# Schedule of the Workshop "Number theory and non-commutative geometry"

## Monday, November 24

 9:30-10:30 Pierre Cartier: Logarithmic divergences, another similarity between quantum field theory and number theory 10:30-11:00 Coffee break 11:00-12:00 Max Karoubi: Hermitian K-theory invariants in Topology and Algebraic Geometry 12:00-14:00 Lunch break 14:00-15:00 Nadia S. Larsen: C*-completions of Hecke algebras associated to Gelfand pairs 15:00-16:00 Marcelo Laca: Bost Connes systems, Hecke algebras, and induction 16:00-16:30 Tea and cake

## Tuesday, November 25

 9:30-10:30 Gunther Cornelissen: The Ihara zeta function and noncommutative boundaries 10:30-11:00 Coffee break 11:00-12:00 Caterina Consani: The Arithmetic Site I 12:00-14:00 Lunch break 14:00-15:00 Alain Connes: The Arithmetic Site II 15:00-16:00 Christopher Deninger: Witt vectors and the relative de Rham Witt complex 16:00-16:30 Tea and cake

## Wednesday, November 26

 9:30-10:30 Jeffrey Lagarias: Splitting measures on polynomials and the field with one element 10:30-11:00 Coffee break 11:00-12:00 Alain Connes: Towards a Weil cohomology 12:00-14:00 Lunch break 14:00-15:00 Lars Hesselholt: The big de Rham-Witt complex 15:00-16:00 Oliver Lorscheid: Tropicalization as base change along a non-archimedean valuation 16:00-16:30 Tea and cake 16:30-17:00 Takuya Takeishi: Irreducible Representations of Bost-Connes systems

## Thursday, November 27

 9:30-10:30 Bram Mesland: Noncommutative geometry of Bianchi groups 10:30-11:00 Coffee break 11:00-12:00 Abhishek Banerjee: Quasimodular Hecke algebras and Hopf actions 12:00-13:45 Lunch break 13:45-14:45 Eric Leichtnam: Deninger's programme: a motivation for developing new mathematics 15:00-16:00 Alain Connes: Quanta of Geometry (Attention! Bethe Colloquium - Nußallee 12, Physikalisches Institut - Hörsaal I) 16:15-16:45 Tea and cake at HIM

## Friday, November 28

 9:30-10:30 Georg Tamme: Differential algebraic K-theory 10:30-11:00 Coffee break 11:00-12:00 Christophe Soulé: Correspondences, K-theory and the cannibalistic class 12:00-14:00 Lunch break 14:00-15:00 Cristian Popescu: Equivariant L-values, Hecke characters and the K-theory of number fields 15:00-16:00 Closing Remarks 16:00-16:30 Tea and cake

# Abstracts

## Abhishek Banerjee: Quasimodular Hecke algebras and Hopf actions

Let $\Gamma=\Gamma(N)$ be a principal congruence subgroup of $SL_2(\mathbb Z)$. We extend the theory of modular Hecke algebras due to Connes and Moscovici to define the algebra $\mathcal Q(\Gamma)$ of quasimodular Hecke operators of level $\Gamma$. Then, $\mathcal Q(\Gamma)$ carries an action of ''the Hopf algebra $\mathcal H_1$ of codimension $1$ foliations'' that also acts on the modular Hecke algebra $\mathcal A(\Gamma)$ of Connes and Moscovici. However, in the case of quasimodular Hecke algebras, we have several additional operators and we can describe them in terms of a new Hopf algebra $\mathcal H$ acting on $\mathcal Q(\Gamma)$. Further, for each $\sigma\in SL_2(\mathbb Z)$, we introduce the collection $\mathcal Q_\sigma(\Gamma)$ of quasimodular Hecke operators of level $\Gamma$ twisted by $\sigma$. Then, $\mathcal Q_\sigma(\Gamma)$ is a right $\mathcal Q(\Gamma)$-module and is endowed with a pairing $(\_\_,\_\_):\mathcal Q_\sigma(\Gamma)\otimes \mathcal Q_\sigma(\Gamma)\longrightarrow \mathcal Q_\sigma(\Gamma)$. We show that there is a ''Hopf action'' of a certain Hopf algebra $\mathfrak{h}_1$ on the pairing on $\mathcal Q_\sigma(\Gamma)$. Finally, for any $\sigma\in SL_2(\mathbb Z)$, we consider operators acting between the levels of the graded module

$\mathbb Q_\sigma(\Gamma)=\underset{m\in \mathbb Z}{\oplus}\mathcal Q_{\sigma(m)}(\Gamma)$, where $\sigma(m)=\begin{pmatrix} 1 & m \\ 0 & 1 \\ \end{pmatrix}\cdot \sigma$

for any $m\in \mathbb Z$. The pairing on $\mathcal Q_\sigma(\Gamma)$ can be extended to a graded pairing on $\mathbb Q_\sigma(\Gamma)$ and we show that there is a Hopf action of a larger Hopf algebra $\mathfrak{h}_{\mathbb Z}\supseteq \mathfrak{h}_1$ on the pairing on $\mathbb Q_\sigma(\Gamma)$.

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## Pierre Cartier: Logarithmic divergences, another similarity between quantum field theory and number theory

By using combinatorial methods in asymptotic expansions (introduced by Metropolis and Rota), we shall be able to explain the " foreign derivations" of Ecalle, and their role in constructing the Riemann surfaces of "resurgent functions". We then show how these methods apply to various classes of polylogarithmic functions, both in the complex domain and in a p-adic version. We end up by showing the similarity between the logarithmic divergences in QFT and the one occurring in number-theoretic situations.

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## Alain Connes: Quanta of Geometry

In the construction of spectral manifolds in noncommutative geometry, a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of real scalar fields naturally appears and implies, by equality with the index formula, the quantization of the volume. We first show that this condition implies that the manifold decomposes into disconnected spheres which will represent quanta of geometry. We then refine the condition by involving the real structure and two types of geometric quanta, and show that connected manifolds with large quantized volume are then obtained as solutions. When this condition is adopted in the gravitational action it leads to the quantization of the four volume with the cosmological constant obtained as an integration constant. Restricting the condition to a three dimensional hypersurface implies quantization of the three volume and the possible appearance of mimetic dark matter. When restricting to a two dimensional hypersurface, under appropriate boundary conditions, this results in the quantization of area and has many interesting applications to black hole physics.

## Alain Connes: The Arithmetic Site II

My first talk is the continuation of the talk of C. Consani on our joint work "the Arithmetic Site" We had shown previously that the non-commutative space quotient of the adele class space of Q by the action of the disconnected part of the idele class group, yields the correct counting function in order to obtain the complete Riemann zeta function as Hasse-Weil zeta function. Now, we have shown that this space is in fact the set of geometric points of the "Arithmetic Site" over the tropical semifield. The action of the multiplicative group of positive real numbers on the adele class space corresponds to the action of the Frobenius on the above geometric points. Moreover, the square of the arithmetic site is meaningful and we obtain the Frobenius correspondences parametrized by positive real numbers, as sub-varieties of this square. We then determine the composition of Frobenius correspondences. Finally I will explain that the arithmetic site is intimately related to the structure of the point in non-commutative geometry.

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## Alain Connes: Towards a Weil cohomology

My second talk is a joint work with C. Consani and provides a first step towards the elaboration of a Weil cohomology theory suitable to recast the local factors of L-functions of arithmetic varieties in a unified manner. It is inspired by our previous work on archimedean factors and cyclic homology and on the results of L. Hesselholt and I. Madsen relating the de Rham Witt complex to topological cyclic homology. We show that the concept of S-algebra as derived from Segal's Gamma-sets provides a unification of most of the various attempts to model algebraic geometry over $F_1$. It unifies, in particular, the various existing approaches using monoids, semirings, hyperrings as well as N. Durov's attempt in the set-up of Arakelov geometry. While the notion of S-algebra is familiar in algebraic topology its use in the above arithmetic framework is new. Finally, we show that cyclic homology over the universal base S can be defined at a purely combinatorial level.

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## Caterina Consani: The Arithmetic Site I

The talk will present a few results obtained in joint collaboration with A. Connes on the geometry of the Arithmetic Site: an algebraic geometric object deeply related to the non-commutative geometric approach to the Riemann Hypothesis.

We shall discuss a classification and an adelic interpretation of the points of this site as well as a semifield characterization of them.

Furthermore, we will present the (semi)ringed structure of the arithmetic site, namely its structure sheaf with its stalks and finally describe the arithmetic site as a geometry in characteristic one.

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## Gunther Cornelissen: The Ihara zeta function and noncommutative boundaries

The Ihara zeta function is a graph theoretical analogue of the Selberg zeta function and can be used to count non-backtracking walks in the graph. I will prove - in a purely combinatorial way - that, if a graph has average degree larger than 4, it can be reconstructed from its edge-deleted subgraphs. The result will then be put in the context of noncommutative geometry on the boundary of the universal covering (joint work with Janne Kool).

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## Christopher Deninger: Witt vectors and the relative de Rham Witt complex

We give a foundation of Witt vector rings which doesn't use universal polynomials to define the ring structure. Using this point of view one gets a very simple construction of the relative de Rham Witt complex of Chatzistamatiou (a quotient of Hesselholt's big de Rham Witt complex) which generalizes the original constructions of Illusie and also the work of Langer and Zink. This is joint work with Joachim Cuntz. We also note that the new Witt vector ring construction gives functorial non-commutative Witt vector rings with Frobenius, Verschiebung and Teichmüller character for arbitrary non-commutative rings.

## Lars Hesselholt: The big de Rham-Witt complex

The cyclotomic trace map of Bökstedt-Hsiang-Madsen, is a map from algebraic $K$-theory to a theory called $\operatorname{TR}$. The big de Rham-Witt complex, introduced in joint work with Ib Madsen, bears the same relationship to $\operatorname{TR}$-theory as Milnor $K$-theory bears to Quillen's algebraic $K$-theory. It generalizes the classical $p$-typical de Rham-Witt complex introduced by Bloch-Deligne-Illusie with the purpose of understanding the structure of the Berthelot-Grothendieck crystalline cohomology of regular schemes over $\mathbb{F}_p$. In this talk, I will present a new and explicit construction of the big de Rham-Witt complex which is based on a theory of modules and derivations over $\lambda$-rings. A surprising outcome of this new construction is an interpretation of the big de Rham-Witt complex as the complex of differentials along the leaves of a foliation.

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## Max Karoubi: Hermitian K-theory invariants in Topology and Algebraic Geometry

Hermitian K-theory (aka Witt groups) is the analog of Algebraic K-theory, defined through the orthogonal or symplectic group in lieu of the general linear group. Although for C* algebras we get essentially the same invariants as for usual K-groups, the theory is quite different for general algebras with involution. A related invariant is Michael Atiyah's KR theory associated to a space with involution. It has connections with real elliptic operators on one hand and real Algebraic Geometry on the other hand. In this lecture, following Atiyah's ideas, we define a map from the Witt group associated to a real algebraic variety to the Witt analog of KR-theory. We show that this map is very close to be an isomorphism between two invariants of seemingly different natures. This is joint work with Charles Weibel.

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## Marcelo Laca: Bost Connes systems, Hecke algebras, and induction

A Bost-Connes type system is a C*-dynamical system, constructed from an algebraic number field, that has a phase transition at low temperature exhibiting spontaneous symmetry-breaking of an action of the idele class group modulo the connected component of the identity on extremal KMS states. We show that this construction is functorial with respect to inclusion of number fields provided that, at the level of C*-dynamical systems, one considers induction through equivariant C*-correspondences with normalized dynamics to be the arrows. We also discuss an application of induction to C*-dynamical systems arising from Hecke algebras of number fields that elucidates their relation to Bost-Connes systems. This is joint work with Sergey Neshveyev and Mak Trifkovic.

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## Jeffrey Lagarias: Splitting measures on polynomials and the field with one element

We analyze the limiting splitting probabilities (mod p) for degree n polynomials with integer coefficients in a box, as the box size go to infinity. With probability one the polynomials have the symmetric group as the Galois group of their splitting field. One can interpret the splitting distributions as distributions on the symmetric group $S_n$ related to splitting of polynomials over finite fields. They vary with $p$, and their values are interpolatable by rational functions of a parameter $z=p$. As p goes to infinity these distributions approach the uniform distribution of $S_n$. As p goes to 1 they approach another distribution with very interesting properties. The talk will describe these properties and discuss how they might be viewed as manifestations of the "field with one element".

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## Nadia Slavila Larsen: C*-completions of Hecke algebras associated to Gelfand pairs

The work of Bost and Connes on a C*-dynamical system with remarkable connections to class field theory brought to attention C*-algebras associated to Hecke pairs. The question arose of whether there is a category equivalence between the nondegenerate *-representations of the Hecke algebra associated to a Hecke pair (G, H) and the unitary representations of G generated by their H-fixed vectors. Hall proposed a condition that would yield an affirmative answer, and also showed that (SL2(Qp), SL2(Zp)) fails to fulfill it. Tzanev, and independently Kaliszewski, Landstad and Quigg, introduced  a topological pair (G, U) associated to a discrete Hecke pair, and discussed relationship between two natural C*-completions, one being the enveloping C*-algebra of the Banach algebra L1(G, U), the other being the corner in the full group C*-algebra C*(G) determined by the self-adjoint idempotent corresponding to U. Here we examine when these completions are isomorphic for (G, U) a Gelfand pair. We show, using property (T), that the isomorphism fails for (SLn(Qp), SLn(Zp)) when n is greater than 3. This is joint work with Rui Palma.

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## Eric Leichtnam: Deninger's programme: a motivation for developing new mathematics

Deninger has developed an infinite dimensional cohomological formalism in order to explain and prove the expected conjectures for the arithmetic zeta functions. He conjectured that these (infinite dimensional) cohomology groups should be constructed as leafwise cohomology groups on suitable foliated spaces; the (Weil type) explicit formulae being interpreted as Atiyah-Bott Lefschetz trace formulae. These foliated spaces are not known to exist and maybe they do not exist at all. But it turns out that, having in mind this motivation, one can find situations where one can prove Atiyah-Bott Lefschetz trace formulae which have interesting similarities with the explicit formulae of Weil type. We shall also see that the Artin conjecture fits naturally as a trace formula in this conjectural picture.

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## Oliver Lorscheid: Tropicalization as base change along a non-archimedean valuation

The understanding of the tropicalization Trop(X) of a classical variety X has seen several revelations in the few past years. Two corner stones are the following descriptions: Trop(X) is a quotient of the Berkovich space associated with X (by Payne and collaborators); and Trop(X) is the set of tropical points of the semiring scheme associated with the bending relations of X (by the Giansiracusa brothers). Both approaches make use of toric geometry or, more generally, F1-geometry.

In this talk, we develop a language that allows us to talk about classical varieties, tropical varieties, Berkovich spaces, F1-schemes and semiring schemes at the same time. This language is based on the notion of an ordered blueprint, and it explains the various tropicalization processes that appear in the literature in a unified way.

In particular, we show that the tropical variety can be defined in terms of the base change of the classical variety along a non-archimedean valuation of the field of definition. The notion of a base change from fields to idempotent semirings might yield a new perspective of the interplay of Arakelov theory, non-archimedean analytic geometry and characteristic 1.

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## Bram Mesland: Noncommutative geometry of Bianchi groups

A Bianchi group is a Kleinian group of the form PSL$(O_{K})$, where $O_{K}$ is the ring of integers in the imaginary quadratic field K. For such groups we establish an explicit relation between the K-homology of the boundary crossed product algebra associated to its action on hyperbolic 3-space the cohomology of of group. This relation is compatible with the index pairing between K-theory and K-homology. We show that the notion of Hecke operator for arithmetic groups has a natural definition in terms of Kasparov's KK-theory, and will discuss their relation to the aforementioned isomorphism. These results are achieved in the context of unbounded Fredholm modules, shedding light on noncommutative geometric aspects of the boundary crossed product. This is joint work with M.H. Sengun at Sheffield.

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## Cristian Popescu: Equivariant L-values, Hecke characters and the K-theory of number fields

In joint work with Banaszak, we use my results with Greither on special values of equivariant L-functions to construct a family of algebraic Hecke characters for an arbitrary CM number field, generalizing Weil's Jacobi sum Hecke characters. Further, we use certain special values of these Hecke characters to construct "Stickelberger splitting" maps for the localization sequences in the Quillen K-theory of CM and totally real number fields. We will review these results and constructions and comment on potential applications to the classical conjectures of Iwasawa and Kummer-Vandiver on class-groups of cyclotomic fields.

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## Christophe Soulé: Correspondences, K-theory and the cannibalistic class.

If X is a smooth projective curve over a finite field, the action of Frobenius on the algebraic K-theory of X can  be viewed both as the action of some Adams operation and as the action of an algebraic correspondence. One gets from this an explicit exponent for the algebraic K-groups of X. When X is the spectrum of the integers, it follows from the known results on the K-theory of X that the action of the cannibalistic class on X vanishes. By analogy with the function field case, one can ask whether this action can also be understood as the action of an "arithmetic correspondence".

## Takuya Takeishi: Irreducible Representations of Bost-Connes systems

The classification problem of Bost-Connes systems was studied by Cornellissen and Marcolli partially, but still remains unsolved. In this paper, we will give a representation-theoretic approach to this problem.

We will generalize the result of Laca and Raeburn, which concerns with the primitive ideal space on Bost-Connes system for $\mathbb{Q}$.

As a consequence, Bost-Connes $C^*$-algebra for a number field $K$ has $h_K^1$-dimensional irreducible representations and doesn't have finite-dimensional irreducible representations for other dimensions, where $h_K^1$ is the strict class number of $K$. In particular, the strict class number is an invariant of Bost-Connes $C^*$-algebras.

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## Georg Tamme: Differential algebraic K-theory

Differential algebraic K-theory is a theory defined on pairs (X,M) where X is a smooth variety over Q and M is a smooth manifold. It combines information about algebraic K-theory classes and differential forms, which represent the images of these classes under regulator maps, in a non-trivial way. For instance, a class in differential algebraic K-theory can contain interesting secondary information. I will explain the construction of this theory and give an application to a conjecture of Lott's.

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