Another look at the index formulas of cyclotomic number theory
The author considers a full cyclotomic field \(K={\mathbb{Q}}(e^{2\pi i/f})\) and introduces a module \(\mathcal L\) over \({\mathbb{Z}}[G]\), where \(G\) is the Galois group of \(K\) over \(\mathbb{Q}\). This module combines in a beautiful way the information afforded by cyclotomic units and the Stickelberger ideal. More concisely: There is an explicit canonical module \(\mathcal M\) containing \(\mathcal L\); the index \([{\mathcal M}:{\mathcal L}]\) equals the class number \(h_K\) (let me call this result ``Anderson's index formula''); and one can extract both the ``Stickelberger quotient'' and the quotient ``units modulo cyclotomic units'' from the pair of modules \(\mathcal M\) and \(\mathcal L\). The author proves his index formula by means of Dedekind's residue formula and a very elegant calculation of this residue which avoids using \(L\)-functions explicitly. The link with universal distributions, and with Sinnott's index formulas for \(h^+\) and \(h^-\), is lucidly explained. It should be mentioned that the ``correcting'' powers of 2 in Sinnott's index formulas arise from cohomology which comes in automatically when one takes plus parts and minus parts, and in this respect Anderson's index formula is simpler, not containing any correction factors at all. The idea behind the construction of \(\mathcal L\) is the following: \(\mathcal L\) is the span of all \(l(r):=-\log \bigl(1-1\otimes e^{2\pi ir/f}\bigr)\) with \(1\leq r \leq f-1\). (Here one works in \(V={\mathbb{R}}\otimes K\), and log\((1-z)\) is defined by the standard series \(-\sum_{n\geq1}z^n/n\).) From this one already ``sees'' the cyclotomic units. The Stickelberger ideal comes in at the back door: If we map \(V\) to \({\mathbb{C}}G\) via \(1\otimes x \mapsto \sum_{(a,f)=1}\sigma_a(x)\cdot \sigma_a^{-1}\), then \(l(r)-l(-r)\) maps to \(2\pi i\) times a Stickelberger element (slightly adjusted). This is stated in the paper as formula 7 (without comment), and it does become clear if one thinks about the analytic properties of the log function. This is a beautiful and witty paper. It has already stimulated further research (P. Cornacchia, to appear), and will certainly continue to do so in the future.
scientific article; zbMATH DE number 934973