This post is based on a recent letter by Matt Papanikolas outlining some results he has discovered whilst writing a (highly anticipated!) monograph on $L$-values in finite characteristic. In staring at Matt's letter, I realized that he allowed one to relate the big 3 matrices (Wronski, Vandermonde and Moore) in one simple formula which I will present below and then pose a related question.
I apologize if this is challenging to read; I am having a devil of a time getting my tex to compile correctly.
Matt's perspicacious insight was based on a paper by Felipe Voloch which was published in the Journal of Number Theory in 1998. (Full disclosure: I had previously played with Voloch's paper and totally missed these ideas...)
Let me begin with some standard notation. As is now becoming standard, following Greg Anderson, let us put $A:= F$_q$[\theta]$. For each nonnegative integer $i$, let's put $A(i):= \{a\in A$ $\mid $ $ \deg (a)\leq i\}$. Moreover, for such $i$, following Carlitz, we set $[i]:=\theta$^{q^i}-$\theta$. These are basic building blocks of $A$; for instance, one easily sees that $[i]$ is the product of all monic primes of degree dividing $i$. Various types of factorials are products of these elements....
Now recall that the Wronskian matrix is made up out of the derivatives of a function. Obviously in finite characteristic one can only differentiate a little bit before ending up with identically zero functions. As such we modify the Wronskian slightly here and use the "divided derivatives" $\partial$_i (i.e., those elements actually giving the coefficients of Taylor series etc.) in order to end up with nontrivial matrices.
Let $x$_0,..., $x$_i be i indeterminates. We define the Vandermonde matrix $V(i)(x$_0,..,$x$_i) as usual by having the $i$-th row consist of $(1$,$a$ ,..., $a$^i$)$, where $a=x$_{i-1}, and so on.
Finally of course the Moore matrix is made up out of the $q$^i powers of elements in a very similar
fashion to both the Wronski and Vandermonde matrices.
Let $g\in F[[\theta]]$. Notice that $g$^{q^i} is simply $g(\theta$^{q^i}) as the elements of $F$ are
fixed. Thus, as Felipe observed, a bit of calculus (e.g., Taylor expansions) immediately gives $g$^{q^i}=$\sum$_j $\partial$_j(g) [i]^j.
Let $a\in A(i)$. Let M(i)=(a, a^q,..., a^{q^i})^t be the column vector consisting of a and its q^j-th powers and let W(i):=(a, $\partial a$, ..., $\partial$_ia)^t be the column vector consisting of the divided derivatives of a. Matt realized that Felipe's result implies
$$M(i)=V(i)([0],[1],...,[i])W(i).$$
Now simply choose a bunch more elements in A(i) and sling the above equations together. We immediately deduce an equality of matrices of the form M=VW where M is a Moore matrix, V is the Vandermonde and W the Wronskian. Thus taking determinants allows us, in this case, to calculate the Wronskian determinant in terms of the Moore and Vandermonde determinants.
It is very wild that one can relate the q-th power mapping, which is a field embedding, in this fashion with the divided derivatives which satisfy the Leibniz identity and I wonder if there is much more here.
In any case, the Moore determinant is intimately related to having elements linearly independent over F_q. In classical theory, the Wronskian of analytic functions determines whether they are linearly independent (a result evidently due to Peano); see arXiv:1301.6598 for a modern approach. In characteristic p, the p-th power mapping clearly wreaks some havoc as usual (think of the Wronskian of x^p and x^{p^2}). But I wonder if there might not be some clear statement about linear dependence involving both the Wronskian and the p-th power mapping...?
(Updated 4-22-13: If you choose as your elements $a\in A(i)$ the set $\{1,\theta,...,\theta$^i}$, you
obtain a Wronskian with determinant 1 thus giving an equality between the determinant of Matt's Vandermonde matrix and the determinant of the Moore matrix. As in Carlitz's original paper back in the 1930's, this Moore determinant is then easily computed to be a Carlitz factorial... It should also be possible to compute the determinant of Matt's matrix directly using the well-known formula for the Vandermonde determinant.)
(Updated 4-23-13: I thank Sangtae Jeong for pointing out the work of F.K. Schmidt, in 1939, and the work of Felipe Voloch and Arnaldo Garcia, in 1987, that goes very far with Wronskians in finite characteristic. See "Wronskians and linear independence in fields of prime characteristic", Manuscripta Math., 59, 1987, 457-469.)
(:1301.6598arXiv:1301.6598
Friday, April 19, 2013
Monday, January 28, 2013
Informal video series on the Carlitz module
Dear All: My student, Rudy Perkins, and his fellow graduate student, Tim All, are creating an informal video lecture series on the Carlitz module. If you are interested, please check out http://rudyperkins.wordpress.com/ . David
Thursday, January 17, 2013
CYCLIC HOMOLOGY AND ARITHMETIC
Cyclic homology has recently revealed its potential in
relation to the description of Serre's Archimedean local factors in the
Hasse-Weil L-function of an arithmetic variety as shown in the paper by
A. Connes and C. Consani : Cyclic homology, Serre's local factors and the lambda-operations.
The elaboration of this topic constitutes one of the two leading themes
of the course that AC is developing at the Collège de France this year.
Cyclic cohomology was introduced and widely publicized in 1981 as an
essential tool in noncommutative differential geometry. The talk Spectral sequence and homology of currents for operator algebras given
by AC at the 1981 Oberwolfach meeting introduced for instance the SBI
long exact sequence and described the cyclic cohomology of the NC torus.
In the context of algebraic geometry instead, the application of cyclic
homology to schemes has a more recent evolution and it has been mainly
promoted in the work of C. Weibel.
In number-theory, there are three fundamental sources of L-functions 1) arithmetic varieties 2) geometric Galois representations 3) automorphic representations.
To a smooth, projective variety X defined over a number field K, corresponds its Hasse--Weil zeta function which is the alternate product of the factors $L(H^m,s)$ attached to the Galois representation on the $\ell$-adic \'etale cohomology $H^m(X_{\bar K},\Q_{\ell})$. The function $L(H^m,s)$ is defined as an infinite Euler product whose non-archimedean factors have an immediate geometric meaning at places $\nu$ of K of good reduction for X (we assume here that $\ell$ does not divide $\nu$), by implementing the action of the geometric Frobenius on the étale cohomology of the reduction of $X$ at $\nu$.
At an archimedean place v instead, the local L-factor is roughly a product of powers of (shifted) Gamma functions whose definition depends upon the Hodge structure on the m-th Betti cohomology of the complex variety $X_{v}=X\times_K\C$.
The main result of the above paper states that the alternate product (as $m$ ranges from $0$ to twice the dimension of X) of Serre's Archimedean factors is the inverse graded determinant of the action on cyclic homology of $X$, with coefficients in infinite adeles $\prod_{v|\infty} K_v$, of the operator $(2\pi)^{-1}(s-\Theta)$, where $\Theta$ generates the lambda operations which are the analogue in cyclic homology of the Adams operations in algebraic K-theory.
Several subtle features ought to be mentioned at this point. One of them is the nuance between cyclic homology and archimedean cyclic homology (this is the one taken up in the above result) which parallels and reflects, in cyclic homology, the difference between reduced and unreduced real Deligne cohomology (we refer to the paper for more details).
This result promotes the development of the archimedean cyclic homology as a theory playing a natural role in the theory of motives in algebraic geometry, in view of its connection to algebraic K-theory by the regulator maps. Moreover, it also suggests the study of a generalization of the above result at the non-archimedean places and the existence of a global Lefschetz formula in cyclic homology.
The second part of the course will focus on the description of the Archimedean counterpart of the rings of periods in p-adic Hodge theory (especially $B_{cris}$ and $B_{dR}$), by elaborating on the results contained in the recent collaborative paper The universal thickening of the field of real numbers. Fontaine's rings of p-adic periods play a fundamental role in arithmetic in view of the comparison theorems relating étale cohomology with coefficients in p-adic numbers, with the other fundamental cohomological theories of arithmetic varieties and in particular with the de-Rham cohomology thus realizing Grothendieck's idea of ``foncteur mysterieux''.
At a real place, the transposition of the construction of the rings of p-adic periods yields non-trivial relevant rings endowed with a canonical one parameter group of automorphisms which replaces the Frobenius in Fontaine's construction. At a complex place, this construction produces fundamental algebraic structures whose applications transcend the realm of arithmetic by producing a natural framework in which Feynman integrals in quantum field theory should be understood.
The class will be entirely given as blackboard-chalk talks.
AC and K. Consani.
In number-theory, there are three fundamental sources of L-functions 1) arithmetic varieties 2) geometric Galois representations 3) automorphic representations.
To a smooth, projective variety X defined over a number field K, corresponds its Hasse--Weil zeta function which is the alternate product of the factors $L(H^m,s)$ attached to the Galois representation on the $\ell$-adic \'etale cohomology $H^m(X_{\bar K},\Q_{\ell})$. The function $L(H^m,s)$ is defined as an infinite Euler product whose non-archimedean factors have an immediate geometric meaning at places $\nu$ of K of good reduction for X (we assume here that $\ell$ does not divide $\nu$), by implementing the action of the geometric Frobenius on the étale cohomology of the reduction of $X$ at $\nu$.
At an archimedean place v instead, the local L-factor is roughly a product of powers of (shifted) Gamma functions whose definition depends upon the Hodge structure on the m-th Betti cohomology of the complex variety $X_{v}=X\times_K\C$.
The main result of the above paper states that the alternate product (as $m$ ranges from $0$ to twice the dimension of X) of Serre's Archimedean factors is the inverse graded determinant of the action on cyclic homology of $X$, with coefficients in infinite adeles $\prod_{v|\infty} K_v$, of the operator $(2\pi)^{-1}(s-\Theta)$, where $\Theta$ generates the lambda operations which are the analogue in cyclic homology of the Adams operations in algebraic K-theory.
Several subtle features ought to be mentioned at this point. One of them is the nuance between cyclic homology and archimedean cyclic homology (this is the one taken up in the above result) which parallels and reflects, in cyclic homology, the difference between reduced and unreduced real Deligne cohomology (we refer to the paper for more details).
This result promotes the development of the archimedean cyclic homology as a theory playing a natural role in the theory of motives in algebraic geometry, in view of its connection to algebraic K-theory by the regulator maps. Moreover, it also suggests the study of a generalization of the above result at the non-archimedean places and the existence of a global Lefschetz formula in cyclic homology.
The second part of the course will focus on the description of the Archimedean counterpart of the rings of periods in p-adic Hodge theory (especially $B_{cris}$ and $B_{dR}$), by elaborating on the results contained in the recent collaborative paper The universal thickening of the field of real numbers. Fontaine's rings of p-adic periods play a fundamental role in arithmetic in view of the comparison theorems relating étale cohomology with coefficients in p-adic numbers, with the other fundamental cohomological theories of arithmetic varieties and in particular with the de-Rham cohomology thus realizing Grothendieck's idea of ``foncteur mysterieux''.
At a real place, the transposition of the construction of the rings of p-adic periods yields non-trivial relevant rings endowed with a canonical one parameter group of automorphisms which replaces the Frobenius in Fontaine's construction. At a complex place, this construction produces fundamental algebraic structures whose applications transcend the realm of arithmetic by producing a natural framework in which Feynman integrals in quantum field theory should be understood.
The class will be entirely given as blackboard-chalk talks.
AC and K. Consani.
Tuesday, October 30, 2012
THE MUSIC OF SPHERES
The title of this post, the music of spheres, refers to a talk The music of shapes which I gave in Lille, on the 26th of September, on the occasion of a joint meeting with the Fields Institute. The talk is an introduction to the spectral aspect of noncommutative geometry and its implications in physics.
The starting point is the naive question "Where are we?", or how is it possible to communicate to aliens our position in the Universe. This question leads, in the Riemannian framework of geometry, to that of determining a complete set of geometric invariants, both for a space and for a point in a space. The theme of Mark Kac, "Can one hear the shape of a drum" associates to a shape its musical scale which is the spectrum of the square root of the Laplacian, or better of the Dirac operator. After illustrating this familiar theme by many concrete examples we give a hint of the additional invariant which allows one to recover the geometric picture, namely the CKM invariant, and illustrate it, in a simplified form, in the simplest possible example of isospectral but non congruent shapes.
What about the relation with music? One finds quickly that music is best based on the scale (spectrum) which consists of all positive integer powers $q^n$ for the real number $q=2^{\frac{1}{12}}\sim 3^{\frac{1}{19}}$. Due to the exponential growth of this spectrum, it cannot correspond to a familiar shape but to an object of dimension less than any strictly positive number. As explained in the talk, there is a beautiful space which has the correct spectrum: the quantum sphere of Poddles, Dabrowski, Sitarz, Brain, Landi et all. Its spectrum consists of a slight variant of the $q^j$ where each appears with multiplicity $O(j)$. (See the original paper of Dabrowski and Sitarz arXiv:math/0209048 (Banach Center Publications, 61, 49-58, 2003) for the precise formula, and the paper of Brain and Landi arXiv:math/1003.2150 for a variant and the many references to the mathematicians involved, my apologies to each of them for not puting the list here.)
We experiment in the talk with this spectrum and show how well suited it is for playing music.
The new geometry which encodes such new spaces, is then introduced in its spectral form, it is noncommutative geometry, which is then confronted with physics. There the core is the spectral Standard Model of A. Chamseddine and the author which goes back to 1996. We tell the tale of the resilience of this model in its successive confrontations with experiments.
Both the start and the end part of the talk are unusual. The previous talk was a talk by Alain Aspect on his recent experiments, with his collaborators, confirming experimentally the "delayed choice" Gedankenexperiment of John Wheeler. So the very beginning of my talk refers to Aspect's point about the subtelty of the concept of "reality" implied by the quantum. The thesis which I defend briefly is that the total lack of control that we have on the outcome of a quantum experiment (we control only the probabilities), is a "variability" which is more primordial than the classical variability governed by the passing of time (on which we have no control either). I also explain briefly why time will emerge from the quantum variability.
The end part, in the question session, is also unusual, it is a long answer to a question which was posed by Alain Aspect.
Update : The talk of Alain Aspect is now also available at the conference website
The starting point is the naive question "Where are we?", or how is it possible to communicate to aliens our position in the Universe. This question leads, in the Riemannian framework of geometry, to that of determining a complete set of geometric invariants, both for a space and for a point in a space. The theme of Mark Kac, "Can one hear the shape of a drum" associates to a shape its musical scale which is the spectrum of the square root of the Laplacian, or better of the Dirac operator. After illustrating this familiar theme by many concrete examples we give a hint of the additional invariant which allows one to recover the geometric picture, namely the CKM invariant, and illustrate it, in a simplified form, in the simplest possible example of isospectral but non congruent shapes.
What about the relation with music? One finds quickly that music is best based on the scale (spectrum) which consists of all positive integer powers $q^n$ for the real number $q=2^{\frac{1}{12}}\sim 3^{\frac{1}{19}}$. Due to the exponential growth of this spectrum, it cannot correspond to a familiar shape but to an object of dimension less than any strictly positive number. As explained in the talk, there is a beautiful space which has the correct spectrum: the quantum sphere of Poddles, Dabrowski, Sitarz, Brain, Landi et all. Its spectrum consists of a slight variant of the $q^j$ where each appears with multiplicity $O(j)$. (See the original paper of Dabrowski and Sitarz arXiv:math/0209048 (Banach Center Publications, 61, 49-58, 2003) for the precise formula, and the paper of Brain and Landi arXiv:math/1003.2150 for a variant and the many references to the mathematicians involved, my apologies to each of them for not puting the list here.)
We experiment in the talk with this spectrum and show how well suited it is for playing music.
The new geometry which encodes such new spaces, is then introduced in its spectral form, it is noncommutative geometry, which is then confronted with physics. There the core is the spectral Standard Model of A. Chamseddine and the author which goes back to 1996. We tell the tale of the resilience of this model in its successive confrontations with experiments.
Both the start and the end part of the talk are unusual. The previous talk was a talk by Alain Aspect on his recent experiments, with his collaborators, confirming experimentally the "delayed choice" Gedankenexperiment of John Wheeler. So the very beginning of my talk refers to Aspect's point about the subtelty of the concept of "reality" implied by the quantum. The thesis which I defend briefly is that the total lack of control that we have on the outcome of a quantum experiment (we control only the probabilities), is a "variability" which is more primordial than the classical variability governed by the passing of time (on which we have no control either). I also explain briefly why time will emerge from the quantum variability.
The end part, in the question session, is also unusual, it is a long answer to a question which was posed by Alain Aspect.
The three speakers, Lille 9/26: E. Ghys, A. Aspect, A. Connes
Update : The talk of Alain Aspect is now also available at the conference website
Saturday, October 13, 2012
Carlitz's formalism and Euler's $\Gamma$-function
It was always my fondest hope that the arithmetic of function fields in finite characteristic would finally become sophisticated enough so that it could be developed somewhat in tandem with classical arithmetic. In the recent past, this hope appears to becoming real. In particular, I would like to draw your attention to the new preprint by Federico Pellarin arXiv:1210.2490 "On the generalized Carlitz-module" where the formalism due to Carlitz is applied to deduce new functional equations for Euler's $\Gamma$-function (as well as applications to other classical special functions).
Let ${\mathbf F}_q$ be the finite field with $q$ elements and let $\theta$ an indeterminate; as usual we put $A:={\mathbf F}_q[\theta]$. When Carlitz was originally developing his module in the 1930's he chose the mapping generated by $\theta \mapsto \theta -\tau$ where $\tau$ is the $q$-th power mapping. I changed this to $\theta + \tau$ simply because this made computing Frobenius elements a bit easier.
As the theory of the Carlitz module, and general Drinfeld modules, began to develop, it was realized that one could abstract from the original settings of function fields over finite fields and this was developed e.g., by Mumford (1977) and Hellgouarch (1992 and 1997). In the paper by Federico this formalism is applied to complex analytic functions in a very concrete fashion.
More specifically, Let $F$ be the field of (complex valued) meromorphic functions in $s$ which are periodic of period $1$. Obviously, the function $s$ is not in $F$ and so one forms the polynomial ring (of functions) ${\mathbf A}:=F[s]$; notice that the huge field $F$ is now playing the role of the finite field $\mathbf F_q$....
The question is what plays the role of the $q$-th power mapping; here Federico makes the crucial choice $\tau (f):=f(s+1)$ where $f$ is a meromorphic function (so that obviously $F$ is the fixed field of this action). One then has the (generalized) Carlitz module associated to $(F[s], \tau)$ simply by following Carlitz's idea of composing actions and his original formalism mentioned above! In particular, one is now able to talk about the torsion elements associated to an element $a\in F[s]$.
So what does this have to do with the $\Gamma$-function? Well, virtually everything! Indeed, of course, everyone knows the (almost!) definitional functional equation
$$\Gamma(s+1)=s\Gamma(s)\,.$$
Federico now simply rewrites this as
$$s\Gamma-\tau(\Gamma)=0\,$$
In other words, the $\Gamma$-function IS an $s$-torsion element for this Carlitz module. In fact, one sees that Carlitz-torsion is intimately connected with the $\Gamma$-function in one form or another. (Indeed, I would not be surprised if there is yet another characterization of the $\Gamma$-function which involves Carlitz-Pellarin torsion.)
But much more is true. In a previous blog I have discussed the applications of the Anderson-Thakur function $\omega(t)$ to $L$-series. The function $\omega(t)$ can abstractly be obtained as the Akheizer-Baker function associated to certain data. In the $F[s]$-case, Federico shows that the Akheizer-Baker function is precisely $\Gamma(s-t)$ and this leads to other functional relations in line with the analogy to $\omega(t)$.
Let ${\mathbf F}_q$ be the finite field with $q$ elements and let $\theta$ an indeterminate; as usual we put $A:={\mathbf F}_q[\theta]$. When Carlitz was originally developing his module in the 1930's he chose the mapping generated by $\theta \mapsto \theta -\tau$ where $\tau$ is the $q$-th power mapping. I changed this to $\theta + \tau$ simply because this made computing Frobenius elements a bit easier.
As the theory of the Carlitz module, and general Drinfeld modules, began to develop, it was realized that one could abstract from the original settings of function fields over finite fields and this was developed e.g., by Mumford (1977) and Hellgouarch (1992 and 1997). In the paper by Federico this formalism is applied to complex analytic functions in a very concrete fashion.
More specifically, Let $F$ be the field of (complex valued) meromorphic functions in $s$ which are periodic of period $1$. Obviously, the function $s$ is not in $F$ and so one forms the polynomial ring (of functions) ${\mathbf A}:=F[s]$; notice that the huge field $F$ is now playing the role of the finite field $\mathbf F_q$....
The question is what plays the role of the $q$-th power mapping; here Federico makes the crucial choice $\tau (f):=f(s+1)$ where $f$ is a meromorphic function (so that obviously $F$ is the fixed field of this action). One then has the (generalized) Carlitz module associated to $(F[s], \tau)$ simply by following Carlitz's idea of composing actions and his original formalism mentioned above! In particular, one is now able to talk about the torsion elements associated to an element $a\in F[s]$.
So what does this have to do with the $\Gamma$-function? Well, virtually everything! Indeed, of course, everyone knows the (almost!) definitional functional equation
$$\Gamma(s+1)=s\Gamma(s)\,.$$
Federico now simply rewrites this as
$$s\Gamma-\tau(\Gamma)=0\,$$
In other words, the $\Gamma$-function IS an $s$-torsion element for this Carlitz module. In fact, one sees that Carlitz-torsion is intimately connected with the $\Gamma$-function in one form or another. (Indeed, I would not be surprised if there is yet another characterization of the $\Gamma$-function which involves Carlitz-Pellarin torsion.)
But much more is true. In a previous blog I have discussed the applications of the Anderson-Thakur function $\omega(t)$ to $L$-series. The function $\omega(t)$ can abstractly be obtained as the Akheizer-Baker function associated to certain data. In the $F[s]$-case, Federico shows that the Akheizer-Baker function is precisely $\Gamma(s-t)$ and this leads to other functional relations in line with the analogy to $\omega(t)$.
Friday, August 10, 2012
A DRESS FOR THE BEGGAR ?
Since 4 years ago I thought that there was an unavoidable incompatibility between the spectral model and experiment. I wrote a post in this blog to explain the problem, on August 4 of 2008, as soon as the Higgs mass of around 170 GeV was excluded by the Tevatron. Now 4 years have passed and we finally know that the Brout-Englert-Higgs particle exists and has a mass of around 125 Gev. In the meantime the problem of this discrepancy in the Higgs mass seemed very hard to resolve and this certainly slowed down quite a bit the interest in the spectral model since there seemed to be no easy way out and whatever one would try would not succeed in lowering the Higgs mass. The reason for this post today is that this incompatibility has now finally been resolved in a fully satisfactory manner in a joint work with my collaborator Ali Chamseddine, the paper is now on arXiv at http://fr.arxiv.org/pdf/1208.1030
What is truly remarkable is that there is no need to modify the spectral model in any way, it had already the correct ingredients and our mistake was to have neglected the role of a real scalar field which was already present and whose couplings (with the Higgs field in particular) were already computed in 2010 as one can see in http://fr.arxiv.org/pdf/1004.0464
This completely changes the perspective on the spectral model, all the more because the above scalar field has been independently suggested by several groups as a way for stabilizing the Standard Model in spite of the low experimental Higgs mass. So, after this fruitful interaction with experimental results, it is fair to conclude that there is a real chance that the spectral approach to high energy physics is on the right track for a geometric unification of all known forces including gravity.
A few words about the picture, the metaphor of the Standard Model as a beggar with a diamond in its pocket was suggested by Daniel Kastler a long time ago, so this explains the character on the right. The character on the left wears the symbols of NCG, ingredients of spectral nature which allow one to reconstruct the geometry from gravitational observables such as the spectrum of the Dirac operator, and to write down the action of the Standard Model coupled to gravity.
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 $A$.....
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 $A$.....
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