{
  "id": "modularity-of-elliptic-curves-over-imaginary-quadratic-fields",
  "title": "Modularity of elliptic curves over imaginary quadratic fields",
  "description": "Every elliptic curve over every imaginary quadratic field is modular, with matching local parameters at every place, after OpenAI's 2026 manuscript. Cyclic covers of a twist branched at two points; integral homology over the group ring and Picard-Lefschetz transvections; congruences between character pieces; local constancy of reduction types; Hilbert irreducibility for a curve and its conjugate; three applications of the Caraiani-Newton lifting theorem; cyclic descent; complex multiplication.",
  "units": [
    {
      "id": "elliptic-curves-over-imaginary-quadratic-fields-and-their-automorphy",
      "title": "Elliptic curves over imaginary quadratic fields and their automorphy",
      "reader": "elliptic-curves-over-imaginary-quadratic-fields-and-their-automorphy.html",
      "source": "src/elliptic-curves-over-imaginary-quadratic-fields-and-their-automorphy.md",
      "source_sha256": "6ce2bd91fdb9c4b4c9d9bc30f51badc85d7fca5764e07b14e10b300bb8abcec0",
      "status": "author_self_checked"
    },
    {
      "id": "cyclic-covers-of-an-elliptic-curve-with-two-branch-points",
      "title": "Cyclic covers of an elliptic curve with two branch points",
      "reader": "cyclic-covers-of-an-elliptic-curve-with-two-branch-points.html",
      "source": "src/cyclic-covers-of-an-elliptic-curve-with-two-branch-points.md",
      "source_sha256": "92c16a633ad9bb7a0e7f80c4fdd0258d00fb43f6ba808dc0a46136437e2163c9",
      "status": "author_self_checked"
    },
    {
      "id": "two-degenerations-and-a-jacobian-of-compact-type",
      "title": "Two degenerations and a Jacobian of compact type",
      "reader": "two-degenerations-and-a-jacobian-of-compact-type.html",
      "source": "src/two-degenerations-and-a-jacobian-of-compact-type.md",
      "source_sha256": "42a276955b4aa1e800486c08b490d9b4001617ed39c2d7f25b6b7c2658d37f8a",
      "status": "author_self_checked"
    },
    {
      "id": "integral-homology-and-picard-lefschetz-transvections",
      "title": "Integral homology and Picard–Lefschetz transvections",
      "reader": "integral-homology-and-picard-lefschetz-transvections.html",
      "source": "src/integral-homology-and-picard-lefschetz-transvections.md",
      "source_sha256": "92b22bad94b52eb11e9632f3d0193a69ea9523e46dfcba21d487fc7ff4dff41e",
      "status": "author_self_checked"
    },
    {
      "id": "rank-two-systems-and-their-congruences",
      "title": "Rank-two systems and their congruences",
      "reader": "rank-two-systems-and-their-congruences.html",
      "source": "src/rank-two-systems-and-their-congruences.md",
      "source_sha256": "2e53ca1f67b3b3777a182fc05b1b53a119a376ca5f05b27a5d5e709c7d3ae5ac",
      "status": "author_self_checked"
    },
    {
      "id": "prescribing-the-local-reduction-type",
      "title": "Prescribing the local reduction type",
      "reader": "prescribing-the-local-reduction-type.html",
      "source": "src/prescribing-the-local-reduction-type.md",
      "source_sha256": "4bebcd37fbac59e98ad9725cb5427ff373d56da4f90633946050344399f5dd6b",
      "status": "author_self_checked"
    },
    {
      "id": "residual-irreducibility-and-one-specialization",
      "title": "Residual irreducibility and one specialization",
      "reader": "residual-irreducibility-and-one-specialization.html",
      "source": "src/residual-irreducibility-and-one-specialization.md",
      "source_sha256": "f45293e30ddd822e20018136afd9f52a54275b60235469ac80675395bc5a3966",
      "status": "author_self_checked"
    },
    {
      "id": "three-lifts-and-descent",
      "title": "Three lifts and descent",
      "reader": "three-lifts-and-descent.html",
      "source": "src/three-lifts-and-descent.md",
      "source_sha256": "69d9e42ea604f27fe291e4117a46b85a80b8f1556739d8632bcbb45ee4b53669",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "modularity lifting in outline (course Global Langlands, LG-GLOB-11)",
    "Tate modules, Weil-Deligne representations and Grothendieck's monodromy theorem, Frobenius elements and determination by traces, residual representations, the Abel-Jacobi uniformization, p-adic Hodge theory (course Galois representations, LG-GAL)",
    "the rank-two local Langlands correspondence with Steinberg twists (course The local Langlands correspondence, LG-LLC-06)",
    "complex representations of GL_2 and their Whittaker models (course Automorphic forms and representations of GL2)",
    "the Picard-Lefschetz formula at a node (course The Lefschetz trace formula, lesson Lefschetz pencils and vanishing cycles)",
    "the Riemann-Hurwitz formula and the Riemann existence equivalence (course The etale fundamental group)",
    "Hensel's lemma in Newton form (course Local fields, NT-LOC-03)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}