{ "arrows": [ { "name": "Psi_1", "from": "L1 (BF family line)", "to": "L0 (C_aux) (x) partner line (x) H_Gamma", "source_type": { "category": "etale (Galois cohomology)", "ring": "O(V_1 x V_2 x W)", "rank": 1, "character": "V_c(f,g)^* up to 1/X", "tate_twist": 0, "orientation": "line" }, "target_type": [ { "category": "crystalline (D_cris of a (phi,Gamma)-submodule)", "ring": "O(V_1)", "rank": 1, "character": "auxiliary F+ line", "tate_twist": -1, "orientation": "line" }, { "category": "crystalline", "ring": "O(V_2)", "rank": 1, "character": "partner F- line", "tate_twist": 0, "orientation": "line" } ], "theorem": "Perrin-Riou regulator L^PR (LZ16) on loc_p; image computed by LR C1.9", "hypotheses": [ "kappa_d exists: C1.4 with (d,6S)=1 (RUN-094)", "f_beta non-critical (RUN-094)", "g noble, ordinary, residually irreducible (RUN-095)" ], "nonzero": { "image": "c_f(k) * C_d * L_p^g", "c*_f(r) in L^x": "C1.13", "C_d": 26, "L_p^g at the point": { "lambda_E(0)": 5, "lambda_0(0)": 4 }, "status": "injective on the line: every factor a non-zero-divisor" }, "frame_action": { "b_f+": "u (RUN-096: Col scales by u)", "omega_f": "canonical replacement (C1.10)", "eta_g": "partner frame" }, "partner_dependence": "ONLY through the separate tensor factor D_cris(F- D(g)^*) and eta_g", "line_level": "class-level (a map of modules)", "partner_slot_separate": true }, { "name": "Psi_2", "from": "L1 (BF family line)", "to": "L2 (BF leading line)", "source_type": { "category": "etale (Galois cohomology)", "ring": "O(V_1 x V_2 x W)", "rank": 1, "character": "V_c(f,g)^* up to 1/X", "tate_twist": 0, "orientation": "line" }, "target_type": { "category": "etale (Galois cohomology)", "ring": "H_Gamma", "rank": 1, "character": "V(g)^*(psi^{-1})", "tate_twist": 0, "orientation": "line" }, "theorem": "C1.6 (tau-quotient image zero) + C1.7 (unique lift to H^1(Q,V(f,g)^*), image in the psi-quotient) + leading term X^n kappa-hat_d mod X", "hypotheses": [ "Prop C1.1 (Selmer vanishing) for the vanishing in C1.6", "Thm A6.1 / section A6.3 for the chain of lattices" ], "nonzero": { "status": "X^n kappa-hat_d is well-defined and non-zero modulo X (LR, C1.7 area)" }, "frame_action": { "b_f+": "none: a quotient projection of Galois modules" }, "partner_dependence": "the target module is V(g)^*(psi^{-1}) (x) H_Gamma", "line_level": "class-level (quotient maps of Galois modules)" }, { "name": "Psi_3", "from": "L2 (BF leading line)", "to": "L3 (Kato line)", "source_type": { "category": "etale (Galois cohomology)", "ring": "H_Gamma", "rank": 1, "character": "V(g)^*(psi^{-1})", "tate_twist": 0, "orientation": "line" }, "target_type": { "category": "etale (Galois cohomology)", "ring": "H_Gamma", "rank": 1, "character": "V(g)^*(psi^{-1})", "tate_twist": 0, "orientation": "line" }, "theorem": "C1.13: kappa-hat*_d(f_beta,g) = [C C_d log L_p(g x tau)/L_p(Ad g)] kappa(g x psi); C1.1's proof: the module is free of rank 1 over H_Gamma", "hypotheses": [ "Kato's reciprocity for g", "Equation 11 and C1.12", "Prop C1.1 for the lift" ], "nonzero": { "status": "both generators nonzero; the scalar C C_d L_p(g x tau)/L_p(Ad g) a non-zero-divisor" }, "frame_action": { "b_f+": "C = c*_f(r) scales by u; the class identity is frame-free" }, "partner_dependence": "L_p(g x tau), L_p(Ad g), kappa(g x psi)", "line_level": "scalar-first, lifted to classes by Prop C1.1", "sharp_question_1": "LR proves equal regulator images, then equality of CLASSES by Prop C1.1 (injectivity): line-level" }, { "name": "Psi_4", "from": "L3 (Kato line)", "to": "L4 (C_arith)", "source_type": { "category": "etale (Galois cohomology)", "ring": "H_Gamma", "rank": 1, "character": "V(g)^*(psi^{-1})", "tate_twist": 0, "orientation": "line" }, "target_type": { "category": "H_Gamma (Iwasawa algebra / distributions)", "ring": "H_Gamma", "rank": 1, "character": "g x psi", "tate_twist": 0, "orientation": "line" }, "theorem": "Kato [Kat04]: localisation at p, Perrin-Riou regulator for F- V(g)^*, lands on L_p(g x psi); and Kato's L_p equals the Mazur-Tate-Teitelbaum L_p built from modular symbols, up to the period frame", "hypotheses": [ "g residually irreducible, ordinary (RUN-095)", "the period frame Omega^+- fixed: RUN-080 (calibrator plus = Neron), RUN-096 C8 (indices 2, 2)" ], "nonzero": { "status": "injective on the line: the p-adic L-function is a non-zero-divisor (C1.1's proof); in-tree the series are nonzero" }, "frame_action": { "Omega^+-": "G (period frame)", "lattice index": "L (2 on both signs of the target, 2 and 1 for the calibrator)" }, "partner_dependence": "entirely partner-side", "line_level": "class-level; the equality of Kato's and MTT's L_p is an equality of elements of H_Gamma", "kato_modsym_status": "theorem", "sharp_question_2": "id(Kato ~ modular symbol) is Kato's theorem that his p-adic L-function is the modular-symbol one up to the stated period frame - a theorem-induced identification of generators of the same line, with the frame named as G; not bookkeeping" } ], "junctions": { "L1 shared by Psi_1 and Psi_2": { "compatible": true, "problems": [] }, "L2: Psi_2 -> Psi_3": { "compatible": true, "problems": [] }, "L3: Psi_3 -> Psi_4": { "compatible": true, "problems": [] } }, "every_arrow_named": true, "every_identity_line_level_or_lifted": true, "partner_slot_separate_everywhere": true, "kato_modsym_is_a_theorem": true, "first_incompatible_junction": null, "positive_control_accepted": true, "composite": "Psi = Psi_4 . Psi_3 . Psi_2 . Psi_1^{-1} : C_aux (x) (partner line) -> C_arith, over H_Gamma-lines", "agrees": true }