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Computing transcendence and linear relations of 1-periods

Emre Can Sertöz, Joël Ouaknine, James Worrell

TL;DR

The paper provides an algorithmic framework to compute all ${ar{f Q}}$-linear relations among a finite set of $1$-periods, thereby deciding transcendence and equality of periods. It achieves this by grounding $1$-periods in the mixed Hodge structures of punctured curves and in push-pull Jacobian motives, ultimately reducing period-relations to explicit endomorphism data and divisor arithmetic on curves. Central contributions include a complete, effective model for relative algebraic de Rham and Betti realizations, a constructive treatment of $1$-motives and their period relations, and a suite of push-pull, correspondences, and supersaturation techniques that render the Kontsevich–Zagier equality problem tractable in this setting. The methods are computationally explicit, rely on divisor theory on curves, and yield practical tools for both period classification and autonomous differential equations, with broad implications for transcendence and arithmetic geometry.

Abstract

A 1-period is a complex number given by the integral of a univariate algebraic function, where all data involved -- the integrand and the domain of integration -- are defined over algebraic numbers. We give an algorithm that, given a finite collection of 1-periods, computes the space of all linear relations among them with algebraic coefficients. In particular, the algorithm decides whether a given 1-period is transcendental, and whether two 1-periods are equal. This resolves, in the case of 1-periods, a problem posed by Kontsevich and Zagier, asking for an algorithm to decide equality of periods. The algorithm builds on the work of Huber and Wüstholz, who showed that all linear relations among 1-periods arise from 1-motives; we make this perspective effective by reducing the problem to divisor arithmetic on curves and providing the theoretical foundations for a practical and fully explicit algorithm. To illustrate the broader applicability of our methods, we also give an algorithmic classification of autonomous first-order (non-linear) differential equations.

Computing transcendence and linear relations of 1-periods

TL;DR

The paper provides an algorithmic framework to compute all -linear relations among a finite set of -periods, thereby deciding transcendence and equality of periods. It achieves this by grounding -periods in the mixed Hodge structures of punctured curves and in push-pull Jacobian motives, ultimately reducing period-relations to explicit endomorphism data and divisor arithmetic on curves. Central contributions include a complete, effective model for relative algebraic de Rham and Betti realizations, a constructive treatment of -motives and their period relations, and a suite of push-pull, correspondences, and supersaturation techniques that render the Kontsevich–Zagier equality problem tractable in this setting. The methods are computationally explicit, rely on divisor theory on curves, and yield practical tools for both period classification and autonomous differential equations, with broad implications for transcendence and arithmetic geometry.

Abstract

A 1-period is a complex number given by the integral of a univariate algebraic function, where all data involved -- the integrand and the domain of integration -- are defined over algebraic numbers. We give an algorithm that, given a finite collection of 1-periods, computes the space of all linear relations among them with algebraic coefficients. In particular, the algorithm decides whether a given 1-period is transcendental, and whether two 1-periods are equal. This resolves, in the case of 1-periods, a problem posed by Kontsevich and Zagier, asking for an algorithm to decide equality of periods. The algorithm builds on the work of Huber and Wüstholz, who showed that all linear relations among 1-periods arise from 1-motives; we make this perspective effective by reducing the problem to divisor arithmetic on curves and providing the theoretical foundations for a practical and fully explicit algorithm. To illustrate the broader applicability of our methods, we also give an algorithmic classification of autonomous first-order (non-linear) differential equations.

Paper Structure

This paper contains 57 sections, 148 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Connections to foundational work
  3. Acknowledgements
  4. The algorithm
  5. Relative algebraic de Rham cohomology of a punctured curve
  6. Representing irreducible curves and their differentials
  7. Representing reducible curves and their differentials
  8. Poles of forms and definition of algebraic de Rham cohomology
  9. Algebraic de Rham cohomology in terms of second type differentials
  10. An effective basis for cohomology
  11. Computing a basis for algebraic de Rham cohomology
  12. Reduction algorithms
  13. Cohomological pullback and the transfer map
  14. Betti homology and the mixed Hodge structure of a punctured marked curve
  15. Representing the mixed Hodge structure
  16. ...and 42 more sections

Figures (1)

  • Figure 1: Making the action on a supersaturated motive effective.