Displaying similar documents to “ p -adic L -functions attached to characters of p -power order”

Congruences involving the Fermat quotient

Romeo Meštrović (2013)

Czechoslovak Mathematical Journal

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Let p > 3 be a prime, and let q p ( 2 ) = ( 2 p - 1 - 1 ) / p be the Fermat quotient of p to base 2 . In this note we prove that ∑ k = 1 p - 1 1 k · 2 k ≡ q p ( 2 ) - p q p ( 2 ) 2 2 + p 2 q p ( 2 ) 3 3 - 7 48 p 2 B p - 3 ( mod p 3 ) , which is a generalization of a congruence due to Z. H. Sun. Our proof is based on certain combinatorial identities and congruences for some alternating harmonic sums. Combining the above congruence with two congruences by Z. H. Sun, we show that q p ( 2 ) 3 ≡ - 3 ∑ k = 1 p - 1 2 k k 3 + 7 16 ∑ k = 1 ( p - 1 ) / 2 1 k 3 ( mod p ) , which is just a result established by K. Dilcher and L. Skula. As another application, we obtain a congruence for the sum ∑ k = 1 p - 1 1 / ( k 2 · 2 k ) modulo p 2 that also generalizes...

On the characteristic power series of the U operator

Fernando Q. Gouvêa, Barry Mazur (1993)

Annales de l'institut Fourier

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We show that the coefficients of the characteristic power series of Atkin’s U operator acting on overconvergent p -adic modular forms of weight k vary p -adically continuously as functions of k . Are they in fact Iwasawa functions of k ?