Results on modularity over imaginary quadratic fields and their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Irreducibility from the residual representation — Lemma 1.1. An l-adic representation whose residual representation is absolutely irreducible is absolutely irreducible, hence semisimple and determined by its Frobenius characteristic polynomials.
• Modularity over imaginary quadratic fields — Theorem 3.1. Every elliptic curve E over an imaginary quadratic field K has an isobaric automorphic representation of GL_2 over K, weight zero and trivial central character, realizing H^1(E) at every prime with matching local parameters, and L(E/K, s) = L(pi_E, s - 1/2). Proved in the eighth lesson, using the named results N1 to N10.
• The Kummer function — Lemma 2.1. On the twist B_t there is a unique function f_t with divisor (S) - (-S) - n(R) + n(-R) and f_t(O) = 1, where R = (t, 1) and S = [n]R; it satisfies f_t o [-1] = 1/f_t.
• Cyclic covers of genus m — Proposition 3.1. For m dividing n, the cover Y^m = f_t is geometrically irreducible of degree m, totally ramified over S and -S and unramified elsewhere, and has genus m.
• Fixed points of the involution — Proposition 5.2. The involution w(Z, Y) = (-Z, 1/Y) has exactly four fixed points, one above each 2-torsion point, so C_m/w has genus (m - 1)/2; for m = 3 it is a curve of genus one.
• A model of compact type — Proposition 2.3. At the origin of the parameter curve, one blow-up gives a regular semistable model whose special fibre is a rational curve with m copies of E attached, permuted cyclically by the rotations.
• Torsion near the origin — Corollary 2.7. For every N, the N-torsion of the Jacobians of the covers near the origin forms a finite etale group scheme with fibre E^m[N] at the origin, compatibly with the rotations. Uses Raynaud's theorem (named result N6).
• The node equation — Lemma 3.1. When [n]R tends to a nonzero 2-torsion point, the cover acquires m nodes, each with local equation XU = -2nt in the additive parameter t.
• A free module over the group ring — Proposition 3.1. The first homology and cohomology of C_m are free of rank two over the integral group ring of mu_m, for every fibre; hence the l-adic cohomology is free over Z_l[mu_m] for every prime l, including l dividing m.
• Picard-Lefschetz transvections — Theorem 5.1. In one integral basis the monodromy contains two transvections whose off-diagonal entries in the group ring are +-(2 - u - 1/u), u a generator of the deck group.
• Rank-two systems — Proposition 4.2. Each character piece V_theta of H^1(C_n) is two-dimensional with determinant the inverse cyclotomic character, potentially semistable with Hodge-Tate weights 0 and 1, and part of a compatible system.
• The congruence chain — Corollary 6.2. With chi = eta psi of orders 3p, 3, p: V_eta = V_chi mod p, V_chi = V_psi mod 3, V_psi = V_1 mod p, with V_eta = H^1(A_t) and V_1 = H^1(B_t).
• Local constancy of the semistable type — Proposition 3.2. In a p-adic family of abelian varieties whose torsion forms finite etale families, the toric rank and the Newton polygon of nearby members equal those of the special member over a fixed finite extension.
• Full monodromy in the toric case — Lemma 4.1. An abelian variety of dimension g with potentially totally toric reduction has monodromy of rank g at every coefficient prime, including the residue characteristic.
• The trivial character — Lemma 2.1. If G_K acts on E[p] through a group containing SL_2(F_p), then H^1(B_t) mod p is absolutely irreducible and decomposed generic over F = K(mu_3p), for every parameter t.
• Specialization with local conditions — Lemma 3.2. A connected Galois cover of an open subset of affine space over Q has a rational point in prescribed p-adic open sets with surjective specialization. Uses the counting form of Hilbert's irreducibility theorem (named result N5).
• One specialization — Proposition 4.1. There is a parameter t in K, inside prescribed local open sets, for which all residual representations of the chain are absolutely irreducible and decomposed generic and A_t has no complex multiplication.
• The three lifts — Proposition 2.1. H^1(B_t) over F is modular: automorphy passes from H^1(A_t) along the congruences at p, 3 and p by the Caraiani-Newton lifting theorem (named result N1).
• Cyclic descent — Lemma 3.2. For a cyclic extension of CM fields of prime degree, a representation that is automorphic and absolutely irreducible on the larger field is automorphic over the smaller one. Uses base change and strong multiplicity one (named result N3).

Reading order

Eight lessons, to be read in order. The first states the theorem and the automorphy inputs. The second constructs the cyclic covers, the third their two degenerations, and the fourth their integral homology and monodromy. The fifth splits the cohomology into rank-two systems and proves the congruence chain. The sixth prescribes the local reduction types, the seventh chooses one parameter with all residual properties, and the eighth moves automorphy along the chain and descends to K. The deep automorphy inputs (the theorems of Caraiani and Newton, Galois representations over CM fields, base change) and several standard results on abelian varieties and p-adic Hodge theory are used without proof, as the named results N1 to N10.