Fontaine-Mazur Conjecture

The Fontaine-Mazur conjecture predicts that irreducible -adic Galois representations of number fields satisfying suitable finiteness and local geometric conditions arise from algebraic geometry, allowing Tate twists. Its regular odd two-dimensional modularity formulation over concerns continuous irreducible representations unramified outside finitely many primes, de Rham at , and with distinct Hodge-Tate weights. Oddness means that the determinant at complex conjugation is . The predicted representation is a Tate twist of the -adic representation associated with a classical cuspidal eigenform.