Caterina (Katia) Consani

My Research:

My research interests lie in noncommutative arithmetic geometry, with a particular focus on forging new connections between noncommutative geometry, algebraic geometry, and number theory. This work is motivated, in part, by the goal of developing a strategy to approach the Riemann Hypothesis.  

Noncommutative arithmetic geometry emerged in the early 1990s with the introduction of the Bost-Connes quantum statistical dynamical system (commonly known as the BC-system). This system features the Riemann zeta function as its partition function, while its zero-temperature vacuum states realize the global class field isomorphism for the rational number field Q.                                                                                                                                                                                                      In an extensive collaborative project with Alain Connes, we have uncovered new connections between the BC-system, p-adic analysis, and the theory of Witt vectors. This work has also led to the development of an archimedean counterpart to the theory of rings of periods in p-adic Hodge theory. Additionally, we have identified the arithmetic significance of the cyclic homology of schemes, which provides a framework to reinterpret Serre's archimedean factors of the Hasse-Weil L-function of a projective algebraic variety over a number field as regularized determinants. 

The study of the adele class space of Q has revealed a correspondence between its algebro-geometric structure and its original analytic definition (due to A. Connes). To establish a connection between this space and the Riemann zeta function, attention is restricted to a specific sector of the adele class space, defined by the maximal compact subgroup of the idele class group. The resulting algebro-geometric space, which describes this (noncommutative) double quotient, is a semi-ringed Grothendieck topos. Its foundational definition—the Arithmetic Site—is constructed as a topos over the Boolean B: the smallest idempotent semifield (also known as the smallest semifield of characteristic one). Our construction is synthesized in the Dialogue with Alain Connes, and in my talk at IHES 2015.  

Extending scalars from B to the multiplicative version of the tropical reals reveals a second significant topos: the Scaling Site. The points of this topos correspond one-to-one with the adele classes of the previously mentioned double quotient. Both the Arithmetic Site and the Scaling Site are defined over algebraic idempotent semi-structures. Consequently, my research also involves developing a general theory of homological algebra for categories such as the category of sheaves of idempotent modules over a topos. A key example is the category of modules over the Boolean semifield, viewed as the characteristic-one analog of the category of abelian groups. This research is described in the Course of Alain Connes at College de France 2016-17, and in my talk at the Conference in honor of Alain Connes 70th Birthday at Fundan University. 

Given the lack of a direct link between the ring of the integers Z and B, this research has recently branched to include a categorical framework of modules and algebras over the categorical sphere spectrum S. This approach, rooted in the theory of Segal's Gamma-sets, offers a characteristic-free, unified perspective on the geometry of the rational primes and the topos-theoretic description of the quotient of the adele class space of Q. This development is explained in the papers:On Absolute Algebraic Geometry the affine case, Segal’s Gamma rings and universal arithmetic, BC-system, absolute cyclotomy and the quantized calculus. 

This research has produced, in Arakelov geometry, a relevant advancement in the Riemann-Roch formula, following the idea that the compactification of the algebraic spectrum of the integers is a curve over the absolute ring S[+-1]. The results of this research are reported in the articles:  Riemann-Roch for the ring Z and On the metaphysics of F1.

Just as Grothendieck's theory of the étale fundamental group extends Galois theory to schemes, the adele class space, viewed as a covering of the Scaling topos, provides a corresponding extension of the class field isomorphism for Q to schemes related to Spec Z. The Scaling topos and its periodic orbits Cₚ, with lengths log p, provide a geometric framework for the well-established analogy between primes and knots. In this framework, the adele class space functions as the maximal abelian cover of the Scaling topos. The preimage of a periodic orbit Cₚ in this cover is canonically isomorphic to the mapping torus of the the multiplication by the geometric Frobenius at p  in the abelianized étale fundamental group of the spectrum of the local ring Zₚ. This construction reveals the linking of the prime p with all other rational primes. The details of this development are contained in the paper Knots, primes and class field theory.

In a natural continuation of the geometric interpretation of the adele class space of the rationals, a recent preprint develops a Picard–Jacobian formalism for the compactified arithmetic curve Spec Z within the adelic framework. In this construction, elements of the adele class space, and specifically its Riemann sector, are described in terms of torsion-free rank-one abelian groups equipped with additional rigidifying data that extend the notion of metrized line bundles in Arakelov geometry. This point of view leads to the definition of a Picard monoid associated with the completed arithmetic curve and to a corresponding arithmetic Jacobian obtained by quotienting by the natural scaling action of the positive real numbers. Within this framework the product of adeles acquires a direct geometric meaning through the tensor product of these rank-one groups, while the Riemann sector of the adele class space appears naturally as a distinguished component of the resulting geometric structure. This approach provides a conceptual interpretation of the geometric strata arising in the spectral realization of L-functions and further clarifies the geometric role of the adele class space in the study of the Riemann zeta function. The details of this development are contained in the preprint On the Jacobian of Spec Z.

In the very recent paper On the absolute geometry of Spec Z we achieved several important steps. First, we upgrade Pic(Spec Z)  to a Grothendieck topos equipped with a structure sheaf defined over F1 (that is the spherical algebra of the multiplicative pointed monoid {0,1}). More precisely, we introduce the F1-arithmetic site as the pair made by the arithmetic topos (i.e. the presheaf topos of sets endowed with an action of the multiplicative monoid of non-negative integers), and the structure sheaf of spherical F1-algebras F1[T]. A central connected definition is the Abel–Jacobi map θ from Spec Z to Pic(Spec Z), which assigns to a prime p the additive abelian group Z[1/p],  and to the generic point of Spec Z the point Z.  The map θ compares well with the previously introduced morphism of toposes Θ from Spec Z to the arithmetic topos, the agreement is achieved through a natural map from the points of the arithmetic topos to Pic(Spec Z). 

A second important step is achieved with the notion of the F1-arithmetic-curve (Spec Z)_F1, namely the pair (Spec Z, ℱ), where the sheaf ℱ is the pullback along the morphism Θ of the structure sheaf of F1-arithmetic site. 

F1-arithmetic-curve provides through its geometric points a realization of  P. Scholze's heuristic idea (as reported in J. Lurie notes), in the form of the following:

Theorem Let F be an algebraically closed perfectoid field of characteristic p. The moduli space of local F-points of (Spec Z)_F1, modulo the canonical symmetries of the stalks, decomposes over the closed points of Spec Z as follows:

(i) At any prime different from p, the moduli space collapses to a single orbit.

(ii) At the prime p, the moduli space is canonically in bijection with the space of all untilts of F modulo the Frobenius.

While this theorem provides a direct geometric realization of Scholze’s original heuristic, the absolute nature of (Spec Z)_F1 leads to a substantially more universal phenomenon. Since the geometry of this curve is defined below Z, it is entirely insensitive to the characteristic of the test field. In particular,  the F1-curve may be evaluated over an arbitrary perfectoid field C of residue characteristic p, without requiring C to have characteristic p or to be algebraically closed. In this general setting, the algebraic structure of the F1-stalk at p intrinsically enforces the extraction of p-power roots, while the locality condition rigidly constrains the image to the subgroup of principal units. This leads to the following generalization of the above theorem and to an absolute interpretation of the (closed points of the) Fargues-Fontaine curve:

Theorem Let C be a perfectoid field of residue characteristic p.

(i) The space of C-points of the F1-curve (Spec Z)_F1 at the prime p is canonically identified with the open unit disk: 1+𝔪_{C♭} inside the tilt of C.

(ii) Assume that C is algebraically closed. Then the space of C-points of (Spec Z)_F1 at p, modulo the intrinsic symmetries of the stalks, is canonically identified with the space of untilts of C♭  modulo Frobenius (ie the closed points of the Fargues--Fontaine curve 𝒳_{C♭,Q_p} together with the distinguished point corresponding to C♭ itself. 

(iii) For every prime ℓ different from p, the moduli space of C-points reduces to a single orbit under the action of the symmetry group (Q_ℓ)^*.

This results shows that the absolute F1-geometry furnishes a universal geometric receptacle for perfectoid fields. It accommodates fields of arbitrary characteristic, canonically recovers their tilt through the local arithmetic of the stalk, and produces the (closed points of the) Fargues–Fontaine curve precisely at the corresponding prime.

To further illustrate the universality of (Spec Z)_F1, one may evaluate its  points in the field C of complex numbers. At a fixed prime p, the absolute geometry naturally detects two distinct places, Σₚ={p,∞}, corresponding respectively to the p-adic and archimedean topologies on the stalk of the structure sheaf.

Theorem Let  (Spec Z)_F1 be evaluated over C at the prime p. For each place v∈ Σₚ={p,∞},the locality condition canonically extends the Frobenius action of 𝐍× to a continuous and well-defined action of the local Weil group W_v. 

Moreover, the space 𝓜ₚᵛ of non-trivial local points is a principal homogeneous space under W_v; equivalently, it consists of a single W_v-orbit.

This unified torsor structure immediately gives rise to the fundamental geometric quotients. At the archimedean place, quotienting the W_∞-torsor 𝓜ₚ∞ by the discrete arithmetic symmetries generated by Frobenius, p^Z ⊂ W_∞, produces the complex Tate curve E_p=𝓜ₚ∞/p^Z≅ C^*/p^Z which canonically carries the structure of an elliptic curve over C. Furthermore, the canonical action of the complex conjugation on the archimedean points induces a natural real structure on E_p. The periodic orbit C_p of the adelic geometry  then appears canonically as the quotient: E_p(R)/⟨ ±1⟩, where the  element -1∈ W_∞ geometrically exchanges the two connected components of the real locus.

Alternatively, one may quotient by the continuous symmetries of the completed stalk. At the level of the Weil group, this corresponds to the subgroup fixed by complex conjugation σ \∈ Aut(W_∞), Quotienting the torsor by this subgroup collapses the geometry to the real projective line.

The overarching aim of this research in number theory is to adapt and extend some of A. Weil's ideas from his proof of the Riemann Hypothesis for function fields to the context of algebraic number fields, leveraging the tools and techniques of noncommutative geometry.  Several recent developments from this joint research with A. Connes (and lately also in collaboration with Henri Moscovici) are discussed in:

• Weil positivity and trace formula: the archimedean place 

• Quasi-inner functions and local factors 

• Spectral triples and zeta cycles 

• Zeta zeros and prolate wave operators: semilocal adelic operators  2025-Best Paper AOFA Award Winner (among 167 papers published in the years 2023 and 2024) in Annals of Functional Analysis

• On q-series and the moment problem associated to local factors 

• Zeta Spectral Triples