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SUMMARY:Stark's Conjectures and Hilbert's 12th Problem
DTSTART;VALUE=DATE-TIME:20210617T120000Z
DTEND;VALUE=DATE-TIME:20210617T130000Z
DTSTAMP;VALUE=DATE-TIME:20210621T030628Z
UID:indico-event-6674@indico.math.cnrs.fr
DESCRIPTION:In this talk we will discuss two central problems in algebraic
number theory and their interconnections: explicit class field theory and
the special values of L-functions. The goal of explicit class field the
ory is to describe the abelian extensions of a ground number field via ana
lytic means intrinsic to the ground field\; this question lies at the core
of Hilbert's 12th Problem. Meanwhile\, there is an abundance of conject
ures on the special values of L-functions at certain integer points. Of
these\, Stark's Conjecture has special relevance toward explicit class fie
ld theory. I will describe two recent joint results with Mahesh Kakde on
these topics. The first is a proof of the Brumer-Stark conjecture away
from p=2. This conjecture states the existence of certain canonical elemen
ts in CM abelian extensions of totally real fields. The second is a proo
f of an exact formula for Brumer-Stark units that has been developed over
the last 15 years. We show that these units together with other easily w
ritten explicit elements generate the maximal abelian extension of a total
ly real field\, thereby giving a p-adic solution to the question of explic
it class field theory for these fields.\n\nhttps://indico.math.cnrs.fr/eve
nt/6674/
LOCATION:
URL:https://indico.math.cnrs.fr/event/6674/
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