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Coleman Integration on Modular Curves

Mingjie Chen, Kiran Kedlaya, Jun Bo Lau

Abstract

Coleman integrals is a major tool in the explicit arithmetic of algebraic varieties, notably in the study of rational points on curves. One of the inputs to compute Coleman integrals is the availability of an affine model. We develop a model-free algorithm that computes single Coleman integrals between any two points on modular curves. Using Hecke operators, any Coleman integral can be broken down into a sum of tiny integrals. We illustrate this using several examples computed in SageMath and Magma. We also suggest some future directions for this work, including a possible extension to iterated Coleman integrals.

Coleman Integration on Modular Curves

Abstract

Coleman integrals is a major tool in the explicit arithmetic of algebraic varieties, notably in the study of rational points on curves. One of the inputs to compute Coleman integrals is the availability of an affine model. We develop a model-free algorithm that computes single Coleman integrals between any two points on modular curves. Using Hecke operators, any Coleman integral can be broken down into a sum of tiny integrals. We illustrate this using several examples computed in SageMath and Magma. We also suggest some future directions for this work, including a possible extension to iterated Coleman integrals.
Paper Structure (19 sections, 6 theorems, 20 equations, 2 tables, 1 algorithm)

This paper contains 19 sections, 6 theorems, 20 equations, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Notations
  3. Modular curves and modular forms
  4. Hecke operators
  5. Coleman integrals
  6. Coleman Integration on Modular Curves
  7. Coleman integrals as sums of tiny integrals
  8. A basis of H0XOmega1
  9. Hecke operators as double coset operators
  10. Tiny integrals from complex approximation
  11. Examples
  12. The modular curve X0N
  13. Example: $X_0(37)$
  14. The Atkin-Lehner quotient X0+N
  15. Example: $X_0^+(67)$
  16. ...and 4 more sections

Key Result

Theorem 2.4

coleman-de-shalitcoleman-ann Let $X/\mathbb{Q}_p$ be a smooth, projective, and geometrically irreducible curve with good reduction at $p$. Let $\mathfrak{X}/\mathbb{Z}_p$ be the smooth model of $X$. Then there is a unique way to assign a $p$-adic integral for every choice of a wide open subspace $U$ of $X^{\mathrm{an}}$, two points $P,Q \in U(\overline{\mathbb{Q}}_p)$, and a $1$-form $\omega \in

Theorems & Definitions (17)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Theorem 2.4
  • Remark 2.5
  • Definition 2.6
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3
  • ...and 7 more