An adequate Riemann–Roch theorem in characteristic one for the arithmetic site and its square: the missing intersection-theoretic input of the Weil-style strategy for the Riemann zeta function

The Arithmetic Site of Connes and Consani (2014) is a semiringed topos in characteristic one — the tropical semiring of integers as a structure sheaf on the topos dual to the multiplicative semigroup of positive integers — whose points over the tropical semifield reproduce the adele class space underlying the noncommutative-geometric approach to the Riemann zeta function, and in whose square the Frobenius correspondences live. The paper states that transposing Weil's proof of the Riemann Hypothesis from function fields to this framework would require in particular an adequate version of the Riemann–Roch theorem in characteristic one. Subsequent work (2014–2026) proved a type-II Riemann–Roch formula with real-valued dimensions on the periodic orbits of the scaling site (2015–2017) and Riemann–Roch theorems of a novel homotopical nature for Arakelov divisors over the sphere spectrum (2022–2023), while the intersection theory and Riemann–Roch theorem on the square of the arithmetic/scaling site — blocked on a workable sheaf cohomology $H^{1}$ for idempotent structures, and by the unclassified tangential deformation $\operatorname{Id}_{\epsilon}$ in the correspondence composition law — remain open.

Background

The Riemann Hypothesis (RH) asserts that every non-trivial zero of the Riemann zeta function has real part $1/2$. For a smooth projective curve over a finite field, the analogous statement was proved by Weil and, in the Mattuck–Tate and Grothendieck reformulations, follows from the Riemann–Roch theorem on the surface $C\times C$ together with the Castelnuovo–Severi inequality for correspondences (a classical positivity inequality for the intersection matrix of correspondences on a surface, in which a correspondence is transposed as one transposes an endomorphism); a correspondence is a divisor (an integral cycle) on the product of the curve with itself, and it acts on the cohomology that carries the zeros of the curve's zeta function. The strategy is thus: use Riemann–Roch on the square of the curve to control the intersection pairing of correspondences and derive the positivity that forces the zeros onto the line.

Connes has pursued an analytic counterpart of this strategy for the Riemann zeta function since the 1990s. The relevant noncommutative space is the quotient of the adele class space of $\mathbb{Q}$ (the adeles $\mathbb{A}_{\mathbb{Q}}$ are a restricted product of all completions of $\mathbb{Q}$; one quotients by the diagonal action of $\mathbb{Q}^{\times}$) by the maximal compact subgroup $\hat{\mathbb{Z}}^{*}$ of the idele class group. A trace formula on this space exhibits a scaling action whose Hasse–Weil counting function (the analogue, in the limit $q\to 1$, of counting points over extensions $\mathbb{F}_{q^r}$) reproduces the complete Riemann zeta function.

Connes and Consani then constructed an algebraic-geometric incarnation of this space, the arithmetic site. It is a semiringed topos. A semiring differs from a ring in that additive inverses need not exist; in the semirings relevant here — semirings of characteristic one, where $1+1=1$, so that addition is idempotent — one is in the tropical (max-plus) world. The underlying topos is $\widehat{\mathbb{N}^{\times}}$, the presheaf topos of the one-object category whose endomorphisms are the multiplicative monoid $\mathbb{N}^{\times}$ of positive integers (equivalently, sets with an action of $\mathbb{N}^{\times}$). The structure sheaf is the tropical semiring of integers $\bar{\mathbb{N}}=(\mathbb{N}\cup\{\infty\},\inf,+)$, with $\mathbb{N}^{\times}$ acting by multiplication. Its set of points over the maximal compact subring $[0,1]_{\max}$ of the tropical semifield $\mathbb{R}^{+}_{\max}$ is canonically the quotient $\mathbb{Q}^{\times}\backslash\mathbb{A}_{\mathbb{Q}}/\hat{\mathbb{Z}}^{*}$, and the action of Frobenius automorphisms $\mathrm{Fr}_{\lambda}$, $\lambda\in\mathbb{R}^{+}_{\times}$, on these points corresponds to the scaling action whose counting function yields the complete zeta function.

The Frobenius correspondences $\Psi(\lambda)$ are realized in the square of the site. The square has an unreduced version, with structure sheaf $\bar{\mathbb{N}}\otimes_{\mathbb{B}}\bar{\mathbb{N}}$ over the Boolean semifield $\mathbb{B}=\{0,1\}$, and a reduced version whose structure sheaf is the multiplicatively cancellative semiring $\operatorname{Conv}_{\geq}(\mathbb{N}\times\mathbb{N})$ of Newton polygons: closed convex subsets $C$ of the quadrant with $C+\mathrm{quadrant}=C$ and extreme points in $\mathbb{N}\times\mathbb{N}$, added by convex hull of the union and multiplied by Minkowski sum. The composition law of correspondences was computed: $\Psi(\lambda)\circ\Psi(\lambda')=\Psi(\lambda\lambda')$ holds except in the special case where $\lambda$ and $\lambda'$ are both irrational while $\lambda\lambda'$ is rational, where the composite acquires the tangential deformation $\operatorname{Id}_{\epsilon}$ of the identity correspondence, built from the semiring of germs at $0$ of continuous functions with values in $\mathbb{R}^{+}_{\max}$ — an infinitesimal $\epsilon$-deformation encoding a flexibility that irrational correspondences have and rational ones lack.

What is missing to complete the Weil-style strategy is exactly the input that Riemann–Roch on $C\times C$ supplies in the function-field case. The source paper closes by stating that the translation of Weil's proof would require in particular an adequate version of the Riemann–Roch theorem in characteristic 1. Connes's later essay makes the target concrete: for divisors of the form

$D(f)=\int \Psi(\lambda)\,f(\lambda)\,d^{*}\lambda$

with $f$ a real-valued test function on $\mathbb{R}^{+}_{\times}$, the right-hand side of the Riemann–Roch formula is expected to be $\frac{1}{2}D.D=s(f,f)$, where $s(\cdot,\cdot)$ is the quadratic form of the explicit formula (the archimedean analogue of the trace pairing on differentials); the self-intersection $D.D$ should be computed as the intersection number of the composite correspondence $D\circ\tilde{D}$ (where $\tilde{D}$ denotes the adjoint, i.e. transposed, correspondence — the analogue, for correspondences, of the transpose of an endomorphism) with the diagonal, realized as a distributional trace analogous to the counting function.

Two bodies of prior art delimit the gap. First, tropical Riemann–Roch theorems exist, but only for curves (metric graphs and tropical curves, in the sense of Gathmann–Kerber), with integer-valued rank invariants; there is no Serre duality (the classical duality on a curve that identifies $H^{1}$ with the dual of the space of differential forms and supplies the degree-two term of Riemann–Roch) and no usable higher sheaf cohomology for idempotent structures. Second, Connes and Consani themselves proved a type-II Riemann–Roch formula on the periodic orbits of the scaling site — the semiringed topos obtained from the arithmetic site by extension of scalars from $\mathbb{B}$ to $\mathbb{R}^{+}_{\max}$: on the quasi-tropical curve $C_{p}=\mathbb{R}^{+}_{\times}/p^{\mathbb{Z}}$, an analogue of Jacobi's description of an elliptic curve with divisor class group $\mathbb{R}\times(\mathbb{Z}/(p-1)\mathbb{Z})$, they develop Cartier divisors, theta functions, and a real-valued continuous dimension $\operatorname{Dim}_{\mathbb{R}}(H^{0}(D))$ (a limit of normalized topological dimensions, in analogy with the continuous dimensions of modules over von Neumann algebras of type II), and prove

$\operatorname{Dim}_{\mathbb{R}}(H^{0}(D))-\operatorname{Dim}_{\mathbb{R}}(H^{0}(-D))=\deg(D).$

This settles the curve case inside the program, but not the square of the site, which is where the Weil strategy operates.

Problem Statement

Construct an adequate and usable Riemann–Roch theory in characteristic one, adapted to the arithmetic site and its square, supplying the missing geometric input of the Weil-style strategy for the Riemann zeta function. The components belong to one scientific objective — transposing the Mattuck–Tate–Grothendieck form of Weil's proof, as the source paper proposes — and are sought as one coherent theory rather than as disconnected constructions:

1. Sheaf cohomology in the idempotent setting: a well-defined cohomology $H^{j}$ (in particular $H^{1}$, e.g. as an idempotent monoid, i.e. a commutative monoid whose addition is idempotent) for modules over the characteristic-one structure sheaves involved (the tropical semiring $\bar{\mathbb{N}}$, the multiplicatively cancellative Newton-polygon semiring $\operatorname{Conv}_{\geq}(\mathbb{N}\times\mathbb{N})$ on the reduced square, the scaling-site sheaf of convex piecewise-affine functions with integral slopes), together with an analogue of Serre duality, in a way that removes or controls the null-element contributions that break the naive Čech complex (the standard complex, built from the sections of the sheaf on the sets of an open cover and on their finite intersections, that computes sheaf cohomology).

2. An intersection theory on the square of the arithmetic site (or of the scaling site) for the Frobenius correspondences: for a divisor or correspondence $D$, the self-intersection $D.D$ must be realized as the intersection number of the composite $D\circ\tilde{D}$ with the diagonal, computed as a distributional trace compatible with the trace-formula counting function, such that for $D(f)=\int\Psi(\lambda)f(\lambda)d^{*}\lambda$ one obtains $\frac{1}{2}D(f).D(f)=s(f,f)$, where $s$ is the quadratic form of the explicit formula.

3. The Riemann–Roch theorem on the square of the site — in the sense of a formula, or at minimum an inequality of the shape $\operatorname{Dim}_{\mathbb{R}}(H^{0}(D))+\operatorname{Dim}_{\mathbb{R}}(H^{0}(-D))\geq\frac{1}{2}D.D$ with the continuous (real-valued) dimensions of the program — adequate to imply the key inequality $s(f,f)\leq 0$ of the Weil strategy for test functions $f$ with $\int f(u)d^{*}u=\int f(u)du=0$.

4. As part of the same intersection-theoretic structure, an account of the tangential deformation $\operatorname{Id}_{\epsilon}$ of the identity correspondence, which appears in the composition law

$\Psi(\lambda)\circ\Psi(\lambda')=\operatorname{Id}_{\epsilon}\circ\Psi(\lambda\lambda')\quad(\lambda,\lambda'\ \text{irrational},\ \lambda\lambda'\in\mathbb{Q}),$

including its structural role in, and compatibility with, the intersection numbers and degrees of the theory.

The theory is sought at the generality of the source (“an adequate version of the Riemann-Roch theorem in characteristic 1”): it is intrinsic to characteristic one, demonstrably adequate for the Weil strategy on the arithmetic site, not confined to curves and not obtained by descent from characteristic zero.

References

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