Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 145 (Iwasawa Algebras)

A filtration on a group is a function $\omega :G\to \mathbb{R}\cup \{\infty \}$ satisfyingIt is a p-valuation when it is separated and, for $x\ne 1$,

Let $p$ be odd and letFor $g=1+A\ne 1$, define $\omega (g)={\min }_{i,j}{v}_p(A_{ij})$. Matrix multiplication and the identitygive the two filtration inequalities. Since $\omega (g)\ge 1>1/(p-1)$, only the p-power condition remains. The binomial theorem givesThe first term has valuation $1+\omega (g)$, while every other term has strictly larger valuation because $p$ is odd and $\omega (g)\ge 1$. Hence $\omega (g^p)=\omega (g)+1$, so this is a p-valuation.

Finally, if ${\omega }_1,{\omega }_2$ are p-valuations, put $\omega =\min ({\omega }_1,{\omega }_2)$. Taking minima preserves both filtration inequalities and the strict lower bound, whileIf $\omega (g)=\infty$, both original valuations force $g=1$. Therefore the pointwise minimum of two p-valuations is again a p-valuation.

For a separated filtered group defineThe associated graded Lie algebra of a filtered group isEach quotient is abelian because $[G_{\lambda },G_{\lambda }]\subseteq G_{2\lambda }\subseteq G_{\lambda +}$. If $x\in G_{\lambda }$ and $y\in G_{\mu }$, setThe standard commutator identities $[xy,z]=[x,z{]}^y[y,z]$ and $[x,yz]=[x,z][x,y{]}^z$ prove well-defined bilinearity, while the Hall-Witt identity gives the Jacobi identity. If $\omega ([x,y])=\omega (x)+\omega (y)$, this bracket has nonzero initial form, so the graded Lie algebra is nonabelian.

For a p-valuation, defineThe p-power axiom and the Hall-Petrescu formula make this well defined and turn each homogeneous component into part of a graded ${\mathbb{F}}_p[t]$-module. The leading-term congruence $[x^p,y]\equiv [x,y{]}^p$ modulo terms of valuation greater than $\omega (x)+\omega (y)+1$ makes the bracket ${\mathbb{F}}_p[t]$-bilinear.

For the given upper-triangular group, write an element of degree $n$ to leading order asLet $X_n,Y_n$ denote the classes with $(\alpha ,\beta )=(1,0),(0,1)$. Matrix commutators giveand pth powers give $t{X}_n=X_{n+1}$ and $t{Y}_n=Y_{n+1}$. Thus

Give the generator $p$ weight one and $g-1$ weight $\omega (g)$. For $\lambda \ge 0$, let ${\mathbb{Z}}_p[G{]}_{\lambda }$ be the ${\mathbb{Z}}_p$-span ofThe identitiesandtogether with the filtration inequalities show that the definition is independent of how an element is expanded and thatCompleteness and an Ordered basis of a complete p-valued group give separatedness through unique noncommutative power-series expansions. This is the Lazard filtration on a group algebra.

Multiplication by $p$ induces a central degree-one element $t$, so $\mathrm{gr}{\mathbb{Z}}_p[G]$ is a graded ${\mathbb{F}}_p[t]$-algebra. Sending the initial form of $g$ to the initial form of $g-1$ respects the Lie bracket, since the displayed algebra commutator has leading term represented by $[g,h]-1$. The universal property of the universal enveloping algebra therefore gives the Lazard enveloping-algebra mapIt is surjective because the defining filtered pieces are spanned by products of $p$ and the elements $g-1$.

For the principal congruence subgroup of ${\mathrm{SL}}_2({\mathbb{Z}}_p)$, let $e=E-1$, $f=F-1$, and $h=H-1$. In the group algebra,The factor $FE$ is congruent to one in filtration degree zero. Direct matrix multiplication, now applied to the group commutator, givesConsequently $ef-fe\equiv H^p-1\equiv p(H-1)=ph$ modulo ${\mathbb{Z}}_p[G{]}_3$. The same argument starts fromUsing $(1+p{)}^{-1}=1-p+p^2+O(p^3)$ in the two matrix commutators givesSince the prefactors $EH,FH$ may again be replaced by one at this precision, we obtain
Therefore

Let $Z=Z(G)$. The inclusion ${\mathbb{F}}_p[[Z]]\subseteq Z({\mathbb{F}}_p[[G]])$ is immediate. For the reverse inclusion, project a central element to every finite group algebra ${\mathbb{F}}_p[G/U]$, where $U$ ranges over open normal subgroups. Its coefficients are constant on conjugacy classes. Compatibility as $U$ shrinks shows that a nonzero coefficient can persist only on an element with finite conjugacy class in $G$: an infinite conjugacy orbit splits into arbitrarily large p-power collections in finer quotients, whose fibre sums vanish in characteristic $p$.

In a p-valued group, an element with finite conjugacy class is central. Indeed, its centralizer is open, so some p-power of every element centralizes it; the p-valuation and the leading commutator identity then force the original commutators to vanish. Thus the finite-conjugacy center is $Z$, and the compatible finite-quotient expansions are supported on $Z$. This proves the Center of an Iwasawa algebra of a complete p-valued group:

For $G=1+p{M}_2({\mathbb{Z}}_p)$, a matrix commuting with every elementary principal-congruence matrix commutes with the full matrix algebra and is scalar. HenceConsequently

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