Def. 2.1.4) and †_{A/L} (Def. 3.3.1) are carefully extended to non-split multiplicative places and are used consistently in the specialization formulae (Prop. 2.1.5, Prop. 4.3.1). The analytic construction via Mazur theta elements (Thm. 3.1.1, §3.2) and the group-ring reduction for d≥3 (Props. 6.1.1, 6.4.1) are self-contained once those local factors are accepted. The main-conjecture cases rest on solid prior results (BSD, constant-curve IMC, constant-field IMC). No internal inconsistency or hidden circularity appears in the central claims.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper constructs a p-adic L-function L_{A/L} in Q_p · Λ for an ordinary elliptic curve A over a global function field K of characteristic p and a Z_p^d-extension L/K (d ≥ 0 allowed) unramified outside ordinary places. The function is built from Mazur’s theta elements and Gauss sums, interpolates twisted Hasse–Weil special values (Lemma 4.1.1), and is shown to satisfy a functional equation (Prop. 4.2.3) and specialization formulae (Prop. 4.3.1) that match those already known for the characteristic ideal of the dual p^∞-Selmer group X_L. The Iwasawa main conjecture (L_{A/L} generates CH_Λ(X_L)) is proved when L = K, when A is constant, when A is semistable and L is the constant-field Z_p-extension, and when X_L is non-torsion; for d ≥ 3 the conjecture for L is equivalent to the conjecture for all intermediate Z_p^{2}-extensions in a nonempty Zariski-open subset of the Grassmannian. A modified form of the conjecture is treated for e = 1.","tokens_in":44036,"tokens_out":831,"duration_ms":15744,"significance":"The work supplies a complete analytic side for the Iwasawa main conjecture in the ordinary function-field setting and systematically extends the local correction factors ϑ_{L/L'} and †_{A/L} to non-split multiplicative places. The group-ring reduction (Props. 6.1.1, 6.4.1) that reduces the d ≥ 3 case to a Zariski-open set of Z_p^{2}-extensions is of independent interest and may apply elsewhere. The results rest on a careful matching of interpolation formulae with previously established algebraic specialization and restriction maps, and they recover or strengthen several earlier theorems (BSD, constant-curve IMC, constant-field IMC). The construction is independent of the Selmer group, so there is no circularity.","major_comments":[],"minor_comments":[{"comment":"After Proposition 1.1.1 the text reads “propersition”; correct to “proposition”.","section":null},{"comment":"In the proof of Proposition 3.3.4 the symbol “tildeL” appears; replace by the consistent notation ˜L.","section":null},{"comment":"Section 5.1.1, product formula (55): the phrase “with multiplicity p. in of both sides” is ungrammatical; rephrase for clarity.","section":null},{"comment":"Definition 2.1.4(c) and the accompanying footnote correct an earlier erratum; a brief parenthetical remark that the present definition supersedes Tan14, Def. 1.3(c) would help the reader.","section":null},{"comment":"The density of notation in §§2–3 (especially the many local factors ϱ, ϑ, †, ∇, t) would be eased by a short “notation summary” table or paragraph at the end of the introduction.","section":null},{"comment":"In Lemma 4.2.2 the comparison with the Stickelberger element of LLTT16a is stated only for L ≠ K; a one-line remark on the L = K case would complete the picture.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is long and technical but the central arguments are carefully written and rest on solid prior work of the author and collaborators. The ordinary-reduction hypothesis is the natural one for the subject and is used consistently. I see no hidden circularity or load-bearing gap. The paper is a natural fit for a strong number-theory journal."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper builds the missing analytic object for the Iwasawa main conjecture of ordinary elliptic curves over global function fields of characteristic p. The new p-adic L-function L_{A/L} is assembled from Mazur theta elements, corrected by local factors (†, ∇, ϱ, ϑ) so that its interpolation, functional equation, and specialization formulae match the algebraic characteristic ideal of the dual Selmer group already known from Tan14 and LLTT. That matching is the real contribution.\n\nWhat works well is the care with the local corrections. The author extends the earlier specialization formulae to non-split multiplicative places (Lemmas 2.1.9–2.2.4) and proves the restriction formula (Prop. 5.1.2) that lets valuations of characters control μ-invariants. The Grassmannian reduction for d ≥ 3 (Props. 6.1.1, 6.4.1) is a clean group-ring argument of independent interest: the main conjecture for a Z_p^d-extension holds if and only if it holds for all intermediate Z_p^{2}-extensions in a non-empty Zariski-open set. The modified conjecture for e=1 is a sensible fix for the counter-examples that appear when one projects to rank-1 subextensions. The cases where IMC is proved (L=K, constant curves, constant-field Z_p-extension under semi-stable reduction, non-torsion Selmer) rest on solid prior results (BSD, LLTT16a,b).\n\nThe soft spot is the ordinary-reduction hypothesis itself: L/K must be unramified outside places of good ordinary or multiplicative reduction. That is the natural setting of the whole subject, not a hidden gap, but it means the construction does not yet cover additive places. The recent preprint ttt26 is cited for the semi-stable case under a μ-hypothesis; the present paper is independent of that and supplies the analytic side cleanly.\n\nThis is for people working on Iwasawa theory or BSD over function fields. The math is standard technical arithmetic geometry, the citations are appropriate, and there is no circularity: the analytic object is built from theta elements and Gauss sums, then matched to the algebraic side. I would send it to referees without hesitation.","headline":"Solid construction of the analytic side of IMC for ordinary elliptic curves over function fields, with a clean Grassmannian reduction for higher-rank towers.","tokens_in":44641,"tokens_out":591,"would_cite":true,"duration_ms":6670,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11R23","11G40","11F67"],"pacs":[],"model":"grok-4.5","headline":"A p-adic L-function for ordinary elliptic curves over function fields interpolates Hasse-Weil values and equals the dual Selmer characteristic ideal in several Iwasawa main-conjecture cases.","keywords":["p-adic L-functions","Iwasawa main conjecture","elliptic curves","global function fields","Selmer groups","characteristic ideals","specialization formulae","ordinary reduction"],"falsifier":"An explicit ordinary elliptic curve over a function field together with a concrete Z_p-extension unramified only at ordinary places for which the constructed L_{A/L} fails to lie in the Iwasawa algebra or fails to generate the characteristic ideal of the dual Selmer group.","tokens_in":44649,"feed_emoji":"∞","tokens_out":989,"duration_ms":17503,"temperature":0.7,"pith_summary":"The paper constructs a p-adic L-function attached to an ordinary elliptic curve over a global function field of characteristic p and a Z_p^d-extension unramified outside ordinary places. This function is defined by interpolating special values of twisted Hasse-Weil L-functions and is shown to obey the functional equation and specialization formulae that match those of the characteristic ideal of the dual p-infinity Selmer group. The Iwasawa main conjecture equating the two sides is proved when the extension is trivial, when the curve is constant, when the extension is the constant-field Z_p-extension under semistable reduction, and when the Selmer group is non-torsion. For rank at least three the conjecture over the full tower is equivalent to the same conjecture over all intermediate rank-two extensions lying in a nonempty Zariski-open subset of the corresponding Grassmannian.","feed_headline":"p-adic L-functions match Selmer ideals for function-field curves","feed_subtitle":"They interpolate Hasse-Weil values and prove the Iwasawa main conjecture for constant curves and constant-field towers.","key_machinery":"The p-adic L-function L_{A/L}, obtained by assembling Mazur theta elements into an inverse-limit element in the completed group ring of the ray-class tower and then correcting by the explicit factors t_{A/L}, nabla_{A/L} and dagger_{A/L}; it supplies both the analytic interpolation and the matching specialization maps used to compare characteristic ideals.","core_discovery":"An element L_{A/L} in the rationalized Iwasawa algebra is built so that it interpolates the special values of twisted Hasse-Weil L-functions of an ordinary elliptic curve A over a function field K relative to any Z_p^d-extension L/K unramified outside ordinary places; after adjustment by explicit local factors it satisfies the same functional equation and specialization formulae as the characteristic ideal of the dual p^infty-Selmer group, and the resulting main conjecture holds in the listed cases with a Grassmannian reduction for higher d.","pith_inferences":["Specialization and restriction maps may transport known main-conjecture or BSD results from the constant-field tower to more general ordinary towers.","The Grassmannian reduction isolates the rank-two case as the essential remaining verification under the ordinary-reduction hypothesis.","Analogous constructions for higher-dimensional ordinary abelian varieties would require only control of local cohomology at multiplicative places."],"forward_implications":["The main conjecture holds for every constant ordinary elliptic curve and every such Z_p^d-extension.","Under global semistable reduction the mu-invariants of L_{A/L} and of the dual Selmer group coincide, and the Selmer group is non-torsion if and only if L_{A/L} vanishes.","For d greater than or equal to 3 the full main conjecture is equivalent to the same statement for all intermediate Z_p^2-extensions in a nonempty Zariski-open subset of the Grassmannian.","A modified form of the conjecture that removes augmentation ideal factors reduces even for rank-one intermediate extensions to a Zariski-open set of Z_p-extensions."],"fun_headline_variants":["p-adic L-functions match Selmer ideals over function fields","Ordinary elliptic curves get interpolating p-adic L-functions","Iwasawa main conjecture proven for function-field curves","p-adic L-functions for Z_p^d-extensions of global function fields","L-functions equal Selmer characteristic ideals in many cases"],"cache_read_input_tokens":32896,"weakest_assumption_plain":"The tower L/K may ramify only at places where the elliptic curve has ordinary (good ordinary or multiplicative) reduction; additive places are forbidden from ramifying.","fun_headline_variants_meta":{"raw":{"variants":["p-adic L-functions match Selmer ideals over function fields","Ordinary elliptic curves get interpolating p-adic L-functions","Iwasawa main conjecture proven for function-field curves","p-adic L-functions for Z_p^d-extensions of global function fields","L-functions equal Selmer characteristic ideals in many cases"]},"model":"grok-4.5","effort":"low","cost_usd":0.004748,"raw_usage":{"total_tokens":1444,"prompt_tokens":883,"num_sources_used":0,"completion_tokens":74,"cost_in_usd_ticks":47480000,"prompt_tokens_details":{"text_tokens":883,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":487,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":883,"tokens_out":74,"duration_ms":3654,"temperature":1.0,"reasoning_tokens":487,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T23:30:59.442337+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"An explicit ordinary elliptic curve over a function field together with a concrete Z_p-extension unramified only at ordinary places for which the constructed L_{A/L} fails to lie in the Iwasawa algebra or fails to generate the characteristic ideal of the dual Selmer group.","supporting_citations":[],"review_version":1}