[Paper Review] The Riemann-Roch strategy, Complex lift of the Scaling Site

Alain Connes, Caterina Consani|arXiv (Cornell University)|May 26, 2018

Quantum chaos and dynamical systems37 references3 citations

TL;DR

This paper introduces a complex lift of the adèle class space of the rationals, realizing it as a moduli space of elliptic curves with a triangular structure, using tropical descent to transfer Riemann-Roch results from complex geometry to characteristic one. The construction involves a ringed topos of analytic functions on the punctured unit disk with a scaling action, and identifies holomorphic functions invariant under Frobenius-like maps, linking to the GL(2)-system and noncommutative geometry.

ABSTRACT

We describe the Riemann-Roch strategy which consists of adapting in characteristic zero Weil's proof, of RH in positive characteristic, following the ideas of Mattuck, Tate and Grothendieck. As a new step in this strategy we implement the technique of tropical descent that allows one to deduce existence results in characteristic one from the Riemann-Roch result over the complex numbers. In order to deal with arbitrary distribution functions this technique involves the results of Bohr, Jessen and Tornehave on almost periodic functions. Our main result is the construction, at the adelic level, of a complex lift of the adele class space of the rationals. We interpret this lift as a moduli space of elliptic curves endowed with a triangular structure. The equivalence relation yielding the noncommutative structure is generated by isogenies. We describe the tight relation of this complex lift with the GL(2)-system. We construct the lift of the Frobenius correspondences using the Witt construction in characteristic one.

Motivation & Objective

• To develop a complex geometry that lifts the tropical Scaling Site from characteristic one to characteristic zero.
• To overcome the obstruction in defining sheaf cohomology in characteristic one by transferring existence results from complex geometry.
• To construct a moduli space of elliptic curves with a triangular structure via adelic geometry and noncommutative geometry.
• To establish a link between the complex lift and the GL(2)-system through Frobenius correspondences and Witt constructions.
• To interpret the complex lift as a noncommutative space where the transverse complex structure arises from almost periodic compactification of R.

Proposed method

• Utilizes tropical descent to transfer Riemann-Roch theorems from complex analytic geometry to the tropical setting of the Scaling Site.
• Constructs a ringed topos as the quotient of the punctured unit disk under the action of N×, with structure sheaf of complex analytic functions.
• Applies the tropicalization map via Jensen’s formula to associate piecewise affine convex functions to analytic functions, linking zeros to valuations.
• Introduces a complex lift of the adèle class space via a topos-theoretic construction involving the pro-étale cover of the unit disk.
• Uses the Witt construction in characteristic one to lift Frobenius correspondences to the complex setting.
• Defines holomorphic functions via a differential condition involving the operator λX + iY, generalizing holomorphicity in the noncommutative setting.

Experimental results

Research questions

• RQ1 Can Riemann-Roch results in complex geometry be used to deduce existence theorems in characteristic one via tropical descent?
• RQ2 How can the adèle class space of the rationals be lifted to a complex geometry that retains arithmetic and noncommutative structure?
• RQ3 What is the role of almost periodic functions in realizing the complex lift as a moduli space of elliptic curves with a triangular structure?
• RQ4 How do Frobenius correspondences in characteristic one lift to the complex setting using the Witt construction?
• RQ5 In what sense does the complex lift provide a quantization of the Scaling Site, and how does it relate to transverse elliptic theory?

Key findings

• The complex lift of the adèle class space is realized as a moduli space of elliptic curves equipped with a triangular structure, parameterized by the almost periodic compactification of R.
• The ringed topos constructed on the punctured unit disk with the N×-action admits a structure sheaf of complex analytic functions, providing a geometric lift of the Scaling Site.
• The function q: Γ_Q,cl → W defined by q(x+iy) = [e^{-2πy}]e^{2πix} is holomorphic in the sense of the differential condition (λX + iY)χ_λ(q) = 0 and is invariant under Frobenius-like maps Fr^a_μ.
• The complex lift fibers over the adèle class space with fiber G, the almost periodic compactification of R, indicating a transverse complex structure in the noncommutative setting.
• The construction of the lift via tropical descent successfully transfers Riemann-Roch existence theorems from C to characteristic one, bypassing the cohomological obstruction in the tropical setting.
• The GL(2)-system is tightly linked to the complex lift, with the Frobenius correspondences lifted via the Witt construction in characteristic one.