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SUMMARY:"CM-theoretic Aspects of the Birch and Swinnerton-Dyer Conjecture 
 and Curve-Based Cryptography"
DTSTART:20170302T101500
DTEND:20170302T111500
DTSTAMP:20260916T210729Z
UID:202dfe089130c2a0efdaefcf946b0f0ead185f5ca78cb617a2f8e199
CATEGORIES:Conferences - Seminars
DESCRIPTION:Prof. Dimitar Jetchev (EPFL) \nThe theory of complex multiplic
 ation (CM theory) has found numerous applications in both modern number th
 eory\, arithmetic geometry and mathematical cryptology. In this talk\, I w
 ill give a basic background on the BSD conjecture and outline the main ide
 as of the recent proof of the conjectural formula for elliptic curves of a
 nalytic rank 1. I will then explain how CM theory provides an algebraic mo
 del for the analytic side of the BSD conjecture via Euler systems and repo
 rt on higher-dimensional constructions from special cycles on unitary Shim
 ura varieties. These constructions are based on local analysis via Bruhat-
 Tits buildings of the corresponding unitary groups. \n\nThe seemingly unr
 elated problem of computing explicit isogenies for principally polarized a
 belian varieties plays a key role in the study of the discrete logarithm p
 roblem (DLP)\, the main computational hardness assumptions for most of the
  existing curve-based cryptographic schemes. I will review recent work on 
 computing isogenies in higher dimensions via theta embeddings and CM theor
 y\, and comment on the implications of these to DLP and parameter selectio
 n. This requires establishing precise structural properties of certain gra
 phs of isogenies of principally polarized abelian varieties - a problem th
 at can be solved using similar local analysis on the Bruhat-Tits buildings
  for symplectic groups as the one used in the construction of Euler system
 s. \n\n 
LOCATION:CIB - BI A0 448 http://plan.epfl.ch/?room=BIA0448
STATUS:CONFIRMED
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