5 research outputs found
Weber's class number problem and its variants
We survey Weber's class number problem and its variants in the spirit of
arithmetic topology; we recollect some history, present a relation to certain
units and generalized Pell's equation, and overview a study of the -adic
limits of class numbers in -towers together with numerical
investigation for knots and elliptic curves
Generalized Pell's equations and Weber's class number problem
We study a generalization of Pell's equation, whose coefficients are certain
algebraic integers. Let and for each . We study the -solutions of the
equation . By imitating the solution to the classical Pell's
equation, we introduce new continued fraction expansions for over
and obtain an explicit solution of the generalized Pell's
equation. In addition, we show that our explicit solution generates all the
solutions if and only if the answer to Weber's class number problem is
affirmative. We also obtain a congruence relation for the ratios of the class
numbers of the -extension over the rationals and show the
convergence of the class numbers in .Comment: 17 page
The -adic limits of class numbers in -towers
This article discusses variants of Weber's class number problem in the spirit
of arithmetic topology to connect the results of Sinnott--Kisilevsky and
Kionke. Let be a prime number. We first prove the -adic convergence of
class numbers in a -extension of a global field and a similar
result in a -cover of a compact 3-manifold. Secondly, we
establish an explicit formula for the -adic limit of the -power-th cyclic
resultants of a polynomial using roots of unity of orders prime to , the
-adic logarithm, and the Iwasawa invariants. Finally, we give thorough
investigations of torus knots, twist knots, and elliptic curves; we complete
the list of the cases with -adic limits being in and find the
cases such that the base -class numbers are small and 's are
arbitrarily large.Comment: 28 pages. new results on lim in Z and large nu in v2. minor
corrections in later version