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SUMMARY:Jean Kieffer (Harvard)
DTSTART:20211005T203000Z
DTEND:20211005T213000Z
DTSTAMP:20260423T125734Z
UID:MITNT/33
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/MITNT/33/">H
 igher-dimensional modular equations and point counting on abelian surfaces
 </a>\nby Jean Kieffer (Harvard) as part of MIT number theory seminar\n\nLe
 cture held in Room 2-143 in the Simons building (building 2).\n\nAbstract\
 nGiven a prime number $\\ell$\, the elliptic modular polynomial of level\n
 $\\ell$ is an explicit equation for the locus of elliptic curves\nrelated 
 by an $\\ell$-isogeny. These polynomials have a large number of\nalgorithm
 ic applications: in particular\, they are an essential\ningredient in the 
 celebrated SEA algorithm for counting points on\nelliptic curves over fini
 te fields of large characteristic.\n\nMore generally\, modular equations d
 escribe the locus of isogenous\nabelian varieties in certain moduli spaces
  called PEL Shimura\nvarieties. We will present upper bounds on the size o
 f modular\nequations in terms of their level\, and outline a general strat
 egy to\ncompute an isogeny $A\\to A'$ given a point $(A\,A')$ where the mo
 dular\nequations are satisfied. This generalizes well-known properties of\
 nelliptic modular polynomials to higher dimensions.\n\nThe isogeny algorit
 hm is made fully explicit for Jacobians of genus 2\ncurves. In combination
  with efficient evaluation methods for modular\nequations in genus 2 via c
 omplex approximations\, this gives rise to\npoint counting algorithms for 
 (Jacobians of) genus 2 curves whose\nasymptotic costs\, under standard heu
 ristics\, improve on previous\nresults.\n
LOCATION:https://researchseminars.org/talk/MITNT/33/
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