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SUMMARY:Leonardo Colò (University of Waterloo)
DTSTART:20260622T200000Z
DTEND:20260622T210000Z
DTSTAMP:20260616T192144Z
UID:CarletonOttawaNT/75
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CarletonOtta
 waNT/75/">From Orientations to p-adic Period Vectors: the Modular Symbol I
 nversion Problem</a>\nby Leonardo Colò (University of Waterloo) as part o
 f Carleton-Ottawa Number Theory seminar\n\nInteractive livestream: https:/
 /uottawa-ca.zoom.us/j/95724297776\nPassword hint: Please contact the organ
 izers for the password.\nLecture held in STEM-664.\n\nAbstract\nOrientatio
 ns of supersingular elliptic curves have come to play a significant role i
 n 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 $\\gamma(\\iota) \\in H_1(X_0(N)\, \\{\\text{cu
 sps}\\}\; \\mathbb{Z})$\, 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 ge
 nus and $c$ the number of cusps.\n\nEach modular symbol $\\gamma$ can be e
 valuated against weight-2 cusp forms via $p$-adic Abelian (Coleman) integr
 als\, producing coordinates $\\langle f\, \\gamma \\rangle_p$. Computing t
 hese on a basis yields a $p$-adic period vector $\\Pi(\\gamma)$\, whose re
 duction modulo $p^m$ provides a discrete invariant.\n\nThis suggests a cor
 respondence\n\\[\n\\text{Orientation } \\iota \\\;\\longmapsto\\\; \\gamma
 (\\iota) \\\;\\longmapsto\\\; \\Pi_m(\\gamma(\\iota))\,\n\\]\nconnecting e
 ndomorphism-theoretic data to homology and then to $p$-adic analytic perio
 ds.\n\nWe investigate the mathematical structure underlying these correspo
 ndences\, discuss the choices and compatibility conditions required to mak
 e them precise\, and explore their potential applications to cryptographic
  constructions.\n
LOCATION:https://researchseminars.org/talk/CarletonOttawaNT/75/
URL:https://uottawa-ca.zoom.us/j/95724297776
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