Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 145 / 5 / Solution

A Qp-Banach algebra is a ${\mathbb{Q}}_p$-algebra with a complete non-Archimedean submultiplicative norm compatible with the p-adic absolute value. A p-valued group is p-saturated when it is complete and every $g$ with $\omega (g)>p/(p-1)$ has a pth root.

In the completed rational Iwasawa algebra $\widehat{{\mathbb{Q}}_{p}G}$, the filtration of $g-1$ is $\omega (g)>1/(p-1)$. Since $v_p(n)=O(\log n)$, the terms ofhave filtrations tending to infinity, so the series converges. The Baker--Campbell--Hausdorff formula expresses $\log (gh)$ in terms of $\log g$ and $\log h$, and its commutator expansion expresses $\log ([g,h])$ with leading term $[\log g,\log h]$. Completeness and p-divisibility allow the higher terms to be removed successively. Integer powers give $\log (g^n)=n\log g$, and continuity extends scalar multiplication to ${\mathbb{Z}}_p$. Thus the Lazard logarithm of a p-saturated group shows that $\log (G)$ is a ${\mathbb{Z}}_p$-Lie subalgebra.

The group-like coproduct identity $\Delta (g)=g\otimes g$ impliesso each logarithm is primitive. If $(g_1,\dots ,g_d)$ is an ordered basis, their initial forms are linearly independent. Conversely, for a primitive element, its lowest initial form must be linear rather than a product; subtracting a ${\mathbb{Q}}_p$-linear combination of the $\log (g_i)$ raises its filtration. Iteration and completeness leave zero. Hence the primitive elements of a completed rational Iwasawa algebra satisfy

For the upper-triangular group, ordinary matrix logarithms givewith the quotient interpreted as $1$ at $a=0$. For odd $p$, both $a\mapsto \log (1+a)$ and $a\mapsto \log (1+a)/a$ are respectively a bijection $p{\mathbb{Z}}_p\to p{\mathbb{Z}}_p$ and a unit-valued function. Therefore