Yet these dual notions are there and are highlighted by the curious cases of similar phenomena in the arithmetic of function fields. This is what I want to discuss here. So, as usual in the characteristic $p$ game, let $A$:=Fq$[\theta]$, $K$:=Fq$(\theta)$ and $K$$\infty$:=Fq$((1/\theta))$. Recall also the Carlitz module $C$ given by $C$$\theta$(z):=$\theta z+z$q; one always views $C$ as the analog of the multiplicative group Gm (indeed its division values generate abelian extensions etc.).
Of course every algebra lies over Z and thus one can always study the corresponding "Z-invariants'' such as class groups or class numbers etc. But the analogy between Z and $A$ really calls out for "$A$-invariants'' also. When I mentioned the possibility of such $A$-objects way back in 1980 at a conference, the participants looked at me like I had lost my mind. Be that as it may, within the past few years such $A$-invariants have indeed been produced in the seminal work of Lenny Taelman; so we now have "class $A$-modules'' and "$A$ class numbers'' (really generators of the Fitting ideals of these finite class $A$-modules). In fact, these notions fit beautifully into the special values of $L$-series in direct analogy with algebraic number theory.
In a similar way, the very notion of "analytic function'' is clearly Z-based; i.e., based on the notion of power series $\sum$ ai xi; built directly into the definition of power series is the standard Z-action arising from multiplication (i.e., the mapping $(i,x)$ $\mapsto x$i).
So the idea of this blog is that there should analogously be "$A$-expansions'' where we now sum over the monic elements of $A$ (and not elements of Z) in the theory and, remarkably, such things do exist. We are, by no means, close to a full theory of such expansions but rather we have a number of highly intriguing results.
Here is a very cool example of what I am talking about (essentially due to Greg Anderson in his famous "log-algebraicity" paper: Journal Number Theory 60, 165-209 (1996)): We begin by recalling, from Calculus 1, the basic expansion
$\log (1+x)=x$-x2/2+x3/3+...
Now let $C$ be the Carlitz module and $\log$C(z) it's logarithm. For i $\geq$ 1, we put $[i]:=\theta$qi-$\theta\in A$ and also Li:=[i][i-1]...[1]. One can easily see that Li is the least common multiple of the monic elements of $A$ of degree $i$. One then has the Z-expansion
$\log$C(z)=$\sum$i zqi/Li.
For the $A$-expansion we have Greg's formula
$\log$C (z)=$\hat\sum$a Ca(z)/a,
where $a$ runs over the monics of $A$ and $\hat \sum$ means that we compute the sum as the limit of {Sd(z)} where Sd(z) is the above sum truncated over the (finite number of) monics of degree $\leq$ d (so, alas, we have not fully removed Z here after all!). Note also that without such a truncation, the convergence of the sum is extremely tricky and rare! The analogies between the $A$-expansion of $\log$C(z) and the usual expansion for $\log (1+x)$ are very clear....
(Greg calls the power series x-x2/2+x3/3+... "log-algebraic" since it is clearly the log of an algebraic
function. Once one views power series this way, many examples spring to mind; indeed Dwork's famous result on points over finite fields can be viewed in this optic. Greg's log-alg ideas are currently having an extremely large impact on research; for more, see Rudy's blog
https://rudyperkins.wordpress.
For a monic $a$, the additive polynomial $C$a(z) has derivative identically equal to $a$. As such one can find a formal composition inverse denoted $C$a-1 (z) as an Fq-linear power series. To obtain an $A$-expansion for the Carlitz exponential we then have the beautiful, unpublished, formula of Federico Pellarin:
$\exp$C(z)=$\hat\sum$a$C$a-1 (az),
where one must now "renormalize'' the sum in two steps: First of all, we truncate the sum over the monics of degree $d$, as before, and then we also truncate the resulting expression (which is an additive power series) to only include the terms of degree $\leq$ qd. Again, as before, without these operations there is no hope of convergence.
Next let's move on to $L$-series in finite characteristic. Again we find that there is a mix between $A$-expansions and Z-expansions. For purposes of illustration we only treat the simplest case; thus given a monic $a$ in $A$ of degree $d$, we set
$\langle a\rangle$:= $a/\theta$d.
Notice that $\langle a \rangle$ is a $1$-unit in K$\infty$, and, as such, the expression $\langle a \rangle$y makes sense for $y \in$ Zpvia the Binomial Theorem (and with the usual exponential properties). We put S$\infty$:=K$\infty$*$\times$ Zp with its obvious abelian group structure, and for s=(x,y)$\in$ S$\infty$, as:=xd$\langle a\rangle$y . One then has the zeta function of $A$ defined by the $A$-expansion
$\zeta$A(s):=$\sum$a a-s.
For x not in Fq[[$1/\theta$]], this expansion converges without further manipulation. For the rest of S$\infty$ we rewrite $\zeta$A(s)=$\hat \sum$a a-s where, to guarantee convergence, we again truncate by the degree $d$ and take the limit....
There is yet a third place where $A$-expansions are now playing a very interesting role and which is presumably somehow related to the above cases. Let $f(\tau)$ be a classical elliptic modular form on the upper half plane associated to SL2(Z). As everybody knows, the form $f$ has an expansion $f=\sum$anqn where q:=e$2\pi i \tau. Now let $g(z)$ be a modular form on the Drinfeld upper half-plane. In particular, $g(z)$ is, by definition, invariant under transformations of the form $z\mapsto z+h$ for $h\in A$; as such I showed long ago that $g(z)$ has a Z-expansion $\sum$cn un where $u(z):=\exp$C$(\pi z)$-1. Noting that
$\sum$ anqn=$\sum$ ane$n 2\pi i \tau$
leads one to suspect that $g(z)$ might also have an expression of the form $\sum$adaua where $a$ runs over the monics and ua:=u(az). In fact, this is almost the correct idea: Let $G(X)$ be a fixed function (so far only polynomials have been considered). Then we call an expansion of the form c0+$\sum$acaG(ua) an "$A$-expansion''. While it turns out that not all forms in finite characteristic have such expansions (at least for the class of functions G considered up till now), it has recently become very clear that a great many important ones do!
For instance, all Eisenstein series have such expansions. More importantly, the two basic cusp forms $\Delta$ and $h$ also have them: $\Delta$=$\sum$a aq(q-1) uaq-1 and $h$ (which is a $q-1$-st root of $\Delta$) has the expansion $h=\sum$a aq ua. These expansions are due to B. L'opez, Arch. Math. 95 (2010), 143–150. Very recently, in Journal of Number Theory 133 (2013) 2247–2266, A. Petrov has shown how to construct families of cusp forms
$\sum$a at Gn(u a)
for certain positive integers $t$ and polynomials {Gn(X)}. Moreover he proves that these forms are, in fact, all Hecke eigenforms with easily computed eigenvalues. In arXiv:1306.4344 I showed how these forms give rise to non-trivial interpolations at the finite primes $\mathfrak v$ of $A$ in the sense of Serre's construction of p-adic modular forms (something I have long wanted to do). This also fits perfectly in to the theory of such forms created by C. Vincent in her 2012 Wisconsin thesis.
I would like to finish by explaining how Petrov's sums, just above, have elliptic modular analogs. Put G(X):=X/(1-X) and qn:=qn. Then, indeed, the normalized Eisenstein series of weight 2k has the Lambert expansion
1+2/$\zeta(1-2k)$ $\sum$n n2k-1 G(qn) .
It is my pleasure to thank Rudy Perkins and Federico Pellarin for their invaluable input.
