From Orientations to p-adic Period Vectors: the Modular Symbol Inversion Problem

Abstract: Orientations of supersingular elliptic curves have come to play a significant role in isogeny-based cryptography. We investigate a framework to associate such orientations not only with class group actions, but with modular symbols on the modular curve $X_0(N)$. More precisely, an orientation determines a relative homology class $\gamma(\iota) \in H_1(X_0(N), \{\text{cusps}\}; \mathbb{Z})$, typically represented as a linear combination of symbol $\{c \to \infty\}$. These symbols live in a high-rank lattice: the relative homology group has rank $2g + (c - 1)$, where $g$ is the genus and $c$ the number of cusps.

Each modular symbol $\gamma$ can be evaluated against weight-2 cusp forms via $p$-adic Abelian (Coleman) integrals, producing coordinates $\langle f, \gamma \rangle_p$. Computing these on a basis yields a $p$-adic period vector $\Pi(\gamma)$, whose reduction modulo $p^m$ provides a discrete invariant.

This suggests a correspondence \[ \text{Orientation } \iota \;\longmapsto\; \gamma(\iota) \;\longmapsto\; \Pi_m(\gamma(\iota)), \] connecting endomorphism-theoretic data to homology and then to $p$-adic analytic periods.

We investigate the mathematical structure underlying these correspondences, discuss the choices and compatibility conditions required to make them precise, and explore their potential applications to cryptographic constructions.

number theory

Audience: researchers in the topic

Carleton-Ottawa Number Theory seminar