{
  "schema": "open-mathematics-proof-routes/v1",
  "course": "LG-MF",
  "lesson": "LG-MF-07",
  "license": "CC0-1.0",
  "source": "src/the-petersson-inner-product-and-poincare-series.md",
  "reader": "LG-MF-07.html",
  "results": [
    {
      "id": "LG-MF-07-modular-local-finiteness",
      "statement": "For any finite-index subgroup Gamma of SL_2(Z), choose D as the finite union of closed level-one modular cells attached to representatives of barGamma\\PSL_2(Z). Projectively distinct Gamma-translates cover H with disjoint cell interiors and meet each compact subset in only finitely many tiles. Their boundaries on each compact positive-height truncation lie in finitely many line/circle arcs, have zero Jordan area, and contribute zero to continuous weighted integrals.",
      "proof": "LG-MF-07.html#continuous-integrals-and-projected-modular-tiles",
      "result": "Lemma 4.0, first three proof paragraphs"
    },
    {
      "id": "LG-MF-07-locally-finite-tile-integration",
      "statement": "For the locally finite line/circle-cell partitions of H, a finite union of modular cells, or a horizontal strip used in Lemma 4.0, compact truncations are Jordan regions and continuous integrals are finitely additive on the finitely many relevant pieces. A nonnegative continuous integrand has improper integral equal to the sum of its piece integrals, with infinity allowed. If its absolute improper integral is finite, the same equality holds for a complex continuous integrand and the piece-integral series converges absolutely. Tile integrals defined by ambient rectangular truncation agree with exhaustion of cell interiors and are independent of boundary assignment.",
      "proof": "LG-MF-07.html#continuous-integrals-and-projected-modular-tiles",
      "result": "Lemma 4.0, finite-partition proof and (4.0b)"
    },
    {
      "id": "LG-MF-07-projected-strip-unfolding",
      "statement": "Let Gamma have finite index in SL_2(Z), D be its finite-cell modular fundamental region, and h be a positive integer with Gamma_infinity,h={plus or minus T^(hr):r in Z} contained in Gamma. For gamma in Gamma_infinity,h\\Gamma, the projected tiles D_gamma=union_r(T^(hr)gamma D intersect S_h), S_h={0<=x<h,y>0}, cover S_h with disjoint cell interiors and locally finite pieces and boundary arcs on compact positive-height truncations. Projection of each tile's cell interiors is injective. For continuous horizontally h-periodic G>=0, sum_gamma integral_(gamma D)G=sum_gamma integral_(D_gamma)G=integral_(S_h)G, with infinity allowed; if G is complex continuous and integral_(S_h)|G| is finite, both equalities hold with absolutely convergent tile-integral series. In particular h=1 applies when {plus or minus T^r} is contained in Gamma; gamma D itself is not asserted to lie in S_h.",
      "proof": "LG-MF-07.html#continuous-integrals-and-projected-modular-tiles",
      "result": "Lemma 4.0 and (4.0a)"
    },
    {
      "id": "LG-MF-07-continuous-poincare-unfolding",
      "statement": "For N>=1, even k>=4, f in S_k(Gamma_0(N)), and the Poincare series P_m of (3.1) summed over {plus or minus T^r}\\Gamma_0(N), the nonnegative and complex continuous-series interchanges are justified by local uniform convergence and finite summed absolute integrals. For every integer m>=1, I_Gamma(f,P_m)=Gamma(k-1)(4*pi*m)^(-(k-1))*a_m(f) and the normalized pairing equals this value divided by d_Gamma; for m=0, I_Gamma(f,P_0)=0. The strip estimate at zero is A*y^(k/2-2), requiring k>2, and cusp exponential decay proves the remaining tail, including m=0.",
      "proof": "LG-MF-07.html#4-unfolding-and-spanning",
      "result": "Theorem 4.1, using Lemma 4.0 and LG-MF-01 Lemma 0.3",
      "retained_prerequisites": [
        "The real plane change-of-variables theorem for the Möbius substitutions; the Jacobian computation alone does not supply that integral theorem."
      ]
    }
  ],
  "earlier_proofs": [
    {
      "lesson": "LG-MF-01",
      "result": "Lemma 0.3, continuous interval and plane series; rectangular exchange",
      "reader": "LG-MF-01.html#analytic-facts-for-subsequent-lessons"
    },
    {
      "lesson": "LG-MF-01",
      "result": "Theorems 2.2 and 3.2, modular reduction and closed fundamental domain",
      "reader": "LG-MF-01.html#2-reducing-a-shape-without-an-infinite-search"
    },
    {
      "lesson": "LG-MF-07",
      "result": "Theorem 3.2, absolute locally uniform Poincare convergence",
      "reader": "LG-MF-07.html#3-averaging-an-exponential"
    }
  ],
  "retained_prerequisites": [
    "Real completeness, elementary continuous Riemann integration and compactness.",
    "The real plane change-of-variables theorem for the Möbius substitutions; the Jacobian computation alone does not supply that integral theorem.",
    "General measurable fundamental-domain integration for the arbitrary-domain assertion in LG-MF-07 Theorem 1.1."
  ],
  "scope": "Continuous series and locally finite modular-cell integration, including projection into strips of the stated actual width. General measurable Tonelli, Fubini and dominated convergence, arbitrary measurable fundamental-domain independence, and the integral change-of-variables theorem are not supplied by these proofs. The full pairing, spanning, Eisenstein-complement statements and their weight restrictions remain in the lesson."
}