From Orientations to $\ell$-adic Period Vectors

TL;DR

This paper introduces a method linking supersingular elliptic curves with modular symbols to compute $ ext{ell}$-adic period vectors, enabling explicit encoding and posing the MSI problem with cryptographic implications.

Contribution

It develops an explicit, computable map from homology classes to truncated $ ext{ell}$-adic period vectors and formulates the Modular Symbol Inversion problem.

Findings

Constructed a map from homology to $ ext{ell}$-adic periods

Formulated the Modular Symbol Inversion (MSI) problem

Discussed connections to isogeny graphs and cryptography

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.