Realizability problem for etale wild kernels of number fields

**Speaker: **Dr. Luca Caputo (UCD)

**Time:** 4:00PM

**Date:** Wed 18th February 2009

**Location:** Mathematical Sciences Seminar Room

**Abstract**

For a number field F, an integer i and a prime p, the i-th etale wild kernels WK_{2i}^{et}(F) is a cohomological generalization of the p-part of the classical wild kernel WK_2(F) (i.e. the subgroup of K_2(F) which is the kernel of Hilbert symbols). The triviality of WK_2(F) is equivalent to interesting arithmetical properties of F, so one is lead to study number fields whose etale wild kernels are trivial. If p is irregular, there exists i such that WK_{2i}^{et}(Q) is nontrivial. In this situation we will show that WK_{2i}^{et}(F) is nontrivial for any number field F containing a particular subextension of Q(mu_p). On the other hand, if something slightly stronger than WK_{2i}^{et}(Q)=0 holds, then every abelian p-group structure appears as WK_{2i}^{et}(F) of some number field F. The way these results are obtained is quite similar to that used to prove analogous results for class groups and Iwasawa modules (e.g. class field towers and genus theory).

(This talk is part of the K-Theory, Quadratic)

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