From Orientations to $\mathrm{\ell }$-adic Period Vectors

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 $\mathrm{\ell }$-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 $(\mathbb{Z}\mathrm{/}{\mathrm{\ell }}^m\mathbb{Z}{)}^d$. Motivated by this encoding, we formulate the Modular Symbol Inversion (MSI) problem – recovering a short homology representative from its truncated $\mathrm{\ell }$-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.