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SUMMARY:Mixed Modular Motives and Modular Forms for SL_2 (\\Z) (4/4)
DTSTART;VALUE=DATE-TIME:20180425T083000Z
DTEND;VALUE=DATE-TIME:20180425T103000Z
DTSTAMP;VALUE=DATE-TIME:20200407T100214Z
UID:indico-event-3349@indico.math.cnrs.fr
DESCRIPTION:In the `Esquisse d'un programme'\, Grothendieck proposed study
ing the action of the absolute Galois group upon the system of profinite f
undamental groups of moduli spaces of curves of genus g with n marked poin
ts. Around 1990\, Ihara\, Drinfeld and Deligne independently initiated the
study of the unipotent completion of the fundamental group of the project
ive line with 3 points. It is now known to be motivic by Deligne-Goncharov
and generates the category of mixed Tate motives over the integers. It
is closely related to many classical objects such as polylogarithms and mu
ltiple zeta values\, and has a wide range of applications from number theo
ry to physics.\n\n \n\nIn the first\, geometric\, half of this lecture se
ries I will explain how to extend this theory to genus one (which generate
s the theory in all higher genera). The unipotent fundamental groupoid mus
t be replaced with a notion of relative completion\, studied by Hain\, whi
ch defines an extremely rich system of mixed Hodge structures built out of
modular forms. It is closely related to Manin's iterated Eichler integral
s\, the universal mixed elliptic motives of Hain and Matsumoto\, and the e
lliptic polylogarithms of Beilinson and Levin. The question that I wish to
confront is whether relative completion stands a chance of generating all
mixed modular motives or not. This is equivalent to studying the action o
f a `motivic' Galois group upon it\, and the question of geometrically con
structing all generalised Rankin-Selberg extensions.\n\n \n\nIn the secon
d\, elementary\, half of these lectures\, which will be mostly independent
from the first\, I will explain how the relative completion has a realisa
tion in a new class of non-holomorphic modular forms which correspond in a
certain sense to mixed motives. These functions are elementary power seri
es in $q$ and $\\overline{q}$ and $\\log |q|$ whose coefficients are perio
ds. They are closely related to the theory of modular graph functions in s
tring theory and also intersect with the theory of mock modular forms.\n\n
\n\nhttps://indico.math.cnrs.fr/event/3349/
LOCATION:IHES Amphithéâtre Léon Motchane
URL:https://indico.math.cnrs.fr/event/3349/
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