On the growth of even K-groups of rings of integers in p-adic Lie extensions

作者：Lim, Meng Fai^*

来源：Israel Journal of Mathematics, 2022, 249(2): 735-767.

DOI：10.1007/s11856-022-2324-4

• IWASAWA INVARIANTS
• CLASS-NUMBERS
• THEOREM
• RANKS

Let p be an odd prime number. In this paper, we study the growth of the Sylow p-subgroups of the even K-groups of rings of integers in a p-adic Lie extension. Our results generalize previous results of Coates and Ji-Qin, where they considered the situation of a cyclotomic DOUBLE-STRUCK CAPITAL Z(p)-extension. Our method of proof differs from these previous works. Their proof relies on an explicit description of certain Galois group via Kummer theory afforded by the context of a cyclotomic DOUBLE-STRUCK CAPITAL Z(p)-extension, whereas our approach is via considering the Iwasawa cohomology groups with coefficients in DOUBLE-STRUCK CAPITAL Z(p) (i) for i >= 2. We should mention that this latter approach is possible thanks to the Quillen-Lichtenbaum Conjecture which is now known to be valid by the works of Rost-Voevodsky. We also note that the approach allows us to work with more general p-adic Lie extensions that do not necessarily contain the cyclotomic DOUBLE-STRUCK CAPITAL Z(p)-extension, where the Kummer theoretical approach does not apply. Along the way, we study the torsionness of the second Iwasawa cohomology groups with coefficients in DOUBLE-STRUCK CAPITAL Z(p) (i) for i >= 2. Finally, we give examples of p-adic Lie extensions, where the second Iwasawa cohomology groups can have nontrivial mu-invariants.