{
  "schema": "open-mathematics-proof-routes/v1",
  "course": "LG-MF",
  "lesson": "LG-MF-10",
  "license": "CC0-1.0",
  "source": "src/oldforms-newforms-and-the-theory-of-atkin-lehner-and-li.md",
  "reader": "LG-MF-10.html",
  "results": [
    {
      "id": "LG-MF-10-weighted-finite-trace",
      "statement": "For N=rL and a character descended to L, the character-weighted trace of f slash diag(1,r) from Gamma_0(L) intersect Gamma^0(r) to Gamma_0(L) is cuspidal of level L. Its exact cosets number r if r divides L and r+1 otherwise.",
      "hypotheses": "Integer k>=2, r prime, N=rL, f in S_k(N,chi_N), chi_N inflated from chi_L; the slash is determinant^(k/2). This includes r=2 and L=1.",
      "proof": "LG-MF-10.html#finite-weighted-trace",
      "locator": "Lemma 3.3, equation (3.1)",
      "scope": "Finite cosets, inverse character weights, representative independence, transformation law and cusp descent."
    },
    {
      "id": "LG-MF-10-square-prime-lowering",
      "statement": "If r^2 divides N and chi_N descends modulo N/r, E_r=U_r maps S_k(N,chi_N) to S_k(N/r,chi_{N/r}); its trace normalization is E_r f=r^(k/2-1) T(f slash diag(1,r)).",
      "hypotheses": "Integer k>=2, r prime, r^2 divides N, chi_N descended to N/r.",
      "proof": "LG-MF-10.html#finite-weighted-trace",
      "locator": "Lemma 3.3, equation (3.2); restated in (4.1)"
    },
    {
      "id": "LG-MF-10-prime-once-lowering",
      "statement": "For q occurring exactly once in N and a character descended to N/q, E_q+q^(k/2-1)W_q^N maps S_k(N,chi_N) to S_k(N/q,chi_{N/q}). The chosen partial matrices normalize Gamma_0(N), preserve the descended character and give a choice-independent operator with square chi_{N/q}(q)I.",
      "hypotheses": "Integer k>=2, q prime, q divides N and q^2 does not divide N, chi_N inflated from chi_{N/q}; matrix [[qx,y],[Nz,q]] with qx-(N/q)yz=1.",
      "proof": "LG-MF-10.html#prime-once-trace",
      "locator": "Proposition 3.4, equations (3.3)–(3.8); lowering restated in (4.2)",
      "scope": "All four conjugation entries, matrix square, choice independence, rational cusp condition and prime-to-q dilation covariance."
    },
    {
      "id": "LG-MF-10-filter-level-control",
      "statement": "The Fourier filter I-V_r E_r applied to level M is cuspidal at level Mr^2 with inflated character, at level Mr when r divides M, and at level M when r^2 divides M and the character descends modulo M/r. Inflation from N to NR with q not dividing R preserves precisely the obstruction to character descent from N to N/q.",
      "hypotheses": "Integer k>=2, r and q prime; for the conductor assertion q divides N and M=NR with q not dividing R. N and R may share other prime factors.",
      "proof": "LG-MF-10.html#support-filters-conductor",
      "locator": "Lemma 3.5, equations (3.9)–(3.11)",
      "scope": "Coefficient telescope, controlled individual-prime level raising, surjective unit reduction and the conductor-kernel CRT lift."
    },
    {
      "id": "LG-MF-10-admissible-support-removal",
      "statement": "If a fixed-character cusp form has each nonzero index divisible by a prime in a finite nonempty admissible set Q, a chosen q in Q can be removed by subtracting V_q phi with phi at level N/q. The residual keeps level N and its character, and is supported on multiples of Q minus q.",
      "hypotheses": "Integer k>=2; f in S_k(N,chi_N); every q in Q divides N and permits character descent modulo N/q. Support means the union of the multiples of primes in Q.",
      "proof": "LG-MF-10.html#common-level-support-extraction",
      "locator": "Lemma 3.6, equations (3.12)–(3.15)",
      "scope": "For q occurring once, M=N times the product of squares of the other primes admits one legal matrix at both N and M. Each intermediate level divides M/r_i and its character descends through M/(r_i q). The common-level trace has the exact remaining coefficient factor 1+q^(-1)."
    },
    {
      "id": "LG-MF-10-fixed-character-auxiliary-support",
      "statement": "For f in S_k(N,chi_N), vanishing of a_n whenever gcd(n,D)=1 gives f=sum_{q|gcd(N,D)} V_q h_q with h_q in S_k(N/q,chi_{N/q}), and only primes admitting character descent may contribute. If there are no such primes, f=0.",
      "hypotheses": "Integer k>=2, N,D>=1, fixed lower-right character; incompatible parity makes the space zero.",
      "proof": "LG-MF-10.html#fixed-character-auxiliary-support",
      "locator": "Theorem 3.7, equation (3.16)",
      "scope": "Arbitrary finite auxiliary prime set, elimination of outside-level and conductor-obstructed primes by Theorem A.1, then admissible-prime extraction at the original level. No arbitrary diamond projection is assumed."
    },
    {
      "id": "LG-MF-10-one-prime-support",
      "statement": "A fixed-character cusp form supported only at indices divisible by p is zero when p is outside the level or prevents character descent, and otherwise is V_p of a cusp form at level N/p.",
      "hypotheses": "Integer k>=2, p prime, fixed character at N.",
      "proof": "LG-MF-10.html#appendix-a-fourier-support-at-one-prime",
      "locator": "Theorem A.1, Sections A.1–A.4",
      "scope": "Conductor-kernel shear, group generation and cusp descent; outside-level vanishing uses the good-prime recurrence and complex coefficient bound."
    },
    {
      "id": "LG-MF-10-trivial-character-bad-coefficients",
      "statement": "For a normalized trivial-character newform, a_p=-w_p p^(k/2-1) when p occurs once in N, and a_p=0 when p^2 divides N.",
      "hypotheses": "Integer k>=2, trivial character, normalized newform and p prime dividing N; newspace stability and positivity are used.",
      "proof": "LG-MF-10.html#5-fricke-signs-and-the-bad-coefficients",
      "locator": "Section 5, proof of the displayed trivial-character formulas, using Theorem 2.3 and Proposition 5.2",
      "scope": "The finite lowering formulas are proved in Lemma 3.3 and Proposition 3.4. This newspace consequence retains the Petersson and spectral prerequisites; it does not prove ramified-character classification."
    }
  ],
  "earlier_proofs": [
    {
      "lesson": "LG-MF-02",
      "result": "Lemmas 1.0–1.1: Chinese remainder construction and determinant-one integral lifting",
      "reader": "LG-MF-02.html#1-a-finite-set-that-remembers-the-level"
    },
    {
      "lesson": "LG-MF-04",
      "result": "Proposition 1.1 and determinant multiplicativity: slash cocycle; cusp and Fourier coordinates",
      "reader": "LG-MF-04.html#1-the-slash-operator-and-the-cusp-condition"
    },
    {
      "lesson": "LG-MF-09",
      "result": "Section 2 after (2.3), Lemma 2.2 and Theorem 2.3: rational-slash cusp preservation, fixed-character prime stability and coefficient formulas",
      "reader": "LG-MF-09.html#2-the-prime-double-cosets-and-their-coefficients"
    },
    {
      "lesson": "LG-MF-06",
      "result": "Theorem 6.1 and its fixed-character extension: complex coefficient bound and finite-dimensional coefficient injection",
      "reader": "LG-MF-06.html#6-how-many-coefficients-determine-a-form"
    }
  ],
  "retained_prerequisites": [
    "The elementary integer, finite-field, finite-dimensional algebra and complex/cusp foundations disclosed in lessons 1, 2, 4, 5, 6 and 9 remain the framework for the fixed-character support proof.",
    "The unrestricted Gamma_1(N) auxiliary-D support statement remains an exact separate proof obligation; arbitrary diamond projection has not been proved to preserve its support hypothesis.",
    "Newspace deductions and multiplicity retain Petersson convergence, integral change of variables, adjoints, positivity and simultaneous finite-dimensional spectral decomposition with their exact programme foundations.",
    "Cross-character prime-value comparison retains the coefficient-field and attached Shimura–Deligne representation construction and finite-extension Chebotarev theorem.",
    "The finite-symbol perfect pairing retains the earlier dim S_2=g identity and its elementary analytic foundations.",
    "Ramified-character bad-prime classification and the local newvector and classical-adelic comparison statements remain outside the supplied finite trace and support proof."
  ],
  "scholarly_sources": [
    {
      "author": "W.-C. W. Li",
      "title": "Newforms and functional equations",
      "publication": "Mathematische Annalen 212 (1975), 285–315",
      "url": "https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0212/LOG_0047.pdf",
      "scope": "Finite weighted traces, partial matrices and fixed-character Fourier support."
    }
  ],
  "figures": [
    {
      "image": "figures/newform-weighted-trace.png",
      "vector": "figures/newform-weighted-trace.svg",
      "source": "figures/draw_newform_weighted_trace.py",
      "caption_locator": "LG-MF-10.html#common-level-support-extraction",
      "mathematical_role": "Finite coset trace and common-level coefficient extraction with exact constants; coefficient equality only at indices prime to the other support primes."
    }
  ],
  "scope": "The finite weighted traces and arbitrary finite auxiliary-support theorem are proved for a fixed character. Existence and same-character comparison use this local support provider; their deeper prerequisites and the unrestricted support statement remain separately recorded."
}