{"id":"011d9d31-216f-4895-8aa8-cb2a924e680d","arxiv_id":"2502.00876","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under two regulator nonvanishing hypotheses, the eigencurve at a p-irregular weight-one point is the four-branch ring with all cross terms zero, and the ordinary etale cohomology of the modular tower is not Hecke-free there.","lead":"This paper works out the local geometry of the p-adic eigencurve at p-irregular weight-one cusp forms, showing that four smooth branches meet with a non-Gorenstein local ring. The result settles the remaining non-CM cases and implies that certain ordinary etale cohomology groups of modular curves are not free over the Hecke algebra.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem A is conditional on unproved regulator nonvanishing: in the exotic case both (sl) and (reg) follow only from the weak p-adic Schanuel conjecture, and in the RM non-CM case (sl) needs the p-adic Four Exponentials conjecture; the abstract omits these conditions.","rationale":"The reader's weakest-assumption analysis points to exactly the right load-bearing spot: the nonvanishing of the two p-adic regulators encoded in (sl) and (reg). My reading of the proof confirms that these hypotheses are not decorative: they control the simplicity of the roots of Q(S), the nonvanishing of denominators in the local description of Selmer classes, and the independence of the four generalized eigenforms used to prove both the existence of four components and the surjectivity of the cotangent map in Theorem A. The paper is internally coherent and carefully labels these hypotheses; I do not see an internal inconsistency or a hidden unstated assumption beyond the transcendence conjectures. The main concern is therefore scope: for exotic representations the theorem is not yet unconditional, and the abstract's wording risks being read as stronger than the proved conditional statement. This does not change the reader's CONDITIONAL verdict; it reinforces it.","tokens_in":41915,"tokens_out":6922,"duration_ms":78226,"concrete_test":"For a concrete p-irregular weight-one newform f with exotic (A4/S4/A5) projective image, explicitly construct the field H, the S-unit modules, the basis B of (24), and the matrices L and M of (25). Then compute Disc(Q(S)) and res_S(Q(S),PL(S)) directly to high p-adic precision and compare with what Conjecture 2.1 would predict. A single example with a zero discriminant or resultant would show that the regulator hypotheses are not vacuous and Theorem A fails at that point; several verified nonzero examples would supply numerical evidence that the hypotheses are satisfiable, though not a proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central conclusion T ≅ Qpbar[[X1,...,X4]]/(XiXj) depends on hypotheses (sl) and (reg) from Section 4.2. These enter at load-bearing steps: (sl), Disc(Q(S)) ≠ 0, makes the residual-slope polynomial Q have four simple roots and excludes the failure of (SV+); (reg), res_S(Q(S),PL(S)) ≠ 0, ensures PL(s) ≠ 0 at those roots so Proposition 4.1 denominators and the determinant nonvanishing in Proposition 6.6 are valid. If a regulator vanished, the four Hida families could collide or fail to be transverse, and Theorem A would fail. For exotic ρ, Lemmas 4.6–4.8 prove (sl) and (reg) only under Conjecture 2.1, the weak p-adic Schanuel conjecture. For RM non-CM ρ, (reg) and (res) are unconditional but (sl) reduces to (36)/(1), which is only shown under Conjecture 2.4, the p-adic Four Exponentials conjecture; the Klein case is the only unconditional RM case. Thus the theorem is honestly stated as conditional, but the main non-CM, non-Klein cases rest on unproved transcendence conjectures, and the abstract presents the result and its application without displaying these hypotheses.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the completed local ring T of the p-adic eigencurve at a p-irregular classical weight-one cusp form f whose attached Galois representation has scalar Frobenius at p. Under two explicit regulator nonvanishing hypotheses, (sl) and (reg), introduced in Section 4.2, the authors prove Theorem A: T is isomorphic to Qpbar[[X1,X2,X3,X4]]/(XiXj) for i<j, with the weight map sending X to X1+X2+X3+X4. This implies that four irreducible components pass through the point, meet transversally, and each is étale over the weight space, and that T is not Gorenstein. The proof proceeds by analyzing the possible ordinary lines through the residual representation, encoding them as roots of a quartic polynomial Q(S), and then using S-unit regulators and p-adic logarithms to show that exactly four Hida families occur. As applications, Theorem B shows that the localized ordinary étale cohomology of the modular tower X1(Np^r) is not free over the localized Hecke algebra, that Ohta's exact sequence does not split, and Theorem 6.7 proves a conjecture of Darmon–Lauder–Rotger on the dimension of the generalized overconvergent eigenspace. The paper is long and technically detailed, with several steps relying on prior work of the authors and of Betina–Dimitrov.","tokens_in":42233,"tokens_out":8544,"duration_ms":89901,"significance":"If correct, this is a significant contribution to the arithmetic of eigencurves and Iwasawa theory. It gives the first complete description of the local geometry at p-irregular weight-one non-CM points, where the usual R=T method fails and the ordinary deformation functor is not representable. The explicit structure T = Qpbar[[X1,...,X4]]/(XiXj) is striking and has concrete consequences: non-Gorensteinness, non-freeness of localized étale cohomology, and a resolution of the Darmon–Lauder–Rotger conjecture in the irregular setting. The hypotheses (sl) and (reg) are not fitted parameters: the regulator matrices L and M are defined from S-units and p-adic logarithms independently of the eigencurve, and the conditional statements in the body are honestly labeled. The paper also connects the hypotheses to Gross–Stark regulators and to the weak p-adic Schanuel conjecture, which is a useful and natural framework. However, the advertised scope in the abstract is broader than what is actually proved unconditionally, since for exotic representations and for RM non-CM representations the hypotheses are only known under unproved transcendence conjectures, except for the Klein case.","major_comments":[{"comment":"The abstract presents the main result and its application as unconditional: it says 'A complete description ... is given' and 'as an application, we show that ... is not free', without displaying the hypotheses (sl) and (reg). In the body, however, Theorem A is explicitly conditional on (sl) and (reg), and those hypotheses are not known unconditionally for the main non-CM cases: for exotic ρ, Lemmas 4.6(ii), 4.7(ii) and 4.8(ii) deduce (res), (sl) and (reg) only from Conjecture 2.1 (the weak p-adic Schanuel conjecture); for RM non-CM ρ, Lemma 4.7(iii) reduces (sl) to the inequality (1), which is unconditional only in the Klein (RM+CM) case and otherwise is a consequence of Conjecture 2.4. Consequently, the 'complete description' advertised in the abstract is not yet established for those cases. I recommend that the abstract and the introductory summary be revised to state the conditional nature of the theorems explicitly and to distinguish the unconditional Klein case from the conjecturally conditional RM and exotic cases.","section":"Abstract and §1; §4.2, Lemmas 4.6–4.8"},{"comment":"The assertion that conditions (sl) and (reg) 'unconditionally hold' in the Klein case is stated without proof; the reader must infer this from the displayed relations in the paragraph. Since this is the only unconditional non-CM case mentioned in the introduction, the verification should be included or at least made explicit enough that the claim can be checked directly from L−(φ), L−(φ), Sφ and the displayed formula for Q(S). As written, the sentence rests on an unshown computation, albeit a short one.","section":"§4.3.2"},{"comment":"The proof that there are exactly four Hida families (Theorem 6.5(i)) uses Proposition 6.6(iii), whose determinant computation is not shown in detail; the text says 'Using explicit expressions provided by Proposition 4.1, we see that' and then gives the final formula. This is acceptable in a research paper provided the formula is correct, but the displayed expression for det(E) has an apparent typo: the denominator in the displayed quotient repeats (s−s''') and omits (s′−s'''). Since the conclusion only needs the slopes to be pairwise distinct, the nonvanishing is unaffected, but the formula should be corrected.","section":"§6, Proposition 6.6 and Theorem 6.5"}],"minor_comments":[{"comment":"The word 'Acknolwedgements' is misspelled; it should be 'Acknowledgements'.","section":"§1, Acknowledgments"},{"comment":"The sentence 'We note that the above inequality is (1) unconditionally satified if ρ has both RM and CM' contains the typo 'satified' and the phrasing is awkward; it should be 'the inequality (1) is unconditionally satisfied'.","section":"§4.3.1, Eq. (1)"},{"comment":"The symbol K is used both for a quadratic field in the earlier dihedral cases and for the quotient field of A0 in Section 5.2; although the context makes the meaning clear, using a different letter for the quotient field would avoid confusion.","section":"§5.2"},{"comment":"The notation M±_{O_tildeC} and M_{O_C} in the statement of Corollary 7.4 is introduced after the statement; a brief indication of the gluing construction before the corollary would improve readability.","section":"§7.2, Corollary 7.4"}],"recommendation":"major_revision","confidential_remarks":"This is a strong conditional result with a substantial technical core. The main concern is that the abstract and introduction blur the boundary between theorem and conjecture: the advertised 'complete description' is conditional on (sl) and (reg), and those hypotheses are genuinely conjectural for the main non-CM, non-Klein cases. The authors are honest in the body, so this is fixable by revision of the framing. I did not find an internal contradiction or a clear derivation error, but I did not verify every long determinant computation; the corrected determinant formula in Proposition 6.6(iii) should be checked carefully during revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this one. It does something genuinely new: it describes the local geometry of the eigencurve at p-irregular weight-one points in the non-CM cases, where the usual R=T methods break down. The four-branch structure T ≅ Qpbar[[X1..X4]]/(XiXj) and the non-freeness of ordinary etale cohomology are new results, and the method—bypassing deformation rings, tracking residual slopes through higher infinitesimal deformations—looks sound. The paper also settles the Darmon–Lauder–Rotger conjecture on the generalized eigenspace and gives a negative answer to Ohta's splitting question. No parameter fitting, no circular reasoning; the regulator matrices L and M are genuinely independent of the eigencurve geometry.\n\nThe soft spot is the conditionality, and it is real. Theorem A depends on (sl) and (reg) from §4.2: a discriminant nonvanishing and a resultant nonvanishing. In the exotic case both are only known under the weak p-adic Schanuel conjecture; in the RM non-CM case (sl) reduces to an inequality that needs the p-adic Four Exponentials conjecture, and the Klein case is the only unconditional RM case. The body is honest about this—the introduction says 'conditionally on the non-vanishing of certain p-adic regulators'—but the abstract presents the result as complete without displaying the hypotheses. That mismatch will mislead casual readers.\n\nThe proofs are long and lean on prior work by Maksoud and Betina–Dimitrov. I didn't find a fatal error, but the length and the delegation mean independent verification will take time. The rigor is high: the hypotheses are stated explicitly, the reduction steps are laid out, and the polynomial Q(S) is explicit.\n\nWho is this for? Arithmetic geometers and Iwasawa theorists working on eigenvarieties, Hida families, and p-adic L-functions. It deserves a serious referee. The referee's main job should be to check Theorems 5.2–5.6 and Proposition 6.6, and to push the authors to state the conditional nature of the main theorems in the abstract and introduction.","headline":"A technically strong paper that gives a new conditional description of the eigencurve at p-irregular weight-one points, but the abstract overstates the unconditional content.","tokens_in":42743,"tokens_out":2859,"would_cite":true,"duration_ms":27459,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F33","11G18","11F80","11R23"],"pacs":[],"model":"deepseek-v4-flash","headline":"At p-irregular weight-one points, the eigencurve is locally four transversally crossing branches, and the localized ordinary étale cohomology is not a free Hecke module.","keywords":["p-adic eigencurve","weight one forms","p-irregular points","Hida families","Gorensteinness","ordinary étale cohomology","Gross-Stark regulators","p-adic transcendence"],"falsifier":"Take an explicit p-irregular weight-one newform and compute the matrices $L$ and $M$ from the S-units of its Galois field; if $\\mathrm{Disc}(Q(S))=0$ for a single such form, the hypotheses (sl) fails and the predicted four-branch geometry cannot hold by this mechanism, while a numerical check of $\\mathrm{res}_S(P_L(S),Q(S))\\neq 0$ would turn the conditional theorem into an unconditional statement for that point.","tokens_in":2164,"feed_emoji":"📐","tokens_out":2580,"duration_ms":113890,"temperature":0.7,"pith_summary":"This paper pins down the local geometry of the p-adic eigencurve at classical weight-one cusp forms that are p-irregular, meaning the two roots of the p-th Hecke polynomial coincide. There the usual deformation-theoretic machinery breaks down: the ordinary deformation functor is not representable, and several Hida families can pass through the same weight-one point. The authors prove that, under two non-vanishing p-adic regulator hypotheses, the completed local ring is the fourfold power series ring $\\overline{\\mathbb{Q}}_p[[X_1,X_2,X_3,X_4]]$ modulo the relations $X_iX_j=0$ for $i\\neq j$; geometrically, four irreducible components cross transversally at the point. The same hypotheses imply that the ordinary p-adic étale cohomology of the tower of modular curves, localized at the corresponding Hecke prime, is not free over the Hecke algebra, so a common freeness assumption in Iwasawa theory fails precisely in this p-irregular situation.","feed_headline":"Eigencurve has four crossing branches at p-irregular weight-one points","feed_subtitle":"Localized étale cohomology of the modular tower is not a free Hecke module there.","key_machinery":"The load-bearing object is a degree-at-most-four polynomial $Q(S)$, built from p-adic logarithms of S-units of the number field cut out by the adjoint representation $\\mathrm{ad}\\,\\rho$; its roots are exactly the possible residual slopes $s$ of the ordinary line $V^+$ at $x$ for which the Selmer group $\\mathrm{Sel}(\\mathrm{ad}\\,\\rho,V^+)$ is one-dimensional. The hypothesis (sl) is the nonvanishing of the discriminant of $Q(S)$, and (reg) is the nonvanishing of the resultant of $Q(S)$ with a second log-polynomial $P_L(S)$; together they make the four roots simple and distinct and force the trace-zero Selmer group $\\mathrm{Sel}(\\mathrm{ad}^0\\rho,V^+)$ to vanish for every line. Higher infinitesimal deformations of the Artin representation $\\rho$ along a candidate component are then analyzed: the first cocycle is forced to be the unique trace-$\\lambda$ Selmer class, and all higher cocycles vanish, proving that each component is étale over the weight space.","core_discovery":"The paper's central claim is Theorem A: if $f_\\alpha$ is the p-stabilization of a p-irregular weight one newform $f$, and if the two regulator non-vanishing conditions (sl) and (reg) hold, then the completed local ring $T$ of the eigencurve at $x$ is isomorphic to $\\overline{\\mathbb{Q}}_p[[X_1,X_2,X_3,X_4]]/(X_iX_j)_{i<j}$, with the weight map sending $X$ to $X_1+X_2+X_3+X_4$. Consequently $T$ is not Gorenstein, and exactly four irreducible components of the eigencurve pass through $x$, each étale over the weight space. The companion Theorem B states that the localized ordinary étale cohomology $H_{\\infty,\\mathfrak{p}_f}$ of the tower $X_1(Np^r)$ is not free over the localized Hida–Hecke algebra $h_{\\mathfrak{p}_f}$, that the standard ordinary cohomology exact sequence does not split as Hecke modules, and that the reduction modulo the maximal ideal is $\\rho\\oplus\\rho$, where $\\rho$ is the Artin representation attached to $f$. The regulator hypotheses are shown to follow from the weak p-adic Schanuel conjecture when $\\rho$ is exotic and from the p-adic four exponentials conjecture when $\\rho$ has real multiplication; in the Klein (RM+CM) case they hold unconditionally.","pith_inferences":["The polynomial $Q(S)$ and the two regulators give a practical algorithm for deciding, on any concrete eigencurve point, whether four branches meet: compute the S-units and the p-adic logarithms, then test simplicity of the roots of $Q(S)$.","The non-freeness of $H_{\\infty,\\mathfrak{p}_f}$ suggests that around p-irregular weight-one points the canonical-period and reciprocity-law constructions that assume freeness will require a Cohen–Macaulay or derived replacement, rather than a free module argument.","One may expect analogous four-branch non-Gorenstein local rings at irregular weight-one points of other eigenvarieties, wherever the same Selmer-group criterion produces four admissible ordinary lines."],"forward_implications":["At any p-irregular weight-one point satisfying (sl) and (reg), the eigencurve is locally a transversal union of four smooth curves, and the completed local ring is not Gorenstein.","The generalized overconvergent eigenspace for the Hecke eigensystem of $f_\\alpha$ inside weight-one overconvergent forms is two-dimensional and isomorphic to $H^1(\\mathbb{Q},\\mathrm{ad}^0\\rho)$, confirming a conjecture on the dimension of this generalized eigenspace.","The localized ordinary étale cohomology $H_{\\infty,\\mathfrak{p}_f}$ and its $\\pm$-parts are not free over the localized Hida–Hecke algebra, so the standard ordinary cohomology exact sequence does not split there; the characteristic-zero fiber is $\\rho\\oplus\\rho$.","The triangulation of $\\varphi,\\Gamma$-modules over the normalization does not descend to any open neighborhood of $x$ in the eigencurve.","For exotic $\\rho$, the regulator hypotheses are consequences of the weak p-adic Schanuel conjecture, so the whole description is unconditional at all weight-one points modulo that conjecture; in the real-multiplication case the analogous role is played by the p-adic four exponentials conjecture, with the Klein case unconditional."],"supporting_citations":[{"why":"Establishes smoothness of the eigencurve at p-regular weight-one points, the baseline case whose local geometry this paper extends to the irregular setting.","marker":"[BD16]"},{"why":"Treats the CM case and shows the hypotheses (sl) and (reg) are equivalent to the anticyclotomic L-invariant conditions used there.","marker":"[BD21b]"},{"why":"Supplies the Fourier-coefficient formula for first-order p-adic deformations of weight-one forms, used to count Hida families through the generalized eigenspace.","marker":"[DLR17]"},{"why":"Provides the p-adic transcendence rank bounds and regulator-map calculations behind Proposition 2.3 and Lemma 2.5, forcing the logarithms entering (sl) and (reg) to be linearly independent.","marker":"[Mak23]"},{"why":"Defines the ordinary étale cohomology module $H_\\infty$ and its exact sequence (63); Theorem B analyzes whether these localize to free Hecke modules.","marker":"[Oht00]"},{"why":"Constructs the triangulation of arithmetic families of $\\varphi,\\Gamma$-modules whose non-descent at irregular weight-one points is proved in Corollary 7.4.","marker":"[KPX14]"}],"fun_headline_variants":["Four components meet at irregular weight-one eigencurve points","Localized Hecke module non-freeness at irregular weight-one points","Non-Gorenstein eigencurve local rings at p-irregular weight one","Eigencurve: four branches, non-free cohomology at irregular cusp forms","Regulator conditions force fourfold eigencurve crossing at weight-one"],"cache_read_input_tokens":44928,"weakest_assumption_plain":"The result stands or falls on the nonvanishing of two p-adic regulator expressions, the discriminant of $Q(S)$ and the resultant of $P_L(S)$ and $Q(S)$; in the exotic case neither is proven, and each would follow from the weak p-adic Schanuel conjecture, while in the real-multiplication case one reduces to the p-adic four exponentials conjecture and is unconditional only in the Klein (RM+CM) case.","fun_headline_variants_meta":{"raw":{"variants":["Four components meet at irregular weight-one eigencurve points","Localized Hecke module non-freeness at irregular weight-one points","Non-Gorenstein eigencurve local rings at p-irregular weight one","Eigencurve: four branches, non-free cohomology at irregular cusp forms","Regulator conditions force fourfold eigencurve crossing at weight-one"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000777,"raw_usage":{"total_tokens":3428,"prompt_tokens":931,"completion_tokens":2497,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":547,"completion_tokens_details":{"reasoning_tokens":2404}},"tokens_in":547,"tokens_out":2497,"duration_ms":17383,"temperature":1.0,"reasoning_tokens":2404,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T17:23:31.891290+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take an explicit p-irregular weight-one newform and compute the matrices $L$ and $M$ from the S-units of its Galois field; if $\\mathrm{Disc}(Q(S))=0$ for a single such form, the hypotheses (sl) fails and the predicted four-branch geometry cannot hold by this mechanism, while a numerical check of $\\mathrm{res}_S(P_L(S),Q(S))\\neq 0$ would turn the conditional theorem into an unconditional statement for that point.","supporting_citations":[],"review_version":1}