Modularity of elliptic curves over imaginary quadratic fields

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.

Eight lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Elliptic curves over imaginary quadratic fields and their automorphy
2. Cyclic covers of an elliptic curve with two branch points
3. Two degenerations and a Jacobian of compact type
4. Integral homology and Picard–Lefschetz transvections
5. Rank-two systems and their congruences
6. Prescribing the local reduction type
7. Residual irreducibility and one specialization
8. Three lifts and descent

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Comparison with the topological fundamental group over the complex numbers · Descent and the étale fundamental group (Integral-descent supporting reading)
• The étale fundamental group · Descent and the étale fundamental group (Integral-descent supporting reading)
• Lefschetz pencils and vanishing cycles (ℓ-adic cohomology and trace formulas)
• Frobenius elements and determination by traces · Galois representations
• Galois representations of weight-two newforms · Galois representations
• p-adic Hodge theory and the Fontaine–Mazur conjecture · Galois representations
• Residual representations and Serre's modularity theorem · Galois representations
• The ℓ-adic Tate module of an elliptic curve · Galois representations
• Weil–Deligne representations and Grothendieck’s monodromy theorem · Galois representations
• Modularity lifting: the Taylor–Wiles method in outline · Global Langlands conjectures and functoriality
• The rank-two correspondence: principal series, Steinberg twists and characters · The local Langlands correspondence for GL_n
• Hensel's lemma, squares and roots of unity in p-adic fields · Local fields
• Real and complex representations: weights, gamma factors and classical forms · Automorphic forms and representations of GL(2)
• Galois representations: Tate modules, Weil–Deligne representations, Frobenius elements and p-adic Hodge theory (course Galois representations); the local Langlands correspondence for GL₂; Whittaker models for GL₂(C); the Picard–Lefschetz formula (course The Lefschetz trace formula); the Riemann–Hurwitz formula and Riemann existence (course The étale fundamental group); Hensel's lemma (course Local fields).