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From Orientations to $\ell$-adic Period Vectors

Leonardo Colò

Abstract

We propose a bridge between oriented supersingular elliptic curves and the arithmetic of modular curves. To an $\mathcal{O}$-oriented supersingular curve, we attach a class in the relative homology group $H(X_0(N),C,\mathbb{Z})$, i.e. modular symbols, compatible with the Hecke action. We then compute vectors of $\ell$-adic periods by pairing with weight $2$ cusp forms via Coleman integration. This yields an explicit, computable map from short combinatorial homology representatives to truncated vectors in $(\mathbb{Z}/\ell^m\mathbb{Z})^d$. Motivated by this encoding, we formulate the Modular Symbol Inversion (MSI) problem -- recovering a short homology representative from its truncated $\ell$-adic period data -- and discuss its arithmetic structure, its relation to path problems on isogeny graphs and Bruhat-Tits trees, and potential applications to cryptographic constructions.

From Orientations to $\ell$-adic Period Vectors

Abstract

We propose a bridge between oriented supersingular elliptic curves and the arithmetic of modular curves. To an -oriented supersingular curve, we attach a class in the relative homology group , i.e. modular symbols, compatible with the Hecke action. We then compute vectors of -adic periods by pairing with weight cusp forms via Coleman integration. This yields an explicit, computable map from short combinatorial homology representatives to truncated vectors in . Motivated by this encoding, we formulate the Modular Symbol Inversion (MSI) problem -- recovering a short homology representative from its truncated -adic period data -- and discuss its arithmetic structure, its relation to path problems on isogeny graphs and Bruhat-Tits trees, and potential applications to cryptographic constructions.

Paper Structure

This paper contains 29 sections, 6 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Supersingular elliptic curves and orientations
  3. Elliptic curves and isogenies
  4. Orders and orientations
  5. Ideal classes and oriented supersingular curves
  6. Oriented isogenies and the Bruhat--Tits tree
  7. Horizontal and vertical isogenies; volcano picture
  8. Modular symbols and relative homology
  9. Modular curves and modular forms
  10. Relative homology and Manin symbols
  11. Class-group representations on modular-symbol homology
  12. Construction 1: Brandt module
  13. Construction 2: Heegner points and geometric geodesic cycles
  14. Construction 3: Bruhat--Tits graph and harmonic cocycles
  15. Equivalence of the three constructions
  16. ...and 14 more sections

Key Result

Theorem 2

The set $\mathrm{SS}^{pr}_{\mathcal{O}}(\rho)$ of optimally $\mathcal{O}$-oriented supersingular elliptic curves with $p$-orientation $\rho$ is a torsor for $\mathrm{Pic}(\mathcal{O})$.

Theorems & Definitions (17)

  • Definition 1
  • Theorem 2
  • Definition 3
  • Proposition 1
  • proof
  • Corollary 1
  • Definition 4
  • Proposition 2
  • Proposition 3
  • proof
  • ...and 7 more