Effective Methods in Tame Geometry and Applications in Arithmetic and Dynamics

Project Information

EffectiveTG

Grant agreement ID: 802107

DOI

10.3030/802107

Project closed

EC signature date 20 August 2018

Start date 1 September 2018

End date 29 February 2024

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 1 155 027,00

EU contribution

€ 1 155 027,00

Coordinated by

WEIZMANN INSTITUTE OF SCIENCE
Israel

Periodic Reporting for period 4 - EffectiveTG (Effective Methods in Tame Geometry and Applications in Arithmetic and Dynamics)

Reporting period: 2023-03-01 to 2024-02-29

Summary of the context and overall objectives of the project

This research project is dedicated to the study of tame geometry and its interaction with other branches of mathematics, notably Diophantine geometry. Tame geometry studies structures in which every definable set has a
finite geometric complexity. The study of tame geometry spans several interrelated mathematical fields, including semialgebraic, subanalytic, and o-minimal geometry. The past decade has seen the emergence of a spectacular link between tame geometry and arithmetic following the discovery of the fundamental Pila-Wilkie counting theorem and its applications in unlikely diophantine intersections. The Pila-Wilkie theorem itself relies crucially on the Yomdin-Gromov theorem, a classical result of tame geometry with fundamental applications in smooth dynamics.
It is natural to ask whether the complexity of a tame set can be estimated effectively in terms of the defining formulas. While a large body of work is devoted to answering such questions in the semialgebraic case, surprisingly little is known concerning more general tame structures - specifically those needed in recent applications to arithmetic. The nature of the link between tame geometry and arithmetic is such that any progress toward effectivizing
the theory of tame structures will likely lead to effective results in the domain of unlikely intersections. Similarly, a more effective version of the Yomdin-Gromov theorem is known to imply important consequences in smooth dynamics.
In this project we approach effectivity in tame geometry from a fundamentally new direction, bringing to bear methods from the theory of differential equations which have until recently never been used in this context. Toward this end, our key goals are to gain insight into the differential algebraic and complex analytic structure of tame sets; and to apply this insight in combination with results from the theory of differential equations to effectivize key results in tame geometry and its applications to arithmetic and dynamics. This allows us to obtain effective algorithms solving some famous problems in Diophantine geometry, where no effective solution procedure was previously known.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

Our work has proceeded along several parallel lines, in anticipation that upon successful maturation of each line it will become possible to combine them in order to obtain stronger results. Below is a brief summary of the main lines we have been pursuing:
1. Complexification of tame structures. The first big part of this work has been successfully completed in my joint paper with D. Novikov titled "Complex cellular structures", published in the Annals of Mathematics 2019. The continuation of this work, extending our results from the subanalytic to the unrestricted-exponential structure, is now the subject of our intensive efforts and I am hopeful this will bear fruit in the next reporting period.
2. Effective counting. This is the subject of my preprint on point-counting with foliations (currently submitted for publication), which establishes many of the main stated goals of the research project in regard to point-counting. In particular this establishes poly-logarithmic counting ("Wilkie's conjecture") as well as polynomial dependence of constants for a wide class of sets, including most sets required in diophantine applications. I consider this to be one of the major achievements of the project.
3. Diophantine applications. We have been pursuing several types of applications in the realm of unlike intersections. These include effective Andre-Oort (in my work on modular curves published in Crelle, and in my joint work with D. Masser on curves in Hilbert modular varieties to appear in Comptes Rendus). Effective applications around torsion points and Pell's equation (in my foliation point-counting preprint). A new approach to Galois-orbit lower bounds (in a paper with H. Schmidt and A. Yafaev). This played a major role in the complete resolution of Andre-Oort by Pila-Shankar-Tsimerman.
4. Applications of differential algebraic methods to weakly-special and weakly-optimal varieties in Shimura varieties (JEMS paper with C. Daw).
5. Effective o-minimality and applications for the Pfaffian class (a joint paper with N. Vorobjov published in IMRN, and a preprint in preparation with G. Jones, H. Schmidt and M. Thomas). This is also the subject of my Ph.D. student B. Zack's research project.
6. Approximate counting and applications to functional transcendence (the subject of a preprint in preparation with my postdoc N. Bhardwaj).
7. Pila-Wilkie counting in non-archimedean and positive characteristics (Duke paper with Cluckers and Novikov, and a preprint with F. Kato).
8. The introduction of "sharply o-minimal geometry" as a framework for pursuing arithmetically tame geometry in an ICM prceedings paper (with Novikov) and a preprint with Novikov and Zak.
9. A proof of Wilkie's full conjecture in the unrestricted setting using methods of sharp o-minimality in an Annals paper with Novikov and Zak.
10. New results applying our counting methods to the classical problem of counting rational points on plane algebraic curves, leading to several sharp bounds with respect to degrees (first with Novikov and Cluckers over the rational numbers, and later with Cluckers and Kato for arbitrary global fields).

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The state of the art in this area has been that very few effective instances of the counting theorem are known, except for some specific low-dimensional cases. A general effective framework, that could potentially lead to an effective counting theorem, has been known in the case of Pfaffian functions (due to Khovanskii, Gabrielov, Vorobjov and others). However, due to technical difficulties in applying this framework to the counting theorem, only the case of curves and surfaces has previously been treated (by Pila, Jones, Thomas). Going beyond the Pfaffian case, essentially nothing was known (effectively) for functions related to modular curves, Shimura varieties, variations of Hodge structures, etc.
The result that we obtained in the project greatly expand the scope of effectivity of the counting theorem. In the Pfaffian case, together with Jones, Schmidt and Thomas we establish essentially a fully generated effective counting theorem and pursue some applications around uniformity issues in the Manin-Mumford conjecture and related topics. In the Shimura context, my paper on point counting with foliations establishes an effective form of the counting theorem leading for instance to the polynomial-time decidability of the Andre-Oort conjecture and the effective polynomial-time decidability of Masser-Zannier's relative Manin-Mumford for elliptic pencils. We have demonstrated the applicability of these results also to effective Andre-Oort in Hilbert modular varieties (with Masser). Additionally, we have applied the basic counting techniques in the algebraic setting to give some new sharp bounds for algebraic cuves.