Fontaine–Mazur modularity at the prime 2

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Fontaine–Mazur modularity at the prime 2

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Recursos

Pick any lemma from the proof that odd, regular 2-adic Galois representations are modular, and say which job it does: getting the target to occur in completed forms, turning that into a classical vector, or bringing the result down to ℚ.

The theorem and why 2 was hard

Here is Theorem 1.1 [0:57]. Let r : G_ℚ → GL₂(ℚ̄₂) be continuous and irreducible. Suppose it is odd, unramified outside finitely many primes, and de Rham at 2 with distinct Hodge–Tate weights. Then r is a Tate twist of the representation attached to a classical cuspidal eigenform. The theorem puts no condition on the residual image. That matters because at 2 the reduction can be scalar or reducible, and in those cases Serre's conjecture gives no modular seed [0:17]. Virtually solvable representations are handled separately at the start.

The three-stage route

Occurrence [1:52]: put the target inside a pro-modular trace locus over a solvable, totally real field F that is split completely at 2. The first step is propagation from a characteristic-2 curve [2:41]. It imposes no local condition at 2. The second step is connectedness [3:18]: marked local lines force a strict Selmer loss, so the reducible locus is too small to disconnect the fiber. A modular seed and the target then lie in one connected piece.
Classicality [3:57]: occurrence does not give you a classical form; this is a separate step. In the reducible local case, a family over a one-dimensional weight space has a dense set of ordinary points with Hodge–Tate weights {0,1}. Thorne's theorem proves those points modular. Specialization through a Tate module and a polynomial-orbit argument then give a locally algebraic vector at the target. Ordinarity is used only for the auxiliary family, never assumed for r.
Descent [4:59]: go down prime-degree cyclic steps. A Hecke-character twist fixes any finite-character mismatch along the way. Only an actual cuspidal packet is descended.

Use it

For each lemma, ask: does it produce occurrence, extract a classical vector, or move down the solvable tower? Practice it at [5:28], then try it on Sections 5–8 of the paper [5:57].

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Selmer converse and sums of two cubes

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Zero-free regions and the quasi-Riemann hypothesis

The quasi-Riemann hypothesis and zero-free regions

Landau–Siegel zeros: how a determinant could forbid a real zero near 1

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

Beating the parity obstruction for primes p with μ(p − 1) = 1

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

Frobenius structures on tame Hecke eigensheaves

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

Equidistribution of prime-degree torus packets

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Bounded Stein degree for log Calabi–Yau boundaries

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Log abundance for compact Kähler spaces

Log abundance for log canonical pairs in characteristic zero

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