Crystalline lifts Sample Clauses

Crystalline lifts. We already saw that for 2 ď k ď p . 1 we have that Symk´2 that 2 Fp is irreducible. Therefore, by Section 2.2.2, we have p Symk´2F2 < W (ρ) ðñ ρ is modular of weight k and level (N, p) = 1. This means, by definition, that there exists a modular form f of weight k and level N (coprime to p) such that ρ – ρf . Then the representation ρf |GQp gives a characteristic zero lift of ρ|GQp . By [Fal87] and [FJ95] the representation ρf |GQp is crystalline (since N is coprime to p) and has Xxxxx– Xxxx weights {k — 1, 0}. We get the implication p GQp Symk´2F2 < W (ρ) ùñ ρ| has a crystalline lift to characteristic zero with Xxxxx–Xxxx weights {k — 1, 0}. Twists by det on the Serre weight side correspond to twists by the mod p cyclotomic character on the representation side – we will see this again later. This means that W (εb b ρ) = {det b b V | V < W (ρ)}. Therefore, including twists in the above, we get the implication Va,b < W (ρ) ùñ ρ|GQp has a crystalline lift to characteristic zero with Xxxxx–Xxxx weights {a . 1, b}. The key idea in the area of the Serre weight conjectures is that this implication is an if and only if! Vaguely speaking the philosophy is as follows: the existence of a local crystalline lift of the prescribed Xxxxx– Xxxx weights should be explained by the fact that there exists a global lift to a representation ρ : Gal(Q/Q) → GL2(Qp) which is unramified almost everywhere and whose restriction to GQp gives such a local lift. Then by the Xxxxxxxx–Xxxxx and Xxxxxxxxx conjectures such representations come from modular forms of the required weight and a level coprime to p. In the case above (that is, the case of GL2 over Q) enough is known about the various conjectures above that we can indeed verify that the implication is an if and only if, but in greater generality we can use this observation as a guiding principle which allows us to give a definition of W (ρ) which generalises more easily. The new definition will be as follows.