Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 27 / 2

Let $u=(u_n)\in U_{\infty }$. The Coleman interpolation theorem for cyclotomic local units gives a unique unit series $f_u\in {\mathbb{Z}}_p[[T]{]}^{\times }$ withSince the entries are principal units, $f_u(0)\equiv 1(\mathrm{mod}p)$. The existence statement can be seen from the norm operator as follows. Choose integral unit polynomials $F_j$ interpolating $u_j$, possible because ${\mathcal{O}}_{K_j}={\mathbb{Z}}_p[{\zeta }_j-1]$. The congruence $NF\equiv F(\mathrm{mod}p)$ and Coleman norm contraction imply that $N^j{F}_{2j}$ differs from $N^{2j-m}{F}_{2j}$ by a factor in $1+p^{j+1}R$ for each fixed $m\le j$. The second series evaluates at ${\zeta }_m-1$ to $u_m$ by repeated finite-level norms. Compactness in the $(p,T)$-adic topology gives a subsequential limit interpolating all the entries. A nonzero difference between two interpolating integral series would have infinitely many zeros in the open unit disk, contradicting the Weierstrass preparation theorem. Applying the norm identity at all levels then proves norm fixation.

Set $D=(1+T)d/dT$. For $k\ge 1$, define the cyclotomic higher logarithmic derivative byThe quotient $D{f}_u/f_u$ belongs to ${\mathbb{Z}}_p[[T]]$, and $D$ preserves that ring, so this indeed takes values in ${\mathbb{Z}}_p$. Equivalently it is $D^k\log f_u$ at zero, where the formal logarithm may be normalized by replacing $f_u$ by $f_u/f_u(0)$. There is no factorial division in this definition. Uniqueness of the Coleman power series gives $f_{uv}=f_u{f}_v$, so ${\delta }_k(uv)={\delta }_k(u)+{\delta }_k(v)$.

Let ${\chi }_{\mathrm{cyc}}:\mathrm{Gal}(K_{\infty }/{\mathbb{Q}}_p)\to {\mathbb{Z}}_p^{\times }$ be the cyclotomic character, with $\sigma ({\zeta }_n)={\zeta }_n^{{\chi }_{\mathrm{cyc}}(\sigma )}$. If $b={\chi }_{\mathrm{cyc}}(\sigma )$, interpolation and uniqueness giveThe binomial series is integral for $b\in {\mathbb{Z}}_p$, and for any formal series $g$ the chain rule givesIteration, followed by evaluation at zero, therefore provesThus these additive Coates–Wiles homomorphisms have cyclotomic weight $k$.

For $a$ prime to $p$, putFor positive $a$, $b_a(X)=X^{1-a}{\sum }_{j=0}^{a-1}{X}^{2j}$ is a Laurent polynomial. For negative $a$, use $b_a=-b_{-a}$. Hence it is regular at $X=1$ with value $a$, and $f_a\in {\mathbb{Z}}_p[[T]{]}^{\times }$ has constant coefficient $a^{p-1}\equiv 1(\mathrm{mod}p)$. Evaluation at ${\zeta }_n$ shows that $c_n(a)$ is a unit and reduces to one in the residue field, so $c_n(a)\in U_n$.

Because $p$ is odd, ${\prod }_{\xi \in {\mu }_p}\xi =1$. Both maps $\xi \mapsto {\xi }^2$ and $\xi \mapsto {\xi }^{2a}$ permute ${\mu }_p$, and thereforeThis rational-function identity also holds for negative $a$ and extends across the removable point $X=1$. It implies $N{f}_a=f_a$. Part 1(b) now proves all the required norm compatibilities, and $c(a)\in U_{\infty }$, with Coleman series $f_a$.

Use the additive formal coordinate $z=\log (1+T)$, in which $D=d/dz$. ThenIts logarithm after normalization by $a^{p-1}$ is even, so all odd derivatives at zero vanish. For the even derivatives, the hyperbolic cotangent generating function for Bernoulli numbers givesHere the Bernoulli numbers use $B_1=-1/2$. Differentiating $k-1$ more times yields the answer for every index:These rational expressions lie in ${\mathbb{Z}}_p$ because they equal the integral logarithmic-derivative definition in part (a). In particular ${\delta }_1=0$; substituting $B_1$ into the even-index formula would give an incorrect answer. For $a=\pm 1$ the sequence is one and every derivative vanishes, as the formula also shows.