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 $\mathcal{O}$-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.

• The paper introduces an explicit algorithm to convert oriented supersingular elliptic curves into truncated ℓ-adic period vectors using modular symbols and Coleman integration.
• It employs modular symbols, ideal class group actions, and Bruhat–Tits tree structures to bridge algebraic geometry with discrete analytic computation.
• The work formulates the MSI problem, a novel cryptographic hardness assumption distinct from traditional lattice-based challenges, paving the way for secure post-quantum protocols.

Algebraic–Analytic Encoding from Orientations to $\ell$-adic Period Vectors

Motivation and Framework

The paper "From Orientations to $\ell$-adic Period Vectors" (2603.29789) presents a novel interface between the arithmetic of oriented supersingular elliptic curves and the discrete topology of modular curves. By integrating period theory, modular symbols, and $\ell$-adic integration, the authors construct an explicit, algorithmically computable map from $\cO$-oriented supersingular curves to truncated $\ell$-adic period vectors. This construction is argued to yield hard computational problems suitable for cryptographic applications and is formalized through the Modular Symbol Inversion (MSI) problem—a path-finding challenge with connections to isogeny graphs, modular symbols, and lattice-based cryptography.

Oriented Supersingular Elliptic Curves and Class Group Actions

The core algebraic input is a supersingular elliptic curve $E$ in characteristic $p$ equipped with an optimal orientation $$\iota : \mathcal O \hookrightarrow \operatorname{End}(E)$$, where $\mathcal O$ is an order in an imaginary quadratic field $K$. Such orientations are parametrized by the ideal class group $\ell$0, enabling a structured sampling of curves and associated isogenies.

Endomorphism rings and orientations are interpreted via embeddings into quaternion algebras, and the action of $\ell$1 on oriented curves is described explicitly through left ideal construction. The mapping from ideal classes to isogenies yields a geodesic path in the quotient of the Bruhat–Tits tree, where path length is determined by the $\ell$2-adic valuation of the ideal norm.

The distinction between horizontal and vertical isogenies is addressed, and the volcano picture emerges naturally when varying the conductor $\ell$3 of the order. The resulting isogeny graphs exhibit complex layered structures, with class group actions interpreted as oriented paths and geodesics in these graphs.

Modular Symbols: Homology, Hecke Actions, and Geometric Representatives

The interface to modular forms is given by modular symbols, which represent relative homology classes in $\ell$4, where $\ell$5 is a modular curve of level $\ell$6 and $\ell$7 is the set of cusps. The modular symbol formalism offers a combinatorial encoding, compatible with Hecke operators.

The paper presents three equivalence constructions for associating a modular-symbol homology class to an oriented supersingular curve:

1. Brandt Module Approach: Ideal classes act via Brandt matrices and the Jacquet–Langlands correspondence, leading to a Hecke-equivariant action on a homology submodule $\ell$8.
2. Heegner Point and Geometric Cycle Construction: Paths between CM points and cusps on the modular curve yield analytic representatives.
3. Bruhat–Tits Tree and Harmonic Cocycle Perspective: Cycles in the quotient graph are specialized to algebraic homology via rigid-analytic uniformization.

These constructions are shown to agree, up to finite ambiguity, in the relevant Hecke-stable quotient.

$\ell$9-adic Period Vectors and Coleman Integration

The analytic output is generated by pairing the constructed homology classes with weight-2 cusp forms using $\ell$0-adic Coleman integration. This map yields truncated period vectors in $\ell$1, where $\ell$2 is the dimension of the eigenform subspace considered.

A practical workflow leverages computational tools such as the algorithm of Chen–Kedlaya–Lau for efficient $\ell$3-adic integration, circumventing the need for explicit projective models. The process consists of translating oriented curves into CM points, computing associated modular symbols, and evaluating integrals against eigenforms to obtain discrete period data.

Modular Symbol Inversion (MSI) Problem and Cryptographic Implications

A key contribution is the formulation of the MSI problem: Given a truncated $\ell$4-adic period vector arising from a short homology class (i.e., a path of bounded combinatorial complexity), recover any preimage of comparable complexity under the period map. MSI is formally distinct from lattice-based SIS/LWE, as the solution space is exponentially sparse and highly structured by path constraints.

The paper conjectures exponential hardness for MSI in parameter regimes where collision probability and generic attacks are suppressed. No efficient reductions to or from classical lattice or isogeny path problems are known, suggesting MSI as a new independent source of cryptographic hardness.

Concrete cryptographic instantiations are provided:

• Identification Protocol: Secret keys are short homology classes, and verification exploits the homomorphism property of the period map.
• Pseudorandom Functions (PRFs): PRFs indexed by short homology classes, with period vectors feeding into standard KDFs.

Parameter selection guidelines address classical and quantum attack surfaces, branching factor, ambient lattice dimension, and output space entropy.

Theoretical and Practical Implications

The algebraic–analytic pipeline established in the paper enables structured sampling from modular curve homology, encoding complex arithmetic into discrete period data. The MSI problem opens the avenue for novel cryptographic primitives, leveraging the arithmetic and topological structure of modular curves beyond traditional isogeny-based paradigms.

Theoretical implications include new perspectives on the interaction between modular symbols, Hecke actions, uniformization, and period theory. Practically, the methods can be used for post-quantum protocol design, assuming parameter choices leading to cryptographic hardness. There is scope for further exploration of algorithms for period computation, stabilization subgroups, and structural attacks exploiting symmetries.

Conclusion

This research provides an explicit map from oriented supersingular elliptic curves to truncated $\ell$5-adic period vectors, bridging arithmetic geometry and modular curve homology via modular symbols and $\ell$6-adic integration. The MSI problem emerges as a new computational hardness assumption with strong cryptographic potential, independent from classical lattice and isogeny-based problems. Future directions include rigorous security reductions, parameter tuning, and empirical validation of period-based cryptosystems (2603.29789).