2026/08/06 by Kazato Nada, Haruma Sasaki, Hiroki Sumida-Takahashi
Mathematics · #math.NT
arxiv created 2026/08/06 · arxiv updated 2026/08/07
Let p be an odd prime number and ζp a primitive p-th root of unity. By computing special arithmetic elements modulo prime ideals, we have investigated the Iwasawa invariants and exceptional pairs (p,χωpk) of Q(√(d),ζp). Here χ denotes the Dirichlet character associated to Q(√(d)), and ωp the Teichmüller character at p. First, we determine \varLambda-isomorphism classes of Iwasawa modules, and describe structural differences between exceptional and non-exceptional pairs in terms of unramified extensions outside p. Next, we report 15 new exceptional pairs in the range |d|<200 (resp.~|d|<10) and p <2, 000, 000 (resp.~p <30, 000, 000), including (p,k,d)=(28, 679, 999, 2, 284, 521,-8) for which the χωp1-k-part of the Iwasawa λ-invariant is equal to two.