Évènement

Leonardo Colò
University of Waterloo
https://www.leonardocolo.com

Date(s) : 09/12/2025   iCal
11h00 - 12h00

Orientations of supersingular elliptic curves have come to play a significant role
in isogney based cryptography. We propose a natural way to associate such orientations
not only with class group actions, but
with modular symbols on the modular curve $X_0(N)$. Concretely,
an orientation determines a relative homology class
$gamma(iota)in H_1(X_0(N),{text{cusps}};mathbb Z)$, typically
represented as a symbol ${ctoinfty}$. These symbols inhabit a high-rank lattice: the relative homology group has
dimension roughly $2g+(c-1)$, 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
$langle f,gammarangle_p$. Computing these on a basis yields a
$p$-adic period vector $Pi(gamma)$, whose reduction modulo $p^m$ provides
a discrete invariant. Thus we obtain a canonical translation
[
text{Orientation } iota ;longmapsto; gamma(iota)
;longmapsto; Pi_m(gamma(iota)),
]
linking endomorphism-theoretic data to homology and then to
$p$-adic analytic periods.

The associated inversion problem, i.e., recovering a short cycle from its
truncated period vector, appears to require exponential time in the
natural parameter (path length), reflecting the combinatorial growth of
the Bruhat-Tits tree.

Crucially, the computational tools involved are already well developed: modular symbols, overconvergent $p$-adic distributions, and harmonic cocycle methods allow one to compute such period vectors efficiently for small analysis primes and moderate precision, without requiring explicit equations of the modular curve. This suggests a new class of hard problems at the intersection of algebraic geometry, $p$-adic analysis, and combinatorics. While feasibility is established by existing algorithms, their inversion hardness motivates cryptographic exploration: the same structures that connect orientations to modular symbols and Abelian integrals can be used to design commitments, signatures, and other primitives.

Emplacement
I2M Luminy - TPR2, Salle 210-212 (2e étage)

Catégories

• Séminaire