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Let 
$F_{\pi }$ be a finite Galois-algebra extension of a number field F, with group G. Suppose that 
$F_{\pi }/F$ is weakly ramified and that the square root 
$A_\pi $ of the inverse different 
$\mathfrak {D}_{\pi }^{-1}$ is defined. (This latter condition holds if, for example, 
$|G|$ is odd.) Erez has conjectured that the class 
$(A_\pi )$ of 
$A_\pi $ in the locally free class group 
$\operatorname {\mathrm {Cl}}(\mathbf {Z} G)$ of 
$\mathbf {Z} G$ is equal to the Cassou–Noguès–Fröhlich root number class 
$W(F_{\pi }/F)$ associated with 
$F_\pi /F$. This conjecture has been verified in many cases. We establish a precise formula for 
$(A_\pi )$ in terms of 
$W(F_{\pi }/F)$ in all cases where 
$A_\pi $ is defined and 
$F_\pi /F$ is tame, and are thereby able to deduce that, in general, 
$(A_\pi )$ is not equal to 
$W(F_\pi /F)$.
${K}_{\textbf {3}}$ for imaginary quadratic fields
We develop methods for constructing explicit generators, modulo torsion, of the 
$K_3$-groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic 
$3$-space or on direct calculations in suitable pre-Bloch groups and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite 
$K_3$-group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for 
$ K_3 $ of any field, predict the precise power of 
$2$ that should occur in the Lichtenbaum conjecture at 
$ -1 $ and prove that this prediction is valid for all abelian number fields.
$p$-ADIC COHOMOLOGY
We investigate the Galois structures of 
$p$-adic cohomology groups of general 
$p$-adic representations over finite extensions of number fields. We show, in particular, that as the field extensions vary over natural families the Galois modules formed by these cohomology groups always decompose as the direct sum of a projective module and a complementary module of bounded 
$p$-rank. We use this result to derive new (upper and lower) bounds on the changes in ranks of Selmer groups over extensions of number fields and descriptions of the explicit Galois structures of natural arithmetic modules.
