{
  "schema": "open-mathematics-proof-unit/v1",
  "lesson": "LG-MF-17",
  "title": "Complex tori, elliptic curves and the moduli interpretation of modular curves",
  "license": "CC0-1.0",
  "source": "src/complex-tori-elliptic-curves-and-the-moduli-interpretation-of-modular-curves.md",
  "reader": "LG-MF-17.html",
  "results": [
    {
      "id": "LG-MF17-MAPS-OF-TORI",
      "statement": "Every holomorphic map C/Lambda -> C/Lambda-prime is induced by z -> alpha z + beta with alpha Lambda contained in Lambda-prime. A pointed isomorphism has nonzero alpha and alpha Lambda = Lambda-prime.",
      "proof_locator": "Theorem 1.1",
      "state": "proof supplied in this lesson; earlier prerequisites remain applicable"
    },
    {
      "id": "LG-MF17-LATTICE-CUBIC",
      "statement": "The normally convergent Weierstrass series satisfies wp-prime^2 = 4 wp^3 - 60G4 wp - 140G6, has nonzero discriminant, and parametrizes the smooth cubic by a pointed biholomorphism preserving the chord-and-tangent group law.",
      "proof_locator": "Theorems 2.1 and 2.3; Lemma 2.2",
      "state": "proof supplied in this lesson; earlier prerequisites remain applicable"
    },
    {
      "id": "LG-MF17-ANALYTIC-GENUS-ONE-CUBIC",
      "statement": "For every compact connected Riemann surface X without boundary of topological genus one, every P in X and every nonzero holomorphic differential omega, the normalized functions x and y=dx/omega give a smooth cubic Y^2Z=4X0^3-aX0Z^2-bZ^3 with a^3-27b^2 nonzero and a pointed biholomorphism Phi:(X,P)->(C_ab,[0:1:0]), satisfying Phi^*(dx/y)=omega.",
      "proof_locator": "Theorem 3.0, equations (G0)-(G13)",
      "proof_anchor": "genus-one-cubic",
      "prerequisite_ids": [
        "LG-MF06-COMPACT-RR",
        "LG-MF06-DIVISOR-DEGREE",
        "LG-MF06-GENUS-BRIDGE",
        "LG-MF06-BOUNDARY-INTEGRATION",
        "LG-MF01-COMPLEX-ANALYSIS",
        "LG-MF04-LAURENT"
      ],
      "state": "proved relative to the listed prerequisite foundations"
    },
    {
      "id": "LG-MF17-CUBIC-UNIFORMIZATION",
      "statement": "Every nonsingular complex Weierstrass cubic y^2=4x^3-a x-b is the cubic associated with a lattice with exactly those coefficients; the lattice is unique up to homothety.",
      "proof_locator": "Theorem 3.1, equations (3.1)-(3.4)",
      "state": "proof supplied in this lesson; earlier prerequisites remain applicable"
    },
    {
      "id": "LG-MF17-ANALYTIC-GENUS-ONE-UNIFORMIZATION",
      "statement": "Every compact connected pointed Riemann surface without boundary of topological genus one admits a pointed biholomorphism U=Phi_Lambda^{-1} composed with Phi to C/Lambda, with Lambda unique up to homothety.",
      "proof_locator": "Corollary 3.1a, equation (3.4b)",
      "proof_anchor": "analytic-genus-one-uniformization",
      "prerequisite_ids": [
        "LG-MF17-ANALYTIC-GENUS-ONE-CUBIC",
        "LG-MF17-CUBIC-UNIFORMIZATION",
        "LG-MF17-LATTICE-CUBIC",
        "LG-MF17-MAPS-OF-TORI"
      ],
      "state": "proved relative to the listed prerequisite foundations"
    },
    {
      "id": "LG-MF17-ELLIPTIC-FUNCTION-FIELD",
      "statement": "Every meromorphic function on C/Lambda is uniquely R(wp)+wp-prime S(wp), with R,S in C(t); torus isogenies induce algebraic maps between the associated smooth Weierstrass cubics.",
      "proof_locator": "Proposition 3.2, equation (3.4a)",
      "state": "proof supplied in this lesson; earlier prerequisites remain applicable"
    },
    {
      "id": "LG-MF17-LEVEL-MODULI",
      "statement": "The analytic moduli sets of elliptic curves marked by an exact-order N point, a cyclic subgroup of order N, or a torsion basis with the specified positive determinant pairing are respectively Gamma1(N)\\H, Gamma0(N)\\H and Gamma(N)\\H for every N>=1.",
      "proof_locator": "Theorem 4.1, equations (4.1)-(4.5)",
      "state": "proof supplied in this lesson; earlier prerequisites remain applicable"
    },
    {
      "id": "LG-MF17-HECKE-ALL-KERNELS",
      "statement": "For n prime to cyclic level N, the normalized T_n is the quotient average 1/n over every actual order-n subgroup, retaining quotient differentials and transported level markings; it equals n^(k-1) times the index-n sublattice sum and the earlier Fourier operator.",
      "proof_locator": "Theorem 5.1; Corollary 5.2, equations (5.1)-(5.14)",
      "state": "proof supplied in this lesson; earlier prerequisites remain applicable"
    },
    {
      "id": "LG-MF17-PRIME-SQUARE-KERNELS",
      "statement": "The noncyclic kernel E[p] contributes p^(k-2); its intermediate subgroups give T_p^2=T_(p^2)+p^(k-1)I with the transported cyclic marking.",
      "proof_locator": "Section 5.2, equations (5.15)-(5.17); Solution 6, equation (8.1)",
      "state": "proof supplied in this lesson; earlier prerequisites remain applicable"
    }
  ],
  "prerequisites": [
    {
      "id": "LG-MF06-COMPACT-RR",
      "lesson": "LG-MF-06",
      "href": "LG-MF-06.html",
      "title": "Dimension formulas for congruence subgroups",
      "proof_locator": "Appendix A.4-A.6, (R11)-(R17)",
      "scope": "Arbitrary holomorphic line bundles on arbitrary compact connected Riemann surfaces: duality, meromorphic-section existence, Riemann-Roch and canonical degree.",
      "remaining_foundations": "Real calculus, continuous-integral existence, coordinate change of variables and Stokes in A.1.1 and R1. A.1.2-A.1.10 supplies smooth Hilbert/Sobolev completions and weak-test maps."
    },
    {
      "id": "LG-MF06-DIVISOR-DEGREE",
      "lesson": "LG-MF-06",
      "href": "LG-MF-06.html",
      "title": "Dimension formulas for congruence subgroups",
      "proof_locator": "Appendix A.6",
      "scope": "The divisor of every nonzero meromorphic function has total order zero; degrees add, negative-degree bundles have no holomorphic section, and holomorphic functions on compact connected X are constant."
    },
    {
      "id": "LG-MF06-GENUS-BRIDGE",
      "lesson": "LG-MF-06",
      "href": "LG-MF-06.html",
      "title": "Dimension formulas for congruence subgroups",
      "proof_locator": "Appendix A.7, (R18)-(R21), with LG-MF-03 Appendix A, Theorem S",
      "scope": "The analytic genus h0(K) equals the topological handle genus; hence topological genus one gives h0(K)=1 and deg K=0."
    },
    {
      "id": "LG-MF06-BOUNDARY-INTEGRATION",
      "lesson": "LG-MF-06",
      "href": "LG-MF-06.html",
      "title": "Dimension formulas for congruence subgroups",
      "proof_locator": "Appendix A.1, (R1)",
      "scope": "Integration over the oriented boundary equals integration of the exterior derivative. Applied in Theorem 3.0 to a meromorphic differential outside small pole disks, it proves the total-residue identity."
    },
    {
      "id": "LG-MF01-COMPLEX-ANALYSIS",
      "lesson": "LG-MF-01",
      "href": "LG-MF-01.html",
      "title": "The upper half-plane and the modular group",
      "proof_locator": "Lemma 0.2 and its Cauchy/Taylor/local-inverse consequences",
      "scope": "Taylor expansions, holomorphic identity and maximum principles, one-variable local holomorphic inverses, and Liouville from Cauchy estimates."
    },
    {
      "id": "LG-MF04-LAURENT",
      "lesson": "LG-MF-04",
      "href": "LG-MF-04.html",
      "title": "Modular forms, lattice functions and Eisenstein series",
      "proof_locator": "Lemma 0.1",
      "scope": "Laurent expansions, pole orders and residues."
    },
    {
      "id": "LG-MF05-MODULAR-J",
      "lesson": "LG-MF-05",
      "href": "LG-MF-05.html",
      "title": "The valence formula and the ring of modular forms of level one",
      "proof_locator": "Theorem 3.1",
      "scope": "Surjectivity and the modular coordinate theorem for j, used by Theorem 3.1."
    },
    {
      "id": "LG-MF02-CONGRUENCE-LIFT",
      "lesson": "LG-MF-02",
      "href": "LG-MF-02.html",
      "title": "Congruence subgroups, cusps and elliptic points",
      "proof_locator": "Lemma 1.1",
      "scope": "Integral determinant-one lifting for every level N, including composite N; used by Theorem 4.1."
    },
    {
      "id": "LG-MF04-05-NORMALIZATION",
      "lessons": [
        "LG-MF-04",
        "LG-MF-05"
      ],
      "proof_locator": "LG-MF-04 Theorem 4.2; LG-MF-05 Theorem 2.1",
      "scope": "Eisenstein constants and E4^3-E6^2=1728 Delta; used in (3.3)."
    },
    {
      "id": "LG-MF08-09-HECKE",
      "lessons": [
        "LG-MF-08",
        "LG-MF-09"
      ],
      "proof_locator": "LG-MF-08 (1.1)-(1.3), (1.8), Lemma 1.1 and Theorem 1.2; LG-MF-09 Theorems 2.3 and 3.1",
      "scope": "Earlier normalized lattice and Fourier Hecke operators and their modularity; used by Theorem 5.1 and Corollary 5.2."
    },
    {
      "id": "LG-MF05-WEIGHT-FOUR",
      "lesson": "LG-MF-05",
      "href": "LG-MF-05.html",
      "title": "The valence formula and the ring of modular forms of level one",
      "proof_locator": "Theorem 2.2",
      "scope": "M4=C E4 and E4 has constant coefficient one; used by the numerical normalization in (5.17)."
    }
  ],
  "remaining_requirements": [
    {
      "id": "LG-MF17-RR-FOUNDATIONS",
      "statement": "Supply or bind exact earlier proofs of the real-integration, weak-derivative and Hilbert-completion foundations of LG-MF-06 Appendix A.",
      "affected_results": [
        "LG-MF17-ANALYTIC-GENUS-ONE-CUBIC",
        "LG-MF17-ANALYTIC-GENUS-ONE-UNIFORMIZATION"
      ],
      "reason": "Those foundations are explicitly retained by the Riemann-Roch provider; the present cubic construction does not prove them."
    },
    {
      "id": "LG-MF17-ALGEBRAIC-GENUS-ONE-COMPARISON",
      "statement": "For every separately given smooth projective connected genus-one algebraic curve over C with a C-point, identify the analytic meromorphic coordinates of Theorem 3.0 with rational functions on that curve and prove that its analytic biholomorphism with the smooth Weierstrass cubic is an algebraic isomorphism.",
      "affected_results": [
        "Extension of the cubic model and its algebraic moduli interpretation to a separately given algebraic genus-one curve"
      ],
      "reason": "Theorem 3.0 proves an analytic biholomorphism; Proposition 3.2 concerns displayed Weierstrass cubics and does not supply this general comparison or GAGA."
    },
    {
      "id": "LG-MF17-WEIERSTRASS-ARBITRARY-FIELD",
      "statement": "Construct the appropriate Weierstrass model for a smooth projective connected genus-one curve with a rational point over an arbitrary field, retaining the characteristic-two and characteristic-three cases.",
      "reason": "The normalization y=dx/omega and the coefficient 4 proof here are over C; arbitrary-field construction is not proved."
    },
    {
      "id": "LG-MF17-WEIL-PAIRING-COMPARISON",
      "statement": "Compare the positive determinant analytic pairing (4.2), with its chosen primitive root, to the divisor-and-function algebraic Weil pairing, retaining sign conventions and arbitrary-field hypotheses.",
      "reason": "Section 4 proves the analytic convention used by Theorem 4.1; it does not prove the algebraic comparison."
    },
    {
      "id": "LG-MF17-ARITHMETIC-MODELS",
      "statement": "Supply algebraic modular-curve models over Q, reduction and Tate-module arguments with their actual base, coefficient and prime hypotheses.",
      "reason": "The lesson establishes analytic marked-curve bijections, rather than the arithmetic extensions."
    },
    {
      "id": "LG-MF17-FAMILIES-AND-CUSPS",
      "statement": "Prove the appropriate family representability or stack statement for level structures, including automorphisms and the compactified cusp fibers.",
      "reason": "Theorem 4.1 classifies analytic isomorphism classes and does not assert a universal family on every coarse quotient."
    }
  ],
  "normalizations": {
    "cubic": "y^2=4x^3-a x-b; O=[0:1:0]",
    "analytic_coordinates": "omega=dt near P; x=t^(-2)+O(t^2), y=dx/omega=-2t^(-3)+O(t)",
    "coefficient_scaling": "omega -> s omega implies (x,y,a,b) -> (s^(-2)x,s^(-3)y,s^(-4)a,s^(-6)b)",
    "lattice": "Lambda_tau=Z+Z tau, Im tau>0; g2=60G4, g3=140G6",
    "discriminant": "D=g2^3-27g3^2=(2pi)^12 Delta",
    "j": "j=1728 g2^3/D",
    "pairing": "Positive determinant in an oriented lattice basis",
    "hecke": "1/n quotient average = n^(k-1) index-n sublattice sum; point marking retains diamonds"
  },
  "figures": [
    {
      "path": "figures/genus-one-pole-bases-and-fibers.png",
      "source": "figures/genus-one-pole-bases-and-fibers.py",
      "license": "CC0-1.0",
      "caption_locator": "Figure 3.1",
      "proof_locators": [
        "Theorem 3.0 (G1)-(G4)",
        "Theorem 3.0 (G9)-(G11)"
      ],
      "scope": "Exact symbolic pole-order bases and regular, branch and infinite fibers of the degree-two map x; schematic, with no metric scale asserted."
    }
  ],
  "exercises": {
    "count": 6,
    "solutions": "Section 8, Solutions 1-6"
  },
  "references": [
    {
      "author": "C. Teleman",
      "title": "Riemann Surfaces",
      "year": 2003,
      "url": "https://math.berkeley.edu/~teleman/math/Riemann.pdf",
      "locators": [
        "Lectures 8-11"
      ]
    },
    {
      "author": "J. Voight",
      "title": "Quaternion Algebras",
      "year": 2021,
      "url": "https://jvoight.github.io/quat.html",
      "locators": [
        "Section 35.3",
        "Section 40.1.11"
      ]
    },
    {
      "authors": [
        "P. Deligne",
        "M. Rapoport"
      ],
      "title": "Les sch\u00e9mas de modules de courbes elliptiques",
      "year": 1973,
      "url": "https://publications.ias.edu/sites/default/files/Number22.pdf",
      "locators": [
        "Chapters II-IV",
        "Chapter VII"
      ],
      "role": "Arithmetic and family questions retained as separate proof requirements"
    },
    {
      "author": "K. Rubin",
      "title": "Elliptic Curves with Complex Multiplication and the Conjecture of Birch and Swinnerton-Dyer",
      "year": 1999,
      "url": "https://swc-math.github.io/aws/1999/99RubinCM.pdf",
      "locators": [
        "Section 2",
        "Theorem 2.3",
        "Proposition 2.6"
      ]
    },
    {
      "author": "S. D. Galbraith",
      "title": "The Weil pairing on elliptic curves over C",
      "year": 2011,
      "url": "https://www.math.auckland.ac.nz/~sgal018/pair-over-C.pdf",
      "locators": [
        "Sections 3-5"
      ],
      "role": "Analytic pairing convention and the separately retained algebraic comparison"
    }
  ]
}