BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//wp-events-plugin.com//7.2.3.1//EN
TZID:Europe/Paris
X-WR-TIMEZONE:Europe/Paris
BEGIN:VEVENT
UID:8938@i2m.univ-amu.fr
DTSTART;TZID=Europe/Paris:20251209T110000
DTEND;TZID=Europe/Paris:20251209T120000
DTSTAMP:20251204T073708Z
URL:https://www.i2m.univ-amu.fr/evenements/from-orientations-to-p-adic-per
 iod-vectors-the-modular-symbol-inversion-problem/
SUMMARY:Leonardo Colò (University of Waterloo): From Orientations to p-adi
 c Period Vectors: the Modular Symbol Inversion Problem
DESCRIPTION:Leonardo Colò: Orientations of supersingular elliptic curves h
 ave come to play a significant role\nin isogney based cryptography. We pro
 pose a natural way to associate such orientations\nnot only with class gro
 up actions\, but\nwith modular symbols on the modular curve $X_0(N)$. Conc
 retely\,\nan orientation determines a relative homology class\n$gamma(iota
 )in H_1(X_0(N)\,{text{cusps}}\;mathbb Z)$\, typically\nrepresented as a sy
 mbol ${ctoinfty}$. These symbols inhabit a high-rank lattice: the relative
  homology group has\ndimension roughly $2g+(c-1)$\, where $g$ is the genus
  and $c$ the number of\ncusps.\n\nEach modular symbol $gamma$ can be evalu
 ated against weight-2 cusp forms via\n$p$-adic Abelian (Coleman) integrals
 \, producing coordinates\n$langle f\,gammarangle_p$. Computing these on a 
 basis yields a\n$p$-adic period vector $Pi(gamma)$\, whose reduction modul
 o $p^m$ provides\na discrete invariant. Thus we obtain a canonical transla
 tion\n[\ntext{Orientation } iota \;longmapsto\; gamma(iota)\n\;longmapsto\
 ; Pi_m(gamma(iota))\,\n]\nlinking endomorphism-theoretic data to homology 
 and then to\n$p$-adic analytic periods.\n\nThe associated inversion proble
 m\, i.e.\, recovering a short cycle from its\ntruncated period vector\, ap
 pears to require exponential time in the\nnatural parameter (path length)\
 , reflecting the combinatorial growth of\nthe Bruhat-Tits tree.\n\nCrucial
 ly\, the computational tools involved are already well developed: modular 
 symbols\, overconvergent $p$-adic distributions\, and harmonic cocycle met
 hods allow one to compute such period vectors efficiently for small analys
 is primes and moderate precision\, without requiring explicit equations of
  the modular curve. This suggests a new class of hard problems at the inte
 rsection of algebraic geometry\, $p$-adic analysis\, and combinatorics. Wh
 ile feasibility is established by existing algorithms\, their inversion ha
 rdness motivates cryptographic exploration: the same structures that conne
 ct orientations to modular symbols and Abelian integrals can be used to de
 sign commitments\, signatures\, and other primitives.
CATEGORIES:Séminaire,Arithmétique et Théorie de l’Information
LOCATION:I2M Luminy - TPR2\, Salle 210-212 (2e étage)\, 163\, avenue de Lu
 miny\, Marseille\, 13009\, France
X-APPLE-STRUCTURED-LOCATION;VALUE=URI;X-ADDRESS=163\, avenue de Luminy\, Ma
 rseille\, 13009\, France;X-APPLE-RADIUS=100;X-TITLE=I2M Luminy - TPR2\, Sa
 lle 210-212 (2e étage):geo:0,0
END:VEVENT
BEGIN:VTIMEZONE
TZID:Europe/Paris
X-LIC-LOCATION:Europe/Paris
BEGIN:STANDARD
DTSTART:20251026T020000
TZOFFSETFROM:+0200
TZOFFSETTO:+0100
TZNAME:CET
END:STANDARD
END:VTIMEZONE
END:VCALENDAR