The minimal weight conjecture Sample Clauses

The minimal weight conjecture. In [DS17], Xxxxxxx and Xxxxxx conjecture the existence of a minimal weight kmin, depending on ρ and a fixed l (Conjecture 7.3.1). In this brief section we take this conjecture and add the algebraic modularity perspective to it. In the following section we assume this conjecture, and determine kmin in some small cases. Conjecture 1.7. Let l ∈ ZΣ, then there exists kmin = kmin(ρ, l) ∈ Ξ+ such min (1) ρ is geometrically modular of weight (k, l) if and only if k ≥Ha kmin; min (2) if k ∈ Ξ+ , then ρ|G has a crystalline lift of weight (kτ , lτ )τ∈Σ for all Fv v v|p if and only if k ≥Ha kmin; (3) if in addition k is regular, then ρ is algebraically modular of weight (k, l) if and only if k ≥Ha kmin.