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.