Computing $p$-adic $L$-functions of totally real number fields

Computing p-adic L-functions of totally real number fields

We prove new explicit formulas for the p-adic L-functions of totally real number fields and show how these formulas can be used to compute values and representations of p-adic L-functions.

INDUCTIVE CONSTRUCTION OF THE p-ADIC ZETA FUNCTIONS FOR NON-COMMUTATIVE p-EXTENSIONS OF TOTALLY REAL FIELDS WITH EXPONENT p

In this paper, we will construct the p-adic zeta function for a non-commutative p-extension F of a totally real number field F such that the finite part of its Galois group G is a p-group with exponent p. We first calculate the Whitehead groups of the Iwasawa algebra Λ(G) and its canonical Ore localization Λ(G)S by using Oliver-Taylor’s theory on integral logarithms. Then we reduce the main con...

p-adic unit roots of L-functions over finite fields

In this brief note, we consider p-adic unit roots or poles of L-functions of exponential sums defined over finite fields. In particular, we look at the number of unit roots or poles, and a congruence relation on the units. This raises a question in arithmetic mirror symmetry.

Perfect Forms over Totally Real Number Fields

A rational positive-definite quadratic form is perfect if it can be reconstructed from the knowledge of its minimal nonzero value m and the finite set of integral vectors v such that f(v) = m. This concept was introduced by Voronöı and later generalized by Koecher to arbitrary number fields. One knows that up to a natural “change of variables” equivalence, there are only finitely many perfect f...

p-ADIC PERIODS, p-ADIC L-FUNCTIONS, AND THE p-ADIC UNIFORMIZATION OF SHIMURA CURVES