arXiv subjects

Alain Connes

Publications and source records attributed to Alain Connes.

At least 19 recordsLinked to original sources

arXiv2026

The Absolute Twistor Line and the Geometry of $\overline{\text{Spec}\, \mathbf Z}$

We construct the absolute algebraic geometry of the compactification $\overline{\text{Spec}\, \mathbf Z}$ by amalgamating the affine absolute curve $(\text{Spec}\, \mathbf Z)_{\mathbf{F}_{1}}$ with an archimedean component defined over the signed extension $\mathbf{F}_{1^2}$ of $\mathbf{F}_1$. By adjoining a formal imaginary unit to the absolute projective line, we obtain an equivariant topos endowed with a canonical geometric inversion symmetry, which induces the twistor real structure on its complex points. This archimedean geometry is incorporated into a global absolute curve defined as an internal object of the odd arithmetic topos, dual to the multiplicative monoid of odd positive integers, and governed by the intrinsic Hopf structure of spherical $\mathbf{F}_{1^2}$-algebras. The restriction of the absolute Frobenius action to odd integers is forced arithmetically by the extension of scalars to $\mathbf{F}_{1^2}$. On complex points, the resulting dynamics simultaneously generates the Adams operations and complex conjugation on real Hodge structures. At the categorical level, the odd arithmetic topos originates in the pericyclic category, whose $\lambda$-operations provide a conceptual interpretation of the local factors of geometric L-functions.

math.AG↗

arXiv2026

On the Absolute Geometry of $\operatorname{Spec}\mathbf{Z}$

A guiding principle in P. Scholze's $p$-adic geometry asserts that the points of $\operatorname{Spec}\mathbf{Z}$ over an algebraically closed perfectoid field of characteristic $p$ are classified, up to equivalence, by its untilts. In this paper, we give a concrete geometric realization and a generalization of this paradigm. We construct the absolute $\mathbf{F}_1$-arithmetic curve $(\operatorname{Spec}\mathbf{Z})_{\mathbf{F}_1}$ by pulling back the $\mathbf{F}_1$-structure sheaf of the arithmetic site to $\operatorname{Spec}\mathbf{Z}$. We demonstrate that $(\operatorname{Spec}\mathbf{Z})_{\mathbf{F}_1}$ provides a common geometric origin for fundamental structures in $p$-adic Hodge theory, complex analytic geometry, and the adelic scaling site. The moduli space of points of $(\operatorname{Spec}\mathbf{Z})_{\mathbf{F}_1}$ over an arbitrary perfectoid field, modulo intrinsic symmetries, canonically parameterizes the space of all perfectoid fields with the same tilt, providing a universal, characteristic-independent geometric realization of Scholze's heuristic. Evaluating the points of $(\operatorname{Spec}\mathbf{Z})_{\mathbf{F}_1}$ over the field $\mathbf{C}$ of complex numbers reveals, at each prime $p$, that the non-trivial points canonically form two principal homogeneous spaces (torsors) over the Weil groups $W_p=\mathbf{Q}_p^\times$ and $W_\infty=\mathbf{C}^\times$. Quotienting the archimedean orbit by the discrete Frobenius symmetries yields the complex Tate curve with modulus $q=p^{-1}$. We show that this elliptic curve canonically decomposes as the product of its real locus, which exactly recovers the adelic periodic orbit $C_p=\mathbf{R}_+^\times/p^\mathbf{Z}$, and a $p$-independent phase space that emerges naturally as a real analogue of the Fargues--Fontaine curve.

math.AG↗

arXiv2026

On the Jacobian of $\overline{{{\rm Spec}\,\mathbb Z}}$

We interpret the structure of the adele class space of the rationals--and specifically its Riemann sector--as the natural monoidal extension of the Picard group of the arithmetic curve $\overline{\operatorname{Spec} \mathbb Z}$. We identify the elements of this space with torsion-free rank-1 abelian groups $L$ endowed with rigidifying data. In the Riemann sector, this data corresponds to a norm, extending the classical notion of metrized line bundles in Arakelov geometry. For the full adele class space, we replace the norm with a group morphism to $\mathbb R$ and a combinatorial datum: a parametrization of the roots of unity associated with the character dual of $L$. We show that the product of adeles is represented geometrically by the tensor product of these rank-1 groups and their rigidifying structures. The resulting monoid space generalizes the Picard group to the full adelic context by incorporating the singular strata required for the spectral realization of $L$-functions.

math.NT↗

arXiv2026

The Riemann Hypothesis: Past, Present and a Letter Through Time

This paper, commissioned as a survey of the Riemann Hypothesis, provides a comprehensive overview of 165 years of mathematical approaches to this fundamental problem, while introducing a new perspective that emerged during its preparation. The paper begins with a detailed description of what we know about the Riemann zeta function and its zeros, followed by an extensive survey of mathematical theories developed in pursuit of RH -- from classical analytic approaches to modern geometric and physical methods. We also discuss several equivalent formulations of the hypothesis. Within this survey framework, we present an original contribution in the form of a "Letter to Riemann," using only mathematics available in his time. This letter reveals a method inspired by Riemann's own approach to the conformal mapping theorem: by extremizing a quadratic form (restriction of Weil's quadratic form in modern language), we obtain remarkable approximations to the zeros of zeta. Using only primes less than 13, this optimization procedure yields approximations to the first 50 zeros with accuracies ranging from $2.6 \times 10^{-55}$ to $10^{-3}$. Moreover we prove a general result that these approximating values lie exactly on the critical line. Following the letter, we explain the underlying mathematics in modern terms, including the description of a deep connection of the Weil quadratic form with the world of information theory. The final sections develop a geometric perspective using trace formulas, outlining a potential proof strategy based on establishing convergence of zeros from finite to infinite Euler products. While completing the commissioned survey, these new results suggest a promising direction for future research on Riemann's conjecture.

math.NT↗

arXiv2025

Quadratic Forms, Real Zeros and Echoes of the Spectral Action

For a real distribution $\mathcal{D}$ on the interval $[0,L]$ with $\tilde{\mathcal{ D}}$ the associated even distribution on the interval $[-L, L]$, we prove that if the associated quadratic form with Schwartz kernel $\tilde{\mathcal{D}}(x - y)$ defines a lower-bounded selfadjoint operator on $L^2([-\frac{L}{2}, \frac{L}{2}])$, whose lowest spectral value $\lambda$ is a simple, isolated eigenvalue with even eigenfunction $\xi$, then all the zeros of the entire function $\widehat \xi(z)$, the Fourier transform of $\xi$, lie on the real line. The proof proceeds in five steps. (1) We give a C*-algebraic proof of a corollary of Carath\'eodory-Fej\'er's 1911 structure Theorem for Toeplitz matrices: if $T \in M_n(\mathbb{C})$ is a Hermitian, positive semidefinite Toeplitz matrix of rank $n - 1$, and $\xi \in \ker T$, then the polynomial $P(z) = \sum \xi_j z^j$ has all its zeros on the unit circle. (2) We formulate and prove a continuous analogue of this result, replacing the Toeplitz matrix with a convolution operator with continuous kernel $h(x - y)$, and the polynomial $P(z)$ with the Fourier transform of the eigenfunction corresponding to the largest eigenvalue. (3) We analyze finite-dimensional truncations of the quadratic forms defined by real, even distributions $\mathcal{D}$ on $[-L, L]$, and observe that the resulting matrices exhibit a structure previously encountered in perturbative expansions of the spectral action. (4) We establish an analogue of Carath\'eodory-Fej\'er's corollary for matrices of this specific structure, thereby extending the zero localization result beyond the classical Toeplitz setting. (5) Finally, we apply a classical theorem of Hurwitz concerning the zeros of uniform limits of holomorphic functions to deduce the general result stated above.

math.OA↗

arXiv2025

Zeta Spectral Triples

We propose and investigate a strategy toward a proof of the Riemann Hypothesis based on a spectral realization of its non-trivial zeros. Our approach constructs self-adjoint operators obtained as rank-one perturbations of the spectral triple associated with the scaling operator on the interval $[\lambda^{-1}, \lambda]$. The construction only involves the Euler products over the primes $p \leq x = \lambda^2$ and produces self-adjoint operators whose spectra coincide, with striking numerical accuracy, with the lowest non-trivial zeros of $\zeta(1/2 + i s)$, even for small values of $x$. The theoretical foundation rests on the framework introduced in "Spectral triples and zeta-cycles" (Enseign. Math. 69 (2023), no. 1-2, 93-148), together with the extension in "Quadratic Forms, Real Zeros and Echoes of the Spectral Action" (Commun. Math. Phys. (2025)) of the classical Caratheodory-Fejer theorem for Toeplitz matrices, which guarantees the necessary self-adjointness. Numerical experiments show that the spectra of the operators converge towards the zeros of $\zeta(1/2 + i s)$ as the parameters $N, \lambda \to \infty$. A rigorous proof of this convergence would establish the Riemann Hypothesis. We further compute the regularized determinants of these operators and discuss the analytic role they play in controlling and potentially proving the above result by showing that, suitably normalized, they converge towards the Riemann $\Xi$ function.

math.NT↗

arXiv2025

Knots, primes and class field theory

In this paper, we present a geometric generalization of class field theory, demonstrating how adelic constructions, central to the spectral realization of zeros of L-functions and the geometric framework for explicit formulas in number theory, naturally extend the classical theory. This generalization transitions from the idele class group, which acts as the adelic analog of Galois groups, to a geometric framework associated with schemes and the ring of integers of global fields. This perspective provides a conceptual explanation for the role of the adele class space in the spectral realization of L-function zeros and identifies the idele class group as a generic point in this context. The sector $X_{\mathbb{Q}}$ of the adele class space corresponding to the Riemann zeta function gives the class field counterpart of the scaling topos. The main result is the construction of a functor mapping finite abelian extensions of $\mathbb{Q}$ to finite covers of $X_{\mathbb{Q}}$, with the monodromy of periodic orbits of length $\log p$ under the scaling action corresponding to the Galois action of the Frobenius at the prime p.

math.NT↗

arXiv2024

Pierre Cartier: A Visionary Mathematician

This article reflects on the life and mathematical contributions of Pierre Cartier, a distinguished figure in 20th- and 21st-century mathematics. As a key member of the Bourbaki collective, Cartier played a pivotal role in the formalization and modernization of mathematics. His work spanned fields such as algebraic geometry, representation theory, mathematical physics, and category theory, leaving an indelible mark on the discipline. Beyond his technical achievements, Cartier was celebrated for his intellectual generosity and humanistic approach to science, shaping not only mathematical thought but also the broader cultural understanding of the field.

math.HO↗

arXiv2024

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

We investigate the moment problem and Jacobi matrix associated -- by the operator theoretic framework of the semilocal trace formula -- to each finite set $S$ of places of $\mathbb Q$ containing the archimedean place. The measure is given by the absolute value squared of the product over $S$ of local factors restricted to the critical line. We treat the case $S=\{p,\infty\}$, where a single prime $p$ is adjoined to the archimedean place. We find that all the key ingredients such as the moments, the orthogonal polynomials and the Jacobi matrices can be expressed as power series in terms of the parameter $q:=1/p$. We show that the series which appear for the moments themselves are Lambert series. The study of the $q$-series for the coefficients of the Jacobi matrix, and for the associated orthogonal polynomials reveals an intriguing integrality result: all those coefficients belong to the ring $\mathbb Z[\frac{1}{\sqrt{2}}]$ obtained by adjoining $1 / \sqrt{2}$ to the ring of integers. The main result of this paper is the conceptual explanation of this integrality property using Catalan numbers.

math.NT↗

arXiv2024

Heat Expansion and Zeta

We compute the full asymptotic expansion of the heat kernel Trace$(\exp(-tD^2))$ where $D$ is, assuming RH, the self-adjoint operator whose spectrum is formed of the imaginary parts of non-trivial zeros of the Riemann zeta function. The coefficients of the expansion are explicit expressions involving Bernoulli and Euler numbers. We relate the divergent terms with the heat kernel expansion of the Dirac square root of the prolate wave operator investigated in our joint work with Henri Moscovici.

math.NT↗

arXiv2024

Knots, Primes and the adele class space

We show that the scaling site $X_{\mathbb Q}$ and its periodic orbits $C_p$ of length $\log p$ offer a geometric framework for the well-known analogy between primes and knots. The role of the maximal abelian cover of $X_{\mathbb Q}$ is played by the quotient map $\pi:X_{\mathbb Q}^{ab}\to X_{\mathbb Q}$ from the adele class space $X_{\mathbb Q}^{ab}:={\mathbb Q}^\times \backslash {\mathbb A}_{\mathbb Q}$ to $X_{\mathbb Q}=X_{{\mathbb Q}}^{ab}/{\hat{\mathbb Z}^*}$. The inverse image $\pi^{-1}(C_p)\subset X_{\mathbb Q}^{ab}$ of the periodic orbit $C_p$ is canonically isomorphic to the mapping torus of the multiplication by the Frobenius at $p$ in the abelianized \'etale fundamental group $\pi_1^{e t}({\rm Spec} \, {\mathbb Z}_{(p)})^{ab}$ of the spectrum of the local ring ${\mathbb Z}_{(p)}$, thus exhibiting the linking of $p$ with all other primes. In the same way as the Grothendieck theory of the \'etale fundamental group of schemes is an extension of Galois theory to schemes, the adele class space gives, as a covering of the scaling site, the corresponding extension of the class field isomorphism for $\mathbb Q$ to schemes related to ${\rm Spec} \,\mathbb Z$.

math.NT↗

arXiv2023

Zeta zeros and prolate wave operators

We integrate in the framework of the semilocal trace formula two recent discoveries on the spectral realization of the zeros of the Riemann zeta function by introducing a semilocal analogue of the prolate wave operator. The latter plays a key role both in the spectral realization of the low lying zeros of zeta--using the positive part of its spectrum--and of their ultraviolet behavior--using the Sonin space which corresponds to the negative part of the spectrum. In the archimedean case the prolate operator is the sum of the square of the scaling operator with the grading of orthogonal polynomials, and we show that this formulation extends to the semilocal case. We prove the stability of the semilocal Sonin space under the increase of the finite set of places which govern the semilocal framework and describe their relation with Hilbert spaces of entire functions. Finally, we relate the prolate operator to the metaplectic representation of the double cover of $\operatorname{SL}(2,\mathbb R)$ with the goal of obtaining (in a forthcoming paper) a second candidate for the semilocal prolate operator.

math.NT↗

arXiv2023

On the metaphysics of $\mathbb F_1$

In the present paper, dedicated to Yuri Manin, we investigate the general notion of rings of $\mathbb S[\mu_{n,+}]$-polynomials and relate this concept to the known notion of number systems. The Riemann-Roch theorem for the ring $\mathbb Z$ of the integers that we obtained recently uses the understanding of $\mathbb Z$ as a ring of polynomials $\mathbb S[X]$ in one variable over the absolute base $\mathbb S$, where $1+1=X+X^2$. The absolute base $\mathbb S$ (the categorical version of the sphere spectrum) thus turns out to be a strong candidate for the incarnation of the mysterious $\mathbb F_1$.

math.NT↗

arXiv2023

Riemann-Roch for the ring $\mathbb Z$

We show that by working over the absolute base $\mathbb S$ (the categorical version of the sphere spectrum) instead of $\mathbb S[\pm 1]$ improves our previous Riemann-Roch formula for $\overline{{\rm Spec\,}\mathbb Z}$. The formula equates the (integer-valued) Euler characteristic of an Arakelov divisor with the sum of the degree of the divisor (using logarithms with base 2) and the number $1$, thus confirming the understanding of the ring $\mathbb Z$ as a ring of polynomials in one variable over the absolute base $\mathbb S$, namely $\mathbb S[X], 1+1=X+X^2$.

math.NT↗

arXiv2022

Cyclic theory and the pericyclic category

We give a historical perspective on the role of the cyclic category in the development of cyclic theory. This involves a continuous interplay between the extension in characteristic one and in S-algebras, of the traditional development of cyclic theory, and the geometry of the toposes associated with several small categories involved. We clarify the link between various existing presentations of the cyclic and the epicyclic categories and we exemplify the role of the absolute coefficients by presenting the ring of the integers as polynomials in powers of 3, with coefficients in the spherical group ring S[C_2] of the cyclic group of order two. Finally, we introduce the pericyclic category which unifies and refines two conflicting notions of epicyclic space existing in the literature.

math.AT↗

arXiv2022

Noncommutativity and Physics: A non-technical review

We give an overview of the applications of noncommutative geometry to physics. Our focus is entirely on the conceptual ideas, rather than on the underlying technicalities. Starting historically from the Heisenberg relations, we will explain how in general noncommutativity yields a canonical time evolution, while at the same time allowing for the coexistence of discrete and continuous variables. The spectral approach to geometry is then explained to encompass two natural ingredients: the line element and the algebra. The relation between these two is dictated by so-called higher Heisenberg relations, from which both spin geometry and non-abelian gauge theory emerges. Our exposition indicates some of the applications in physics, including Pati--Salam unification beyond the Standard Model, the criticality of dimension 4, second quantization and entropy.

hep-th↗

arXiv2022

Charmless two-body $B$ meson decays in perturbative QCD factorization approach

The perturbative QCD (PQCD) approach based on $k_T$ factorization has made a great achievement for the QCD calculation of the hadronic B decays. Regulating the endpoint divergence by the transverse momentum of quarks in the propagators, one can do the perturbation calculation for kinds of diagrams including the annihilation type diagrams. In this paper, we review the current status of PQCD factorization calculation of two-body charmless $B\to PP, PV, VV$ decays up to the next-to-leading order (NLO) QCD corrections. two new power suppressed terms in decaying amplitudes are also taken into account. By using the universal input (non-perturbative) parameters, we collected the branching ratios and ${\bf CP}$ asymmetry parameters for all the charmless two body $B$ decays, calculated in the PQCD approach up to the NLO, including some power suppressed terms in decaying amplitudes. The results are compared with the ones from QCD factorization approach, soft-collinear effective theory approach and the current experimental measurements. For most considered B meson decays, the PQCD results for branching ratios agree well with other approaches and the experimental data. The PQCD predictions for the ${\bf CP}$ asymmetry parameters for many of the decay channels do not agree with other approaches, but have a better agreement with the experimental data. The longstanding $K \pi $ puzzle about the pattern of the direct CP asymmetries of the penguin-dominated $B \to K \pi $ decays can be understood after the inclusion of the NLO contributions in PQCD. The NLO corrections and power suppressed terms play an important role in the color suppressed and pure annihilation type $B$ decay modes. These rare decays are more sensitive to different types of corrections, providing opportunity to examine the factorization approach with the more precise experimental measurements.

hep-ph↗

arXiv2022

Riemann-Roch for $\overline{\text{Spec}\mathbb Z}$

We prove a Riemann-Roch theorem of an entirely novel nature for divisors on the Arakelov compactification of the algebraic spectrum of the integers. This result relies on the introduction of three key concepts: the cohomologies (attached to a divisor), their integer dimension, and Serre duality. These notions directly extend their classical counterparts for function fields. The Riemann-Roch formula equates the (integer valued) Euler characteristic of a divisor with a slight modification of the traditional expression in terms of the sum of the degree of the divisor and the logarithm of 2. Both the definitions of the cohomologies and of their dimensions rely on a universal arithmetic theory over the sphere spectrum that we had previously introduced using Segal's Gamma rings. By adopting this new perspective we can parallel Weil's adelic proof of the Riemann-Roch formula for function fields including the use of Pontryagin duality.

math.AG↗