Tuesday, July 31, 2012

Another occurence of the quasi-character $\chi_t$

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 $(f)$ ($f\in {\mathbf F}_q[\theta]$) 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 $f^j\chi_t(f)^e$ for various $t$ integral over $A$.....

Wednesday, July 11, 2012

Operator/scalar fusion in finite characteristic and remarkable formulae

Let $E$ be a curve of genus $1$ over the rational field $\mathbf Q$. One of the glories of mathematics is the discovery that (upon choosing a fixed rational point "$\mathbf O$") $E$ comes equipped with an addition which makes its points over any number field (or $\mathbf R$ or $\mathbf C$) a very natural abelian group. (In the vernacular of algebraic geometry, one calls $E$ an "abelian variety" of dimension $1$ or an "abelian curve".)

Built into this setup is a natural tension between the two different avatars of the integers $\mathbf Z$ which now arise. On the one hand, an integer $n$ is an element of the scalars $\mathbf Q$ over which our curve $E$ lies; on the other hand, $n$ is also an operator on the group formed  by the elliptic curve (and, in fact, it is well known that this operator is actually a morphism on the elliptic curve).

One would, somehow, like to form a ring that encompasses both of these avatars. An obvious way to do this would be to form ${\mathbf Z}\otimes {\mathbf Z}$ but, alas, this fails as this tensor product is simply $\mathbf Z$. I have always thought, perhaps naively, that one of the motivations in studying ${\mathbf F}_1$ was the hope that progress could be made here....

In any case, in finite characteristic we are blessed with more flexibility. Let $q$ be a power of a prime $p$ and let ${\mathbf F}_q$ by the field with $q$-elements with $A:={\mathbf F}_q[\theta]$ the polynomial ring in the indeterminate $\theta$. In the 1970's, soon after he defined elliptic modules (a.k.a., Drinfeld modules) Drinfeld was influenced by the work of Krichever to define an associated vector bundle called a "shtuka". In order to do so, Drinfeld worked with the $2$-dimensional algebra $A\otimes_{\mathbf F_q} A$ which precisely combined the roles of operator and scalar. Soon after that, Greg Anderson used this algebra to develop his higher dimensional analog of Drinfeld modules (called "$t$-modules"); in particular, Anderson's theory allowed one to create a good category of "motives" out of Drinfeld modules which is, itself, equipped with a good notion of a tensor product.

One can associate to Drinfeld modules analogs of classical special functions such as $L$-series, gamma functions; etc. Classical theory leads to the expectation that these gamma functions should somehow be related to the $L$-series much as gamma functions are "Euler-factors at infinity" in classical algebraic number theory. But so far that has not been the case and the connection, if one exists, remains unknown.

The basic Drinfeld module is the rank $1$ module $C$ discovered by L. Carlitz in the 1930's (in a triumph of old school algebra!); it is a function field analog of the algebraic group ${\mathbf G}_m$ and its exponential is a function field analog of the classical exponential function.  Let $\tau(z):=z^{q}$ be the $q$-th power mapping with $\tau^i$ defined by composition; the Carlitz module is then the $\mathbf F_q$-algebra map defined by $C_\theta:=\theta \tau^0+\tau$. Using Anderson's notion of a tensor product, Greg and Dinesh Thakur rapidly defined, and studied, the $n$-tensor power $C^{\otimes n}$ of the Carlitz module in "Tensor powers of the Carlitz module and zeta values," Ann. of Math. 132 (1990), 159–191.  In particular, they defined the following marvelous function
$$\omega (t):=\theta_1 \prod_{i=0}^\infty \left(1-\frac{t}{\theta^{q^i}}\right)^{-1}\,,$$
where $\theta_1$ is a fixed $(q-1)$-st root of $-\theta$. Notice that $\omega(t)$ is obviously the reciprocal of an entire function and, in that, it reminds one of Euler's gamma function.

However, much more profound is the result of Anderson/Thakur (loc. cit.) that $\lim_{t\mapsto\theta}(t-\theta)\omega(t)$ is the period $\tilde{\xi}$ of the Carlitz module. Here one can't help but be reminded of the famous equality $\Gamma(1/2)=\sqrt \pi$; so one is led to view $\omega(t)$ as yet another function field manifestation of the notion of a gamma function. Indeed, in a tour de force, "Determination  of the algebraic relations among special $\Gamma$-values in positive characteristic," (Ann. of Math. (2) (2004), 237-313), Anderson, Dale Brownawell, and Matt Papanikolas used $\omega(t)$ to establish virtually all the transcendence results one would want of the geometric gamma function.

So it was apparent, to me anyway, that this magical $\omega(t)$ should also make itself known in the theory of characteristic $p$ $L$-series. However, I simply did not see how this could happen. This impasse was recently broken by some fantastic results of Federico Pellarin ("Values of Certain $L$-series in positive characteristic," Ann. of Math. to appear, http://arxiv.org/abs/1107.4511) and these results precisely provide the operator/scalar fusion mentioned in the title of this blog!

So I would like to finish by describing some of Federico's results, and also those of my student Rudy Perkins in this regard. They both are obtaining all sorts of beautiful formulae of the sort one might find in the famous book by Whittaker and Watson which is very exciting and certainly bodes very well for the future of the subject. But before doing so, we do need one more result of Anderson/Thakur.

As in my previous blog put $K:={\mathbf F}_q((1/\theta))$ with the canonical absolute value. Put ${\mathbf T}:=\{\sum_{i=0}^\infty a_it^i\}$ where $\{a_i\}\subset K$ and $a_i\to 0$ as $i \to \infty$; so $\mathbf T$ is simply the Tate algebra of functions with coefficients in $K$ converging on the closed unit disc.

The algebra $\mathbf T$ comes equipped with two natural operators: First of all, the usual hyperdifferential operators act on $\mathbf T$ via differentation with respect to $t$ in the standard fashion.  Now let $f(t)=\sum a_it^i\in \mathbf T$; we then set $\tau (f):=\sum a_i^qt^i$ and call it the
"partial Frobenius operator" (in an obvious sense). Note that, in this setting, $\tau$ is actually $\mathbf F_q[t]$-linear. Note also, because we are in characteristic $p$ these operators commute.

Anderson and Thakur look at the following partial Frobenius equation on $\mathbf T$: $\tau \phi=(t-\theta)\phi$ (N.B.: $t-\theta$ is the "shtuka function" associated to the Carlitz module). The solutions to this equation clearly form an $\mathbf F_q[t]$-module and the remarkable result of A/T is that this module is free of rank $1$ and generated by $\omega(t)$.

One can rewrite the fundamental equation $\tau \omega=(t-\theta)\omega$ as
$$(\theta \tau^0+\tau)\omega=t\cdot \omega\,;$$
in other words, if we use the partial Frobenius operators to extend the Carlitz module to $\mathbf T$ then $\omega$ trivializes this action. So if $f(\theta)\in A$ one sees immediately that  $C_f\omega=f(t)\omega$.

Abstracting a bit, if $t$ is a scalar, then one defines the "quasi-character" $\chi_t(f):=f(t)$ simply by evaluation. It is Federico's crucial insight that this quasi-character is exactly the necessary device to fuse both the scalars and operators in the theory of characteristic $p$ $L$-series by defining the associated $L$-series $L(\chi_t,s)$ (in the standard fashion). These functions have all the right analytic properties  in the $s$-variable and also have excellent analytic properties in the $t$-variable!

(The reader might have imagined, as I did at first, that the poles $\{\theta^{q^i}\}$ of $\omega(t)$ are too specialized to be associated to something canonical. However, we now see that these poles correspond to the quasi-characters $f(\theta)\mapsto f(\theta^{q^i})=f(\theta)^{q^i}$ and so are completely canonical...)

The introduction of the variable $t$ is, actually, a realization of the notion of "families" of $L$-series. Indeed, if $t$ belongs to the algebraic closure of $\mathbf F_q$, then $\chi_t$ is a character modulo $p(\theta)$, where $p(\theta)$ is the minimal polynomial of $t$.

Theorem: (Pellarin) We have $(t-\theta)\omega(t)L(\chi_t,1)= -\tilde{\xi}\,.$

And so $\omega(t)$ makes its appearance in $L$-series! (One is  also reminded a bit of Euler's famous formula $e^{\pi i}=-1$.) Now let $n$ be a positive integer $\equiv 1$ mod $(q-1)$.

Theorem: (Pellarin) There exists a rational function $\lambda_n\in {\mathbf F}_q(t,\theta)$ such that
$$(t-\theta)\omega(t)L(\chi,n)=\lambda_n \tilde{\xi}^n\,.$$
In "Explicit formulae for $L$-values in finite characteristic" (just uploaded to the arXiv as http://arxiv.org/abs/1207.1753), my student Rudy Perkins gives a simple closed form expression for these $\lambda_n$ as well as all sorts of connections with other interesting objects (such as the Wagner expansion of $\mathbf F_q$-linear functions, recursive formulae for Bernoulli-Carlitz elements, etc.).

So the introduction of $\chi_t$ has opened the door to all sorts of remarkable results. Still, the algebraic closure of $K$ is such a vast thing (with infinitely many extensions of bounded degree etc.), that there may be other surprises we do not yet know. Moreover, we do know that the algebras of measures can be interpreted as hyperdifferential operators on $\mathbf T$. Where are they in the game Federico started?

Thursday, July 5, 2012

Habemus Higgs

Yesterday, the ATLAS and CMS experiments at CERN announced the discovery of a Higgs boson at 125 GeV. Surely, this will become one of the most important discoveries of the century. It also caused quite a few interesting 4th of July parties (for once, with a good justification for the fireworks).

For theoretical physicists, this is as much a good reason of excitement as for the experimentalists. Although a measurement of the Higgs self interaction will only come after the upgrade at 14 TeV of the LHC, the current measurement already suggests interesting questions (for example, there appears to be a deficit in the WW channel of decay, which may be an accident, or an indication of something more interesting).

As it is well known, the noncommutative geometry models of particle physics generally give rise to a heavier Higgs (originally estimated at around 170 GeV, then lowered in more recent versions of the model, but still well above 125 GeV). The usual method, in these models, to obtain estimates on the Higgs, is to impose some boundary conditions at unification energy, dictated by the geometry of the model, and running down the renormalization group equations (RGE). The geometric constraints impose some exclusion curves on the manifold of possible boundary conditions, but do not fix the boundary conditions entirely: in fact, recent work on the NCG models observed a sensitive dependence on the choice of boundary conditions (within the constraints imposed by the geometry). Moreover, the renormalization group flow typically used in these estimates is the one provided by the one-loop beta function of the minimal Standard Model (or in more recent versions, of effective field theories obtained from extensions of the MSM by right handed neutrinos with Majorana mass terms, that is, the RGEs considered in hep-ph/0501272v3), rather than a renormalization group flow directly derived from a quantum field theoretic treatment of the action functional of the NCG model, the spectral action.

Perhaps more interestingly (as what one is after, after all, are extensions of the MSM by new physics), while the original NCG models of particle physics focussed on the MSM, there are now variants that include new particle: a first addition beyond the MSM was a model with right handed neutrinos with Majorana mass terms, which accounts for neutrino mixing and a see-saw mechanism.

More recently, a very promising program for extending the NCG model was developed by Thijs van den Broek and Walter van Suijlekom (arXiv:1003.3788), for versions with supersymmetry. While their first paper on the subject deals only with the QCD sector of the model, they are now well on their way towards including the electroweak sector.

I apologize for the plot spoiler, but given the occasion I think it is worth mentioning: the model that van den Broek and van Suijlekom are currently developing appears to be fairly close to the MSSM, although it is not the MSSM. In particular, the renormalization group equations in their model are going to be different than the equations of MSSM. In particular this means that the "cheap trick" used so far in the NCG models, of importing RGE equations of known particle physics models and running them with boundary conditions imposed by NCG, will not apply to the supersymmetric version and Higgs estimates within this model will involve a genuinely different RGE analysis. It will be interesting to see how the Higgs sector changes in their version of the NCG model, and whether it gives a more realistic picture close to the observed results.

Falsifiability is the most important quality of any scientific theory. Indeed, having explicit experimental data that point out the shortcomings of a theoretical model is the best condition for a serious re-examination of assumptions and techniques used in model building.

Cheers to the LHC, the ATLAS and CMS collaborations, for a great job!