Some quadratic cases Clause Samples

Some quadratic cases. First we consider an explicit example retrieved from [DDR16] which is further discussed in [DS17, Section 11.5]. !~ Example 2.1. Let F = Q(√5) and p = 3. We consider Example IIIb1 from [DDR16, Section 9]. In the example ρ : GF → GL2(F9) is irreducible and such that ρ|IF3 ω0 ∗ , 0 ω3 where ω0 is the level two fundamental character corresponding to some choice of τ0. By Appendix B we find that ρ is algebraically modular of weights ((2, 4), (0, —1)) and ((4, 4), (—1, —1)) By Theorem 1.3, we find that ρ has a crystalline lift of the same weights. Now upon twisting, by Lemma 4.8, Chapter 5 we can transfer to irregular weights and find ρ has a crystalline lift of weight ((3, 1), (0, —1)). Thus kmin(ρ, (0, —1)) = (3, 1). We discuss one further example, where the minimal weight turns out to be regular and equal the minimal algebraic weight.
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