2019/04/18 by Mabud Ali Sarkar, Sarkar, Mabud Ali, Absos Ali Shaikh +1
Mathematics · #11F85 #11R11 #11R18 #11S15 #11Y40 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT) #math.NT #msc:11F85 #msc:11R11 #msc:11R18 #msc:11S15 #msc:11Y40
paper · pdf · doi:10.48550/arxiv.1904.09850
published as Houston Journal of Mathematics, 2024 · 31 pages. An Erratum has been accepted by the journal to correct an error identified by an independent reviewer following publication. Comments are welcome
openalex publication_date 2019/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28 · arxiv created 2026/07/31 · arxiv updated 2026/08/03
The p-adic logarithm appears in many places in number theory. Therefore, a comprehensive description of the image of the p-adic logarithm would be beneficial. In particular, it is important to figure out the image of 1 + \mathfrakmK, where K denotes an algebraic extension of ℚp and \mathfrakmK represents its maximal ideal. If the ramification index of K is strictly less than p-1, then it is known that the p-adic logarithm serves as a bijection from 1+\mathfrakmK to \mathfrakmK. If the ramification index is equal to or greater than p-1, then the p-adic logarithm is no longer a bijection, and the situation is more complicated. Our main result is the computation of logp(1+\mathfrakmK) in two distinct cases: first, when K=ℚp(ζp), a totally ramified p-cyclotomic extension of ℚp with a ramification index of p-1; and second, when K is a quadratic extension of ℚ2, characterised by a ramification index of either 1 or 2. As an application, we compute the normalised p-adic regulator of the number field ℚ(ζp) by utilising the image of the p-adic logarithm on the units.