My first introduction to the theory of Drinfeld modules was in the mid 1970’s when I was a graduate student at Harvard. My advisor, Barry Mazur, had heard about them from lectures by Deligne (who, I believe, had previously met Drinfeld in Moscow). In any case, based on his knowledge of elliptic modular curves, Barry asked me whether the difference of two cuspidal points would be of finite order in the Jacobian of the modular curves of rank two Drinfeld modules (it is). He expected that showing this would involve Eisenstein series and then said, “But I don’t know how to construct them.” I went home and wrote down the obvious formula from $SL_2({mathbf Z)$ which clearly converged and I was off; it took me a little while to realize that, in fact, the convergence was indeed strong enough to define a “rigid analytic function” in the sense of John Tate – such rigid functions play the role in nonArchimedean analysis that holomorphic functions do in complex analysis. The glorious point to Tate’s idea was that by drastically reducing the number of “admissable” open sets (via a Grothendieck topology), one could actually force analytic continuation, “GAGA” theorems (which basically say that anything done analytically on a projective variety actually ends up in the algebraic category), and so on…..

Anyway, once one had Eisenstein series, the definitions of general modular forms were completely straightforward. What was not obvious was eastablishing that they possessed expansions at the cusps in analogy with the “$q$-expansions” of elliptic modular froms; but one can in fact do this with a little rigid geometry. The resulting expansions arise from the appropriate Tate objects in the theory also in analogy with the classical elliptic theory. Coherent chomology then shows that the forms of a given weight, which are also holomorphic at the cusps, form finite dimensional spaces and so on. Moreover, one could readily define the Hecke operators with the obvious definition and see that the Eisenstein series are eigenforms with eigenvalues associated to a prime $(f)$ ($fin {mathbf F}_q[theta]$) of the from $f^i$.

However, there were some issues that immediately arose which vexed me greatly then, and still do even now with a good deal of progress on them. They are:

1. The Hecke operators are associated to ideals $(i)subset {mathbf F}_q[theta]$ whereas the expansions at cusps are of the form $u^j$ for $j$ an integer and $u$ the local parameter; an obvious mismatch very much unlike classical theory!

2. A simple combinatorial calculation shows that the Hecke operators are *totally* multiplicative in obvious distinction from what happens with elliptic modular forms.

3. There is a form $Delta$ highly analogous to its elliptic cousin. Very early on, Serre asked me to compute its eigenvalues and I was surprised that I could show $Delta$ has the same eigenvalues as an Eisenstein series. In fact, there are all sorts of forms that have the same eigenvalues, which is, from a classical point of view, very concerning!!

Since then, there has been a lot of great work on these rigid modular forms by Gekeler, Reversat, Teitelbaum, Böckle, Pink, Bosser, Pellarin, Armana and others. I want to focus here on the recent work of Bartolomé López and, in particular, Aleks Petrov (who is a student of Dinesh Thakur); see http://arxiv.org/abs/1207.6479 . Remarkably there appears to be a very serious connection with my last post (on the work of Federico Pellarin and Rudy Perkins).

More precisely, as above, let $u$ be the parameter at the cusp $infty$ that we are expanding our forms about. Now when one computes the expansion of the Eisenstein series at the cusps, one passes through an intermediate expansion of the form $sum_a c_a g_a$ where $a$ runs over the monic elements in ${mathbf F}_q[theta]$ and $g_a$ is an easily specified function depending on $a$. Such expansions are called “$A$-expansions” by Petrov and can be seen to be unique. The first example, as mentioned, are the Eisenstein series, but Lopez showed more remarkably that the form $Delta$ has an $A$-expansion as does Gekeler’s function $h$ (which is a root of $Delta$).

Petrov shows the existence of infinitely many forms with such $A$-expansions. Moreover, these expansions also work very well with the Hecke operators and, in fact, one can see that they give rise to eigenforms with very simple eigenvalues (like those mentioned for Eisenstein series). Indeed a form with such an $A$-expansion is essentially determined by its eigenvalues and weight and this is a very positive development!

Since one has so many forms with such simple eigenvalues, it is natural to wonder if *all* the Hecke eigenvalues are of the same simple form, and so I asked Aleks what examples he had of Hecke eigenvalues. Now recall that in my last post, if $t$ is a scalar, we defined the quasi-character $chi_t$ by $chi_t(f)=f(t)$ for $f in {mathbf F}_q[theta]$. Well, remarkably, Aleks sent me some tables where, for the primes $f$ calculated, the eigenforms indeed have associated eigenvalues of the form $f^jchi_t(f)^e$ for various $t$ integral over $A$…..