Determination of certain mod p Galois representations using local constancy

Let p ≥ 5 be a prime and k be an integer in [ 2 p + 2 , p 2 - p + 3 ] . Let b be in [ 2 , p ] such that k - 2 ≡ b mod p - 1 and c : = k - 2 - b p - 1 . We give an explicit radius of local constancy in the weight space of the mod p reduction of the two dimensional crystalline representations V k , a p of Gal ( ℚ ¯ p / ℚ p ) , where the slope ν ( a p ) is constrained to be in ( 1 , c ) and non-integral. We use the mod p local Langlands correspondence for GL 2 ( ℚ p ) to compute the mod p reductions explicitly under additional conditions on the slope. We show that the reduction in the disk depends only on k and ⌊ ν ( a p ) ⌋ . As an application, we obtain explicit mod p reductions at many new values of k and a p . We also obtain an explicit radius of local constancy in the a p space (for a fixed k as above) which is bigger than the explicit radius given in a result of Berger.