(by A.C. and K. Consani)

Right after the end of the Sixth Annual Spring School/Conference on Noncommutative Geometry and Operator Algebras, a second meeting took place at Vanderbilt University, on May 15-18. This workshop has been dedicated to explore some aspects of several emerging relations linking Noncommutative Geometry and the geometry over the field with one element.

In the next weeks, we expect to post a more elaborate and thoughtful overview on this new interesting development in noncommutative/arithmetic geometry. In the meanwhile, the following is a first outline of the topics covered in several of the main talks. Some of the speakers have supplied us with the files of their transparencies and/or with an abstract of their presentation. When available, we added the abstracts within quotation marks “… “. The pdf files presently available have also been included here below.

In the next weeks, we expect to post a more elaborate and thoughtful overview on this new interesting development in noncommutative/arithmetic geometry. In the meanwhile, the following is a first outline of the topics covered in several of the main talks. Some of the speakers have supplied us with the files of their transparencies and/or with an abstract of their presentation. When available, we added the abstracts within quotation marks “… “. The pdf files presently available have also been included here below.

The first day of the meeting was dedicated to the review and the discussion of the following 5 papers whose main subject has an evident connection with the field with one element.

1) Lisa Carbone gave in her talk “Kac-Moody groups, finite fields and Tits geometries” an overview of the seminal paper by Jacques Tits “Sur les analogues algebriques des groupes semi-simples complexes” (Colloque d’Algebre superieure, 1956, Bruxelles). The following is her review.

“Motivated by trying to find a “geometric” interpretation of a finite dimensional simple Lie group G in contrast to the “algebraic” version of G proposed by Chevalley the previous year, Tits introduced a “geometry” X which has G as its automorphism group. Tits’ geometry X also had the mysterious property that when constructed over a finite field F_q , one could take the limit q –> 1 in which the group G tends to the discrete subgroup W (the Weyl group of G) and the geometry X tends to the geometry of W.

Tits’ examples are quite sketchy in the 1956 paper. I would like to propose however that this was the seed of a deep circle of ideas that Tits cultivated and developed over three decades. The 1956 paper seems to have been a precursor to the notion of a Bruhat-Tits building for a Chevalley group over a finite field, or a simple algebraic group over a nonarchimedean local field. These constructions also evolve naturally into the notion of a Tits building for a Kac-Moody group over a finite field associated to the Tits functor for Kac-Moody groups.

In all of these subsequent constructions of Tits, the notion of a “field with 1 element” is present, both on the group level, and in the associated Tits geometry.

In my talk, I attempted to give an overview of examples from each of the classes described above, and to indicate what happens as we try to take a limit F_q –> F_1. This viewpoint has been very useful in my work, and I indicated a number of things I have been able to prove using the Tits geometries over F_1.”

“Motivated by trying to find a “geometric” interpretation of a finite dimensional simple Lie group G in contrast to the “algebraic” version of G proposed by Chevalley the previous year, Tits introduced a “geometry” X which has G as its automorphism group. Tits’ geometry X also had the mysterious property that when constructed over a finite field F_q , one could take the limit q –> 1 in which the group G tends to the discrete subgroup W (the Weyl group of G) and the geometry X tends to the geometry of W.

Tits’ examples are quite sketchy in the 1956 paper. I would like to propose however that this was the seed of a deep circle of ideas that Tits cultivated and developed over three decades. The 1956 paper seems to have been a precursor to the notion of a Bruhat-Tits building for a Chevalley group over a finite field, or a simple algebraic group over a nonarchimedean local field. These constructions also evolve naturally into the notion of a Tits building for a Kac-Moody group over a finite field associated to the Tits functor for Kac-Moody groups.

In all of these subsequent constructions of Tits, the notion of a “field with 1 element” is present, both on the group level, and in the associated Tits geometry.

In my talk, I attempted to give an overview of examples from each of the classes described above, and to indicate what happens as we try to take a limit F_q –> F_1. This viewpoint has been very useful in my work, and I indicated a number of things I have been able to prove using the Tits geometries over F_1.”

2) Christophe Soule review in his talk “Algebraic varieties over F_1” the main aspects developed in his paper “Les varietes sur le corps a un element” (Mosc. Math. J. 4 (2004), no. 1, 217–244, 312). An abstract of his presentation is downloadble here and in clear below:

Define a gadget over F_1 to be the pair

**X**=(*X*, A_X) where X is a covariant functor from the category*F*of finite abelian groups to the category of sets, and A_X is a complex algebra. Given a finite abelian group G, a point x in X(G), and a character**s**of G, we assume given a character, e_{x,s} of A_X. If f: G-> G’ is a morphism and y belongs to X (G’), the following equality is supposed to be satisfied : e_{f(y),s}= e_{y , s circ f} for any character s.An affine variety V over Z defines a gadget X over F_1 by letting X(G) be the set of points of V in the group algebra of G and by defining the algebra A_X to be the ring of regular functions on the complex points of V (with the obvious evaluation maps).

A *morphism* **X** -> **Y**, between two gadgets over F_1 consists of a natural transformation from the functor X to Y and a morphism of algebras from A_Y to A_X compatible with evaluation maps. It is called an *immersion* when both maps are injective.

An *affine* variety over F_1 is a gadget **X** such that

– For every G the set X(G) is finite;

– The complex algebra A_X is a commutative Banach algebra;

– There exists an affine variety X_Z over Z and an immersion i: X -> X_Z of gadgets satisfying the following property:

for any affine variety V over Z and any morphism of gadgets h: X -> V, there exists a unique algebraic morphism h_Z: X_Z -> V such that h equals h_Z composed with i.

Examples of varieties X_Z, where X is an affine variety over F_1, include smooth toric varieties and the algebraic group-schemes GL_2 and GL_3.

3) Niranjan Ramachandran gave in his talk “Zeta functions and motives (d’apres Manin)” an overview of the paper by Y. Manin “Lectures on zeta functions and motives (according to Deninger and Kurokawa)” (Columbia University Number-Theory Seminar, New-York 1992, Asterisque No. 228 (1995), 4, 121–163).

The following is his review.

“The aim of my talk was to provide a brief introduction to the beautiful paper of Yuri Manin (Lectures on motives and zeta functions – to be found on Katia’s website www.math.jhu.edu/~kc) on the fascinating ideas of Christopher Deninger and Nobushige Kurokawa on zeta functions and F1.

The basic analogy between number fields and function fields has driven much of 20th century arithmetic geometry. This leads one to the desire to view Spec Z as a curve, but over which field? Of course, F1.

Deninger has expressed the completed Riemann zeta as R divided by s.(s-1)

where R is a regularized determinant to be viewed as infinite-dimensional analogue of a determinant of an endomorphism of a finite dimensional vector space. Compare with the zeta function of a smooth projective curve (of genus g) over a finite field F_q: a polynomial of degree 2g divided by (1-t) (1-qt) where t is the variable q^{-s}.

Manin provides an overview of the theory of motives over a finite field. He comments that even though we may not be able to define F1 or the category of varieties or motives over F1, we can certainly discuss zeta functions of motives over F1. The discussion strongly suggests that the only zeta functions that one obtains are generated by (s-n) for an integer n. Classical groups G are supposed to define varieties over F1 (original insight of Jacques Tits that G(F1) = W_G the Weyl group of G) as are projective spaces P^n; the zeta function of P^n over F1 is supposed to be s.(s-1)….(s-n). Thus the denominator in Deninger’s expression for the completed Riemann zeta function is the zeta function of P1 over F1 which exactly parallels the function field case.

Manin also points out that the stable homotopy groups of spheres should be viewed as the algebraic K-theory of F_1 and the classical map J in algebraic topology from the stable homotopy groups to the algebraic K-theory of the integers is the one induced by the map F1 –> Z. This is a very important observation. The order of the image of J involves Bernoulli numbers (and hence zeta values!).

Manin also discusses the Kurokawa product of zeta functions and provides many examples from arithmetic and geometry (Selberg zeta, multiple gamma functions, ..) which could not be covered in this lecture. In particular, Kurokawa has defined the zeta function of (Spec Z) x_{Spec F1} (Spec Z) even though a mathematical definition of the fibre product is still lacking.”

The following is his review.

“The aim of my talk was to provide a brief introduction to the beautiful paper of Yuri Manin (Lectures on motives and zeta functions – to be found on Katia’s website www.math.jhu.edu/~kc) on the fascinating ideas of Christopher Deninger and Nobushige Kurokawa on zeta functions and F1.

The basic analogy between number fields and function fields has driven much of 20th century arithmetic geometry. This leads one to the desire to view Spec Z as a curve, but over which field? Of course, F1.

Deninger has expressed the completed Riemann zeta as R divided by s.(s-1)

where R is a regularized determinant to be viewed as infinite-dimensional analogue of a determinant of an endomorphism of a finite dimensional vector space. Compare with the zeta function of a smooth projective curve (of genus g) over a finite field F_q: a polynomial of degree 2g divided by (1-t) (1-qt) where t is the variable q^{-s}.

Manin provides an overview of the theory of motives over a finite field. He comments that even though we may not be able to define F1 or the category of varieties or motives over F1, we can certainly discuss zeta functions of motives over F1. The discussion strongly suggests that the only zeta functions that one obtains are generated by (s-n) for an integer n. Classical groups G are supposed to define varieties over F1 (original insight of Jacques Tits that G(F1) = W_G the Weyl group of G) as are projective spaces P^n; the zeta function of P^n over F1 is supposed to be s.(s-1)….(s-n). Thus the denominator in Deninger’s expression for the completed Riemann zeta function is the zeta function of P1 over F1 which exactly parallels the function field case.

Manin also points out that the stable homotopy groups of spheres should be viewed as the algebraic K-theory of F_1 and the classical map J in algebraic topology from the stable homotopy groups to the algebraic K-theory of the integers is the one induced by the map F1 –> Z. This is a very important observation. The order of the image of J involves Bernoulli numbers (and hence zeta values!).

Manin also discusses the Kurokawa product of zeta functions and provides many examples from arithmetic and geometry (Selberg zeta, multiple gamma functions, ..) which could not be covered in this lecture. In particular, Kurokawa has defined the zeta function of (Spec Z) x_{Spec F1} (Spec Z) even though a mathematical definition of the fibre product is still lacking.”

4) Jack Morava presented in his talk “K-theory of ring objects in homotopy theory” some relevant aspects of the paper by D. Quillen “On the cohomology and K-theory of the general linear groups over a finite field” (Annals of Mathematics, 2nd Ser., Vol.96, No.3, 1972, 552–586). An overview of his presentation is downladable here. Here is his abstract:

“Direct sum gives the category of finitely generated projective modules over a ring R (together with their isomorphisms) a symmetric monoidal structure. In 1972, Quillen defined the algebraic K-theory of R in terms of the best approximation to the geometric realization of this category by an abelian object in the homotopy category: an infinite loop-space or, in topologists’ contemporary language, a

These ideas have been vastly extended in the four decades since, in particular to general symmetric monoidal categories (Segal, eg finite sets) or to `categories with cofibrations and weak equivalences’ (Waldhausen, eg finite cell complexes). Relatively recent developments (eg the theory of symmetric spectra) in our understanding of commutative ring objects in homotopy theory provide a unified approach to these generalizations and to related constructions (eg `topological’ Hochschild and cyclic (co)homology).

My talk was basically historical; I tried to sketch the development of this language, and to use it to compare the category of vector spaces over a finite field and the category of finite sets. I wanted to clarify the extent to which the K-theory of the latter can be viewed as a limit, as q->1, of the K-theory of F_q.”

**spectrum.**These ideas have been vastly extended in the four decades since, in particular to general symmetric monoidal categories (Segal, eg finite sets) or to `categories with cofibrations and weak equivalences’ (Waldhausen, eg finite cell complexes). Relatively recent developments (eg the theory of symmetric spectra) in our understanding of commutative ring objects in homotopy theory provide a unified approach to these generalizations and to related constructions (eg `topological’ Hochschild and cyclic (co)homology).

My talk was basically historical; I tried to sketch the development of this language, and to use it to compare the category of vector spaces over a finite field and the category of finite sets. I wanted to clarify the extent to which the K-theory of the latter can be viewed as a limit, as q->1, of the K-theory of F_q.”

5) Eugene Ha lectured on the main parts of the theory developed by Nikolai Durov in his preprint “New Approach to Arakelov Geometry” (arXiv:0704.2030). Here is the Eugene’s review:

“Arakelov geometry is an amalgam of scheme-theoretic algebraic geometry and complex differential geometry that allows one to do intersection theory on models of algebraic varieties over the “compactification” of Spec(Z). However, missing from Arakelov’s theory is a direct definition of the fiber at archimedean infinity, and in particular, of the notion of an “archimedean valuation ring.” Recently, this situation has been rectified by Nikolai Durov who has created a full-fledged theory of generalized (commutative) rings and schemes that provides a framework for treating compactified arithmetic varieties in an uniform scheme-theoretic manner [A New Approach to ArakelovGeometry, 2007]. The “classical” algebraic geometry of Grothendieck is preserved in Durov’s theory since the category of generalized rings contains the classical commutative rings as a full subcategory.

To develop a theory of “rings,” like the “localization of Z at the infinity prime,” that are monoidal but not additive, one can try to first frame classical ring theory in categorical terms,which has the advantage of allowing one to think of additivity as a monoidal structure. (For example, a ring is simply a monoid in the monoidal category of abelian groups.) It is well-known that the categorical notion that enables this transition is that of a monad in sets, i.e., a monoid in the monoidal category of endofunctors of sets. For a (classical) commutative ring R, the monad M_R attached to R is the functor that maps a set S to the set underlying the free R-module generated by S. The category of R-modules is then the category of modules of the monad M_R, and the ring R itself can be recovered from this category in the usual way (take the center of the endomorphism ring of the identity functor of R-modules).

This motivates the definition of a generalized commutative ring as a monad in sets which is moreover algebraic (commutes with filtered inductive limits of sets) and commutative (an algebraic monad in sets determines a family of n-ary operations on its modules, and commutativity for the monad means, roughly, that all these n-ary operations commute).

To see how one might arrive at a “correct” notion of the “local ring of Z at infinity” (or rather of its completion) suitable for a scheme-theoretic Arakelov geometry, Durov considers the notion of a “Z_infinity-lattice” in a real vector space. In the p-adic case, Z_p-lattices in a finite-dimensional p-adic vector space V correspond (up to similitude) to the maximal compact submonoids of End(V). This leads to the definition of Z_infinity-lattices (again, up to similitude) in a finite-dimensional real vector space E as the compact convex symmetric bodies in E. Further comparison with the p-adic case leads to the definition of the set underlying the free Z_infinity-module with basis S as the standard octahedron in R^{(S)}, and hence to the definition of Z_infinity as the generalized ring corresponding to this endofunctor.

In particular, Z_infinity is a generalized subring (i.e., algebraic submonad) of the real numbers R, as is Z_+, the generalized ring that maps a set S to the set of formal finite non-negative-integral linear combinations of elements of S. Thus one can take the intersection of Z_+ and Z_infinity: this is Durov’s definition of F_1, the so-called field of one element. One can also describe F_1 as the free algebraic monad in sets with a single 0-“arity” generator. Modules over F_1 are simply sets with a marked point.

Going far beyond generalized commutative algebra, Durov has also developed a rather complete theory of spectra and generalized schemes. In his theory the “affine line” Spec(Z) is an affine scheme defined over F_1, and the compactification of Spec(Z) is a pro-generalized scheme.

Finally, while many of the motivations and constructionsof Durov’s theory are very natural, the results of some of his computations differ from various widely-held expectations. Forexample, Durov has computed the Picard group of the compactification of Spec(Z) and has found that it is the multiplicative group of positive rational numbers, whereas the function field-number field analogy suggests that it should be be the positive real numbers. Moreover, the product S of Spec(Z) with itself over F_1 is shown to be Spec(Z) in Durov’s theory, which is inconsistent with Kurokawa’s definition of the zeta function of S.”

To develop a theory of “rings,” like the “localization of Z at the infinity prime,” that are monoidal but not additive, one can try to first frame classical ring theory in categorical terms,which has the advantage of allowing one to think of additivity as a monoidal structure. (For example, a ring is simply a monoid in the monoidal category of abelian groups.) It is well-known that the categorical notion that enables this transition is that of a monad in sets, i.e., a monoid in the monoidal category of endofunctors of sets. For a (classical) commutative ring R, the monad M_R attached to R is the functor that maps a set S to the set underlying the free R-module generated by S. The category of R-modules is then the category of modules of the monad M_R, and the ring R itself can be recovered from this category in the usual way (take the center of the endomorphism ring of the identity functor of R-modules).

This motivates the definition of a generalized commutative ring as a monad in sets which is moreover algebraic (commutes with filtered inductive limits of sets) and commutative (an algebraic monad in sets determines a family of n-ary operations on its modules, and commutativity for the monad means, roughly, that all these n-ary operations commute).

To see how one might arrive at a “correct” notion of the “local ring of Z at infinity” (or rather of its completion) suitable for a scheme-theoretic Arakelov geometry, Durov considers the notion of a “Z_infinity-lattice” in a real vector space. In the p-adic case, Z_p-lattices in a finite-dimensional p-adic vector space V correspond (up to similitude) to the maximal compact submonoids of End(V). This leads to the definition of Z_infinity-lattices (again, up to similitude) in a finite-dimensional real vector space E as the compact convex symmetric bodies in E. Further comparison with the p-adic case leads to the definition of the set underlying the free Z_infinity-module with basis S as the standard octahedron in R^{(S)}, and hence to the definition of Z_infinity as the generalized ring corresponding to this endofunctor.

In particular, Z_infinity is a generalized subring (i.e., algebraic submonad) of the real numbers R, as is Z_+, the generalized ring that maps a set S to the set of formal finite non-negative-integral linear combinations of elements of S. Thus one can take the intersection of Z_+ and Z_infinity: this is Durov’s definition of F_1, the so-called field of one element. One can also describe F_1 as the free algebraic monad in sets with a single 0-“arity” generator. Modules over F_1 are simply sets with a marked point.

Going far beyond generalized commutative algebra, Durov has also developed a rather complete theory of spectra and generalized schemes. In his theory the “affine line” Spec(Z) is an affine scheme defined over F_1, and the compactification of Spec(Z) is a pro-generalized scheme.

Finally, while many of the motivations and constructionsof Durov’s theory are very natural, the results of some of his computations differ from various widely-held expectations. Forexample, Durov has computed the Picard group of the compactification of Spec(Z) and has found that it is the multiplicative group of positive rational numbers, whereas the function field-number field analogy suggests that it should be be the positive real numbers. Moreover, the product S of Spec(Z) with itself over F_1 is shown to be Spec(Z) in Durov’s theory, which is inconsistent with Kurokawa’s definition of the zeta function of S.”