Derivative of power series attached to Γ-transform of p-adic measures

Abstract

We extend the result of Anglès (2007) [1], namely f ′ ( T ; θ ) ≢ 0 ( mod p ) for the Iwasawa power series f ( T ; θ ) ∈ Z ¯ p 〚 T − 1 〛 . For the derivative D = T d d T , a numerical polynomial Q on Z p , and a prime π in Z ¯ p over p, we show that Q ( D ) f ( T ; θ ) ≡ 0 ( mod π ) if and only if Q ≡ 0 ( mod π ) i.e. Q ( x ) ≡ 0 ( mod π ) for all x ∈ Z p . This result comes from a similar assertion for the power series attached to the Γ-transform of a p-adic measure which is related to a certain rational function in Z ¯ p 〚 T − 1 〛 .