Katia Consani has kindly written the following post for the NCG blog. Many thanks to Katia!

I like to think that from his grave in St. Petersburg, Euler is probably quite pleased to see how far his studies on “Analysis Infinitorum” have been successfully developed in these past 224 years from his death. The book “Introductio in analysis infinitorum”(1748) has been translated into English in 2 books, a few years ago, by John D. Blanton. Stimulated by the nice post of Alain Connes reporting on the “Euler Festival”, Euler’s arithmetical and geometrical achievements and re-interpretation of the concept of infinitesimal variables in quantized calculus, I rushed to the library to glance again into these books and pleasantly re-discover some parts.

I find these 2 books truly exciting and inspiring for the clarity of the style and of course for the original concepts developed by Euler. These include for example, ideas “on series which arise from products”, or “on continued fractions”, just to mention the title of some of the chapters. It is also funny to report that in the translator’s introduction, Blanton notes that the (final)encouragement for the translation of that book came from a remark he heard from Andre Weil, in 1979, when Weil spoke on the Life and Works of Leonhard Euler at the University of Rochester. Apparently, Weil claimed that he was trying to convince the mathematical community that the students of mathematics would profit much more from a study of Euler’s “Introductio in analysis infinitorum”, rather than of the available modern textbooks…

Andre Weil wrote also a nice and instructive long chapter commenting on the life and the scientific achievements of Euler in his book “Number Theory, an approach through history” (Birkhauser). Quoting from Weil’s book (cfr. par. XVIII, p. 261): “…So much work has been done on the series zeta(n)”–Euler writes in 1735– “that it seems hardly likely that anything new about them may still turn up…” Nevertheless as Weil remarks, for the next ten years Euler never relaxed his efforts to put his conclusions, for example on zeta(2), on a sound basis. His attention had been drawn also by his earlier work on the gamma-function (the factorial function in Euler’s words) to infinite products in their relation to infinite series “…in them [infinite products] the factors proceed accordingto the prime numbers, whose progression is no less irregular [than the terms in Goldbach’s series]…”. A few years later, Riemann would take the “eulerian products” for zeta(s) as the starting point in his studies in number-theory and he would eventually introduce a”complete” version of zeta(s), inclusive of a gamma factor and satisfying a functional equation.

The definition and the study of what is today called the gamma-function is in fact one of the great achievements of Euler’sstudies in arithmetic. He extended the factorial function n! from the naturals to all real numbers greater than -1 by writing the formula

and observing that the integral on the right of this formulaconverges for non integers values of n, provided that . Later on, Riemann proved that the gamma function (i.e. theEuler’s integral on the right side of the formula) which is in fact defined for all complex numbers s in the half-plane , canbe analytically extended to define an analytic function of a complexvariable s, with simple poles at the negative integers and no zeroes. The description of this function has apparently nothing in common with the “eulerian factor” , nonetheless it is only when one completes with a suitable gamma factor that the resulting function behaves very nicely (i.e. satisfies a functional equation). One says that the gamma function is the contribution arising from the “archimedean prime infinity” (i.e. from the archimedean evaluation of Q). Legendre, subsequently introduced the notation in place of the original . This notation prevailed in France and, by the end of the nineteenth century, in the rest of the world as well. With this modification, the set of zeroes of is the set of the natural numbers.

This fact should bring us back on some of the earlier posts on thisBlog, on operators whose spectrum is the set of the natural numbers.In fact, it turns out that the the archimedean factor that enters inthe definition of the complete zeta function, i.e. what is today written as

and also the archimedean factor

that is included, together with , in the description of the complete Hasse-Weil L-function of algebraic varieties defined over number fields (i.e. higher arithmetic generalizations of the Riemann zeta function), both have a nice description as the inverse of a regularized determinant det_infty(s-T) for the action of a self-adjoint operators T (archimedean frobenius…) on an infinite-dimensional real vector space H (cfr. C. Deninger “On theGamma-factors attached to motives”, Invent. Math. 104 (1991) and C.Consani “Double complexes and Euler L-factors”, Compositio Math. 111(1998)).

At this point, one naturally wonders whether a precise geometri cconstruction is lurking from such a re-interpretation of the gammafactors and in particular whether NCG, with its sophisticated machinery of spectral triples, can help in proving this expectation. In other words, is there a noncommutative space endowed with a representation in a Hilbert space and a Dirac operator supportingt his classical arithmetic theory? Of course, the first natural candidates for the Hilbert space and the Dirac operator are the aforementioned real vector space H and the unbounded self-adjoint operator T. However, we know that from the mere knowledge of these two objects one cannot expect to reconstruct a noncommutative manifold: the relevant information arises from the definition of an involutive algebra (acting on a Hilbert space).

It turns out though, that the definition of the real vector space H is in fact quite geometrical! H is an infinite direct sum, indexed over the natural numbers, of real de-Rham cohomology groups associated to smooth, projective algebraic varieties (a point for the case of which are naturally related to the definition of these zeta-functions. Moreover, with a bit more of technical effort (cfr.C. Consani and M. Marcolli “Noncommutative geometry, dynamics and infinitely-adic Arakelov geometry” Selecta Math. 10 (2004)) one can show that the archimedean part of the zeta-function of an arithmetic surface of genus at least 2, indeed coincides with a precise zeta-function in a family of zetas associated to a noncommutative manifold. This manifold supplies the arithmetical data in two possible ways, either with a construction performed in its interior or with a second construction carried out on its boundary.

I like to conclude this post by saying that these results give evidence to the statement that NCG successfully carries on Euler’s geometrical legacy!

Is there a relation between Eulerian factors at finite and infinite places? We know both are determinants

but apparently of very different operators on very different (conceptually I mean) spaces! Is it too much to hope to unify the too?

Dear Chris,

your comment and questions bring up a quite interesting and central topic in arithmetic geometry: a coherent unification in the definition of the local Euler’s factors (archimedean and non)

is one of the `dreams’ in number-theory, one of my dreams, at least…

The interpretation of the archimedean factor by means of a regularized determinant for the action of a self-adjoint operator on an infinite dimensional vector space of geometric nature, as I wrote in my recent post, suggests the possibility,

or at least raises the question, on the existence of a similar interpretation for the non-archimedean Euler’s factors.

To be precise, one should talk of Euler’s factors

in the Hasse-Weil L-function associated to a smooth, projective algebraic variety X over a global field (here I mean either a number field or the function field of an algebraic curve defined over a finite field).

These functions are, as I said, higher dimensional geometric generalizations of the Riemann zeta function, whose definition is recovered by working

with the algebraic variety defined by the prime spectrum of the integers.

The hope for a unification is also motivated by a classical and famous result in the arithmetic geometry over function fields, where the truthfulness of the Grothendieck-Lefschetz trace formula for the action of the (geometric) Frobenius operator in etale cohomology H_{et} (= a suitable replacement of the Betti=singular cohomology theory for these frameworks) supplies indeed, a global geometrical/cohomological expression for the L-function.

So, the question is whether such cohomological methods (trace formula, for instance) can be extended to cover the number-field case. Here, the obvious obstacle lies, as we know, in the different

natures of the local factors.

For the non-archimedean factors, which by

the way are defined by the following (schematic) formula:

L_p(X,s) = det(1-Fr_p(p)^{s}|(H_{et})^I_p)^{-1}

(here, for simplicity, I have restricted myself to the case of an algebraic variety X over the rationals, so p denotes a fixed prime in Z, s is a complex variable and I_p is the inertia group at p: a subgroup of the absolute Galois group of Q)

there are some positive and encouraging results supporting, under suitable conditions, the expectation that also such factors can be described as regularized determinants for an operator acting

on an infinite dimensional complex vector space

(cfr. C. Deninger “Local L-factors of motives and regularized determinants”, Invent. math (1992)).

Also in this case the powerful machinery of the spectral triples in NCG has been put at work and with success! Once again, as for the archimedean case, the motivation for using noncommutative tools came from the hope to understand the geometry underlying and supporting the cohomological description of the local Euler’s factors.

(cfr. C. Consani, M. Marcolli “Spectral triples from Mumford curves”, IMRN (2003)).

Well, this answer/comment is getting rather long… You may have guessed my excitement in talking and working on this very intriguing subject!

Katia Consani