p-adic valuation of harmonic sums and their connections with Wolstenholme primes

Wolstenholme Type Theorem for Multiple Harmonic Sums

for complex variables s1, . . . , sl satisfying Re(sj) + · · · + Re(sl) > l − j + 1 for all j = 1, . . . , l. We call |s| = s1 + · · · + sl the weight and denote the length by l(s). The special values of multiple zeta functions at positive integers have significant arithmetic and algebraic meanings, whose defining series (1.1) will be called MZV series, including the divergent ones like ζ(. . ....

SUPERSINGULAR PRIMES AND p-ADIC L-FUNCTIONS

We discuss the problem of finding a p-adic L-function attached to an elliptic curve with complex multiplication over an imaginary quadratic field K, for the case of a prime where the curve has supersingular reduction. While the case of primes of ordinary reduction has been extensively studied and is essentially understood, yielding many deep and interesting results, basic questions remain unans...

On Wieferich primes and p-adic logarithms∗†

We shall make a slight improvement to a result of p-adic logarithms which gives a nontrivial upper bound for the exponent of p dividing the Fermat quotient x − 1.

On Sums of Powers of the p-adic Valuation of n!

T-ADIC L-FUNCTIONS OF p-ADIC EXPONENTIAL SUMS

The T -adic deformation of p-adic exponential sums is defined. It interpolates all classical p-power order exponential sums over a finite field. Its generating L-function is a T -adic entire function. We give a lower bound for its T -adic Newton polygon and show that the lower bound is often sharp. We also study the variation of this L-function in an algebraic family, in particular, the T -adic...