A question about the definition of $p$-adic pseudo-measure.

Question

Let $\mathfrak B$ be a profinite abelian group and let $\Lambda(\mathfrak B)$ be defined as the inverse limit $\varprojlim \mathbb Z_p[\mathfrak B/ \mathcal H]$ where the inverse limit is taken with respect to all the open normal subgroups $\mathcal H$ of $\mathfrak B$. In the book Cyclotomic fields and zeta values, it is said that an element of $\Lambda(\mathfrak B)$ defines a $p$-adic integral measure on $\mathfrak B$. I think it is fairly intuitive why an element of $\Lambda(\mathfrak B)$ should define such a measure, since a measure is basically a way of assigning numbers ($p$-adic in this case) to open subgroups of $\Lambda(\mathfrak B)$ and their cosets in such a way that they are disjoint additive. If the image of an element $\lambda \in \Lambda(\mathfrak B)$ under the projection to $\mathbb Z_p[\mathfrak B/\mathcal H]$ is written as $\sum\limits_{x \in \mathfrak B/\mathcal H} c_{\mathcal H}(x)x$, then we can think of this notation to mean that $\lambda$ assigns the $p$-adic integer $c_{\mathcal H}(x)$ to the coset $x$, and the inverse limit condition gives disjoint additivity. Next, the author defines what is called a $\textit{pseudo-measure}$. An element $\lambda$ in the total ring of fractions of $\Lambda(\mathfrak B)$ (localisation outside all zero-divisors) is said to a pseudo-measure if we have $$(g-1)\lambda \in \Lambda(\mathfrak B)$$ for all $g \in \mathfrak B$. I didn't understand why this definition makes sense or how it is motivated, i.e., how this definition accounts for the pole of the zeta function. Kindly help. Thank you!!

Answer 1

You acknowledge that there is no mystery in the definition of p-adic measures attached to a profinite abelian group $G$, which are simply elements of the complete group algebra $\Lambda(G)$. As for p-adic pseudo-measures,they are elements of the total ring of fractions $Q(G)$ satisfying a certain technical condition described at the bottom of p.35 of the Coates-Sujatha book. So I guess that your question is actually about the arithmetic motivation for this machinery.

We must go back to the 1950’s, when Kubota and Leopoldt discovered the p-adic analogues $L_p(s, \chi)$ of the complex functions $L(s, \chi)$, which are characterized by an interpolation property in the following sense : roughly speaking, for any integer n > 0 and any even Dirichlet character $\chi$, the special value $L_p(1 - n, \chi)$ is equal, up to a precise Euler factor, to the corresponding special value $L(1 - n, \chi\omega^{- n})$ , where $\omega$ is the so called Teichmüller character (denoted $\theta $ in the C-S. book) which gives the Galois action on the group $\mu_p$ of p-th roots of unity. This is thm.5.11 in Washington’s « Introduction to Cyclotomic Fields », but Wash. himself admits that the direct proof given at this point is not particularly enlightening. To get additional insights, we must appeal to Iwasawa’s theory of $Z_p$-extensions : an extension of fields is called a $Z_p$-extension if it is Galois and its Galois group is isomorphic to the additive group of the ring $Z_p$ of p-adic integers. The prototypical example is the cyclotomic extension $Q(\mu_p^{infty})/Q(\mu_p)$; denote its Galois group by $\Gamma$, and $\Lambda = \Lambda (\Gamma)$ for short. Note that $\Lambda$ is isomorphic to the ring of formal power series $Z_p[[T]]$ via $\gamma$ -->1 + T, where $\gamma$ is a topological generator of $\Gamma$. In the algebraic part of the theory, one studies e.g. the inverse limit X of the p-class groups $A_n$ of the intermediate fields $Q(\mu_{p^n})$. This X is naturally a compact torsion $\Lambda$- module, to which one can attach, via the action of $\gamma$, a « characteristic power series » char(X) in $Z_p[[T]]$ . Now, in the classical theory of a cyclotomic field $Q(\mu_m)$, the Stickelberger theorem gives a specific element of the group algebra of $Gal(Q(\mu_m)/Q)$ which annihilates the class group of $Q(\mu_m)$. Basing on this, one can show, if $\chi$ is not the trivial character, that $L_p(s,\chi)$ is given by a p-adic power series (W., thm.7.10) which comes essentially, via the identification $\Lambda \cong Z_p[[T]]$ , from the inverse limit of the Stickelberger elements along the cyclotomic tower ; if $\chi$ is trivial, one must divide by a certain simple polynomial. The so called Iwasawa Main Conjecture (now a theorem of Mazur-Wiles) also appears naturally : for any non trivial $\chi$, the power series defining $L_p(s,\chi)$ and the $\chi\omega^{-1}$- part of char(X) are « the same » - an astonishing encounter between algebra and analysis, « unquestionably one of the great discoveries in number theory » (C-S., p. 3). Note that all the above notions and properties have been extended to the case of a totally real number field instead of $Q$, but the theory is much more difficult.

It remains to explain the « what for » of pseudo-measures. Away from the trivial character, $L_p(s,\chi)$ is an analytic p-adic function. But at the trivial character, it’s the p-adic zêta function, which has a (simple) pole at s = 1, hence the denominator which appears in thm.7.10 op. cit. At the beginning of W.’s chapter 12 , it is noted that « the concept of a measure [due to Mazur] yields a p-adic integration theory which allows us to interpret the p-adic L-function as a Mellin transform, as in the classical case », and thm.12.2 gives such a formula, but only for a non trivial character because the LHS is analytic. The extension to pseudo-measures [due to Serre] allows to dispense with characters. Actually, from the point of view adopted by C-S., $\zeta_p$ is no longer a p-adic function, but a p-adic pseudo-measure on $G$ = $Gal(Q(\mu_p^{infty})/Q)^+$. It encompasses all the $L_p(s,\chi)$, for all even $\chi$ (including the trivial character), as is shown that the interpolation formula of thm.1.4.2 which is reproduced on the front cover (NB : the character $\chi$ there is the cyclotomic character). Even the Main Conjecture has a simpler "global" expression (thm.1.4.3), without characters.