2016/02/08 by Emmanuel Lecouturier · 18 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Combinatorics #Conjecture #Discrete mathematics #Field (mathematics) #Galois group #Galois module #Group (periodic table) #Logarithm #Mathematical analysis #Mathematics #Modulo #Physics #Prime (order theory) #Pure mathematics #Rank (graph theory) #math.NT
paper · pdf · doi:10.1112/jlms.12123
published in Journal of the London Mathematical Society 98(1), 35-58 (Wiley)
arxiv created 2016/02/08 · openalex publication_date 2018/03/28 · arxiv updated 2018/04/04 · openalex created_date 2021/02/01 · openalex updated_date 2026/07/29
Let p ⩾ 5 and N be prime numbers such that p divides N − 1 . We estimate the p-rank of the class group of Q ( N 1 / p ) in terms of the discrete logarithm, with values in Z / p Z , of certain units. Using the Gross–Koblitz formula and identities on the N-adic Gamma function, we explicitly compute these logarithms. In particular, we give a new proof which does not use modular forms of a result of Calegari and Emerton. Using the same method, we prove a special case of a twisted form of a conjecture of Gross about the relation between some Stickelberger element and the Galois structure of the class group of the cyclotomic field Q ( ζ p , ζ N ) . A special case of our formulae, for which we do not have an elementary proof, is the following: Assume there are some integers a, b such that N = ( a p + b p ) / ( a + b ) . Then ( a + b ) · ( ∏ k = 1 ( N − 1 ) / 2 k 8 k ) is a pth power modulo N.