{ "attack": "the Loeffler-Rivero hypotheses instantiated; C4's direction closed; C1's partner-independence identified as a theorem", "source": { "paper": "D. Loeffler and O. Rivero, 'Eisenstein degeneration of Euler systems'", "arxiv": "2201.02078v2", "journal": "J. reine angew. Math. 814 (2024), 241-282", "read": "2026-09-15", "extracts": "data/external/literature/loeffler-rivero-2201.02078v2/" }, "correction_to_RUN_093": "RUN-093 reported C4's nonvanishing as UNMEASURED on the grounds that every witness needs the overconvergent theta-critical eigenspace. That was wrong: the reducibility-ideal witness is a theorem (LR Theorem A6.1 = Bellaiche-Chenevier [BC06, Thm 1]), and its hypotheses are computable.", "C4_STATUS": { "membership": "CLOSED", "membership_source": "RUN-093, COMPUTED_HERE", "nonvanishing": "CLOSED", "nonvanishing_source": "CITED_THEOREM", "nonvanishing_chain": [ "LR Theorem A6.1 (= Bellaiche-Chenevier [BC06, Thm 1]): reducibility ideal is exactly m", "LR section A6.2: hence all four representations are non-split", "LR Corollary A6.3(i): V(f_beta) is case (3) of Theorem A2.1 — non-split and CRYSTALLINE at p", "case (3)'s class lies in H^1(Q, Q_11(chi_8)(1)); crystalline puts it in H^1_f; non-split makes it nonzero", "RUN-092: dim H^1_f = 1, spanned by kappa(epsilon)" ], "direction": "CLOSED", "scalar": "OPEN", "conclusion": "[Theta_aux] = c_ES/Kum * kappa(epsilon), c_ES/Kum in Q_11^x" }, "C1_STATUS": { "partner_independence": "CITED_THEOREM", "why": "C_pi = C = c*_f(r) is a functional of the auxiliary family alone: PARTNER-INDEPENDENT, and nonzero. RUN-090 carried this as certificate item C1's unproved core.", "hypotheses_instantiated": { "f_beta non-critical": true, "partner ordinary": true }, "still_missing": "the BF classes' coordinates and any finite-level representative" }, "B1_B2": { "verdict": "PASS-EXISTENCE", "theorem": "LR Theorem C1.4", "hypothesis_on_d": { "d": 5, "condition": "d in Z_{>1} with (d, 6S) = 1", "gcd_with_6S": 1, "holds": true }, "label": "CITED_THEOREM for the existence; COMPUTED_HERE for the hypothesis on d", "note": "the same class serves both partners: kappa_d(f, g) is built for the PAIR, so BF_E and BF_0 are the same theorem applied with g = the two curves" }, "B3_B4": "UNMEASURED", "hypotheses_computed_here": { "identification_in_LR_notation": { "psi": "1 (trivial)", "tau": "chi_8", "r": 0, "N": 8, "alpha": 1, "beta_critical": -11, "all_q_expansion_coefficients_match": true }, "11_decency": { "per_prime": { "2": { "divides_conductor_of_psi_over_tau": true, "psi_over_tau_at_ell": 0, "value_not_one": false, "passes": true }, "11": { "divides_conductor_of_psi_over_tau": false, "psi_over_tau_at_ell": -1, "value_not_one": true, "passes": true } }, "decent": true }, "non_criticality": true, "ext_group": { "theorem": "LR Theorem A2.1 (Soule): exactly three classes — semisimple; non-split with tau eps^{-1-r} as sub (not de Rham at p); non-split with psi as sub (crystalline at p)", "case_3_is_crystalline": true, "case_3_ext_group": "H^1(Q, Q_11(chi_8)(1))", "case_2_ext_group": "H^1(Q, Q_11(chi_8)(-1))", "target_line_RUN092": "H^1(Q, Q_11(chi_8)(1))", "case_3_is_the_target_line": true, "cases_2_and_3_differ": true, "convention_LR": { "trace_normalisation": "tr rho(Frob_l^{-1}) = a_l", "second_character": "chi_8 * eps^{-1-r}", "ext_group": "H^1(Q, psi * (tau eps^{-1-r})^{-1}) = H^1(Q, chi_8 eps^{1+r})" }, "convention_RUN091": { "trace_normalisation": "tr rho(Frob_l) = a_l", "second_character": "chi_8 * eps^{1+r}", "ext_group": "H^1(Q, (chi_8 eps^{1+r}) * 1^{-1}) = H^1(Q, chi_8 eps^{1+r})" }, "conventions_dual_but_same_ext_group": true, "independent_confirmation": "LR Theorem A4.5 states non-criticality as the non-vanishing of H^1_f(Q, psi tau^{-1} eps^{1+r}) -> H^1_f(Q_p, ...), i.e. in this very group", "label": "COMPUTED_HERE for the bookkeeping; CITED_THEOREM for A2.1 and A4.5", "agrees": true }, "partner_ordinarity": { "389.a1": { "a_11": -4, "unit_root": 7, "ordinary": true }, "19.a1": { "a_11": 3, "unit_root": 3, "ordinary": true } } }, "c113_vs_pc001_dictionary": { "theorem": "LR Theorem C1.13", "proof_opens": "Taking C = c*_f(r)", "C_is_nonzero": true, "C_lives_on": "V_1, the AUXILIARY family's disc (Theorem C1.9: 'a meromorphic function on V_1 alone')", "uniqueness": "Remark C1.11: although g is used in the construction of omega_f, 'it is uniquely determined by f'", "dictionary": { "C": "C_pi", "C_d(f_beta, g, j)": "S_5(t)", "L_p(g tensor tau, j-1-r)": "lambda_i(t)", "L_p(Ad g)": "D_i", "log^{[r+1]}": "j(t) = log(1+t)/log_11(12)", "kappa(g x psi)": "z_i / the Kato class" }, "every_lr_factor_matched": true, "every_pc001_factor_matched": true, "consequence": "C_pi = C = c*_f(r) is a functional of the auxiliary family alone: PARTNER-INDEPENDENT, and nonzero. RUN-090 carried this as certificate item C1's unproved core.", "partner_independence_label": "CITED_THEOREM", "label": "DERIVED_FROM_VERIFIED_INPUTS for the dictionary; CITED_THEOREM for C1.9/C1.11/C1.13", "agrees": true }, "notation_trap": "LR's f is the EISENSTEIN family and LR's g is the CUSPIDAL partner; PC-001 uses the opposite letters. LR normalises by tr rho(Frob_l^{-1}) = a_l, so LR's second character is chi_8 eps^{-1-r} where RUN-091's is chi_8 eps^{1+r}; the conventions are dual and give the SAME Ext group.", "claim_labels_legend": [ "COMPUTED_HERE", "CITED_THEOREM", "DERIVED_FROM_VERIFIED_INPUTS", "UNMEASURED", "FAILED" ], "status": { "C1": "partner-independence CITED_THEOREM with hypotheses instantiated; BF coordinates still UNMEASURED", "C2": "UNMEASURED — Pi_E/Pi_0 still open", "C3": "PASS", "C4": "direction CLOSED, exact scalar OPEN", "C5-C7": "unchanged", "C8": "PARTIAL", "C9-C10": "NOT YET READY", "BSD": "OPEN" } }