Modularity of elliptic curves over imaginary quadratic fields

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Modularity of elliptic curves over imaginary quadratic fields

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Read the geometric prime switch that proves every elliptic curve over every imaginary quadratic field is modular. For each link in the chain, say which prime it is congruent modulo and why, then follow automorphy from an auxiliary curve to E.

The statement and the gap

For any imaginary quadratic field K and any elliptic curve E/K, there is an automorphic representation π_E of GL₂(𝔸_K) with r_ℓ(π_E) ≃ H¹(E). The local parameters match at every place, and L(E/K,s) = L(π_E, s−½). Before this paper, Caraiani–Newton's lifting theorem needed residual hypotheses in characteristic 3 or 5. That left gaps, for example fields where X₀(15)(K) has positive rank [0:37].

The prime switch, built from geometry

Take a non-CM E, a large prime p with mod-p image containing SL₂(𝔽_p), n = 3p and F = K(μ_n). A genus-n cyclic cover of a twist B_t splits under rotation into rank-two pieces V_θ that each look like the cohomology of an elliptic curve. The trivial piece is B_t and an order-3 piece is a new elliptic curve A [1:20].

The chain is V_η ≡ (mod p) V_χ ≡ (mod 3) V_ψ ≡ (mod p) V_1 [2:20]. The rule: a root of unity of ℓ-power order reduces to 1 in characteristic ℓ, so the order-p part of a character dies mod p and the order-3 part dies mod 3. Freeness of the integral cohomology over the group ring turns equal residual characters into equal residual representations, even though p divides n [2:33]. Try the middle link yourself at [3:10].

Moving automorphy

A is modular using the prime 5 (ζ₅ ∉ F). Three lifts follow: at p, then at 3, then at p again, which reaches B over F. You then descend through cyclic prime-degree steps to K and untwist to get E [3:40]. Each lift needs absolute irreducibility. Vanishing cycles supply two opposite unipotents with entry 2 − λ − λ⁻¹ ≠ 0, and no line is fixed by both [4:24].

When you read a prime switch, ask two questions: which part of the character dies modulo which prime, and what keeps the residual image large [5:05].

Research preprints

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The quasi-Riemann hypothesis and zero-free regions

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Converse theorem for elliptic curves with 2-torsion

Hilbert's tenth problem over the rationals

Why Catalan's constant G is irrational

Goldfeld's conjecture and the 2∞-Selmer converse

Average rank of elliptic curve twists

Chowla–Elliott correlations via divisibility graphs

The Grothendieck–Teichmüller Lie algebra and the Deligne–Drinfeld conjecture

Milnor K-theory and field reconstruction mod p

Milnor K-theory and birational anabelian reconstruction

Bogomolov–Pop reconstruction: the uniform bound along a pencil

Fontaine–Mazur modularity at the prime 2

The dimension of the two-adic Hecke algebra

Pro-modularity at 2 without de Rham conditions

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Poisson-Dirichlet law for prime factors of p−1

Totient function fibers and smooth shifted primes

Independence of largest prime factors in consecutive integers

Ostmann's inverse Goldbach problem and prime indecomposability

Hecke eigensheaves with tame ramification

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Global Arthur parameters via cuspidal excursion data

Nilpotent orbit decomposition of cuspidal automorphic functions

Rationality of the canonical Arthur filtration

Hecke eigensheaves with ramification at marked points

Temperedness at ramified places for exceptional groups

Restricted geometric Langlands in positive characteristic

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Zilber–Pink for curves in A2: elliptic squares

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Log abundance for compact Kähler spaces

Log abundance for log canonical pairs in characteristic zero

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