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Rigorous expansions of modular forms at CM points, I: Denominators

Chris Xu

TL;DR

This work develops a rigorous algorithm to compute the power series expansion of a weight $2$ cusp form at a CM point on a modular curve, with denominators controlled by ramification data and without requiring an explicit model of the modular curve. It combines an analytic coefficient computation with a delicate ramification analysis (horizontal and vertical) to bound denominators via a global scaling $C$ and to recover coefficients as algebraic integers in the CM field $F_{R,N}$. The paper delineates how to exploit Newton polygons of the formal group law and endomorphism data to bound and compute the needed denominators and to determine the necessary ramification exponents, including special handling for $j_E\in\{0,1728\}$. As part I of a series toward an equationless Chabauty framework, the results enable rigorous local expansions that underpin p-adic intersection computations on modular curves, with the second paper promised to complete precision analysis and integral differential recovery over $\mathbb Z$.

Abstract

We describe an algorithm to rigorously compute the power series expansion at a CM point of a weight $2$ cusp form of level coprime to $6$. Our algorithm works by bounding the denominators that appear due to ramification, and without recourse to computing an explicit model of the corresponding modular curve. Our result is the first in a series of papers toward an eventual implementation of equationless Chabauty.

Rigorous expansions of modular forms at CM points, I: Denominators

TL;DR

This work develops a rigorous algorithm to compute the power series expansion of a weight cusp form at a CM point on a modular curve, with denominators controlled by ramification data and without requiring an explicit model of the modular curve. It combines an analytic coefficient computation with a delicate ramification analysis (horizontal and vertical) to bound denominators via a global scaling and to recover coefficients as algebraic integers in the CM field . The paper delineates how to exploit Newton polygons of the formal group law and endomorphism data to bound and compute the needed denominators and to determine the necessary ramification exponents, including special handling for . As part I of a series toward an equationless Chabauty framework, the results enable rigorous local expansions that underpin p-adic intersection computations on modular curves, with the second paper promised to complete precision analysis and integral differential recovery over .

Abstract

We describe an algorithm to rigorously compute the power series expansion at a CM point of a weight cusp form of level coprime to . Our algorithm works by bounding the denominators that appear due to ramification, and without recourse to computing an explicit model of the corresponding modular curve. Our result is the first in a series of papers toward an eventual implementation of equationless Chabauty.

Paper Structure

This paper contains 28 sections, 15 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Future work
  3. Leitfaden
  4. Acknowledgements
  5. Notation and setup
  6. Modular curves
  7. Basepoint
  8. Good reduction models of $E$ at each prime
  9. Level structure
  10. Computation of power series coefficients
  11. Ramification as the source of denominators
  12. Ramification scenarios
  13. Eliminating ramification: the smooth case
  14. Eliminating ramification: the singular case
  15. Effects of eliminating ramification on our power series
  16. ...and 13 more sections

Key Result

Proposition 2.1

For any $\mathbf Z[1/\gcd(6,N)]$-scheme $S$, the canonical "coarse base change" map $Y_{H,S} \to Y_H \times S$ is an isomorphism.

Theorems & Definitions (24)

  • Proposition 2.1: Ces16
  • Remark 2.2
  • Proposition 3.1
  • Remark 4.1
  • Proposition 4.2
  • Remark 4.3
  • Remark 4.4
  • Proposition 4.5
  • Proposition 4.6
  • Proposition 4.7
  • ...and 14 more