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SUMMARY:Nicolas Bergeron (École normale supérieure\, Paris France)
DTSTART:20201019T133000Z
DTEND:20201019T143000Z
DTSTAMP:20260422T212932Z
UID:Cohomology/1
DESCRIPTION:Title: Sczech cocycles and hyperplane arrangements 1\nby Nicolas Bergeron
(École normale supérieure\, Paris France) as part of CRM workshop: Arith
metic quotients of locally symmetric spaces and their cohomology\n\nLectur
e held in Virtual.\n\nAbstract\nMany authors\, among which Nori\, Sczech\,
Solomon\, Stevens\, or more recently Beilinson—Kings—Levin and\nCharo
llois—Dasgupta—Greenberg\, have constructed different\, but related\,
linear groups cocycles that are\nusually referred to as « Eisenstein cocy
cles. » In these series of lectures I will explain a topological\nconstru
ction that is a common source for all these cocycles. One interesting feat
ure of this construction is that\nstarting from a purely topological class
it leads to the algebraic world of meromorphic forms on hyperplane\ncompl
ements in n-fold products of either the (complex) additive group\, the mul
tiplicative group or a (family of)\nelliptic curve(s). We will see that ev
entually our construction reveals hidden relations between products of\nel
ementary (rational\, trigonometric or elliptic) functions) governed by rel
ations between classes in the\nhomology of linear groups.\nThis is based o
n a work-in-progress with Pierre Charollois\, Luis Garcia and Akshay Venka
tesh\n
LOCATION:https://researchseminars.org/talk/Cohomology/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paul Gunnells (University of Massachusetts Amherst)
DTSTART:20201019T144500Z
DTEND:20201019T154500Z
DTSTAMP:20260422T212932Z
UID:Cohomology/2
DESCRIPTION:Title: Modular symbols over function fields\nby Paul Gunnells (University
of Massachusetts Amherst) as part of CRM workshop: Arithmetic quotients of
locally symmetric spaces and their cohomology\n\nLecture held in Virtual.
\n\nAbstract\nModular symbols\, due to Birch and Manin\, provide a very\nc
oncrete way to compute with classical holomorphic modular forms.\nLater mo
dular symbols were extended to GL(n) by Ash and Rudolph\, and\nsince then
such symbols and variations have played a central role in\ncomputational i
nvestigation of the cohomology of arithmetic groups\nover number fields\,
and in particular in explicitly computing the\nHecke action on cohomology.
\n\nA theory of modular symbols for GL(2) over the rational function fiel
d\nwas developed by Teitelbaum and later applied by Armana. In this talk\
nwe extend this construction to GL(n) and show how it can be used to\ncomp
ute Hecke operators on cohomology. This is joint work with Dan\nYasaki.\n
LOCATION:https://researchseminars.org/talk/Cohomology/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Romyar Sharifi (UCLA)
DTSTART:20201019T160000Z
DTEND:20201019T170000Z
DTSTAMP:20260422T212932Z
UID:Cohomology/3
DESCRIPTION:Title: Eisenstein cocycles in motivic cohomology\nby Romyar Sharifi (UCLA)
as part of CRM workshop: Arithmetic quotients of locally symmetric spaces
and their cohomology\n\nLecture held in Virtual.\n\nAbstract\nI will desc
ribe joint work with Akshay Venkatesh on the construction and study of GL_
2(Z)-cocycles valued in\nsecond K-groups of the function fields of squares
of the multiplicative group over the rationals and of a\nuniversal ellipt
ic curve over a modular curve. I will also describe the pullbacks of their
restrictions to\ncongruence subgroups under torsion sections\, relating t
hese specializations to explicit maps on homology\ngroups of modular curve
s\n
LOCATION:https://researchseminars.org/talk/Cohomology/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicolas Bergeron (École normale supérieure\, Paris France)
DTSTART:20201020T133000Z
DTEND:20201020T143000Z
DTSTAMP:20260422T212932Z
UID:Cohomology/4
DESCRIPTION:Title: Sczech cocycles and hyperplane arrangements 2\nby Nicolas Bergeron
(École normale supérieure\, Paris France) as part of CRM workshop: Arith
metic quotients of locally symmetric spaces and their cohomology\n\nLectur
e held in Virtual.\n\nAbstract\nMany authors\, among which Nori\, Sczech\,
Solomon\, Stevens\, or more recently Beilinson—Kings—Levin and\nCharo
llois—Dasgupta—Greenberg\, have constructed different\, but related\,
linear groups cocycles that are\nusually referred to as « Eisenstein cocy
cles. » In these series of lectures I will explain a topological\nconstru
ction that is a common source for all these cocycles. One interesting feat
ure of this construction is that\nstarting from a purely topological class
it leads to the algebraic world of meromorphic forms on hyperplane\ncompl
ements in n-fold products of either the (complex) additive group\, the mul
tiplicative group or a (family of)\nelliptic curve(s). We will see that ev
entually our construction reveals hidden relations between products of\nel
ementary (rational\, trigonometric or elliptic) functions) governed by rel
ations between classes in the\nhomology of linear groups.\nThis is based o
n a work-in-progress with Pierre Charollois\, Luis Garcia and Akshay Venka
tesh\n
LOCATION:https://researchseminars.org/talk/Cohomology/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emmanuel Lecouturier (Yau Mathematical Sciences Center & IAS)
DTSTART:20201020T150000Z
DTEND:20201020T160000Z
DTSTAMP:20260422T212932Z
UID:Cohomology/5
DESCRIPTION:Title: On Sharifi's conjecture and generalizations\nby Emmanuel Lecouturie
r (Yau Mathematical Sciences Center & IAS) as part of CRM workshop: Arithm
etic quotients of locally symmetric spaces and their cohomology\n\nLecture
held in Virtual.\n\nAbstract\nRomyar Sharifi made beautiful explicit conj
ectures relating the K-theory of cyclotomic field to modular\nsymbols modu
lo some Eisenstein ideal.\nWe report on some partial results on these conj
ectures and their implication for Mazur's Eisenstein ideal. We\nalso discu
ss some ongoing project exploring the analogue of these conjectures for Bi
anchi manifolds. All this\nis joint work with Jun Wang\n
LOCATION:https://researchseminars.org/talk/Cohomology/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicolas Bergeron (École normale supérieure\, Paris France)
DTSTART:20201021T133000Z
DTEND:20201021T143000Z
DTSTAMP:20260422T212932Z
UID:Cohomology/6
DESCRIPTION:Title: Sczech cocycles and hyperplane arrangements 3\nby Nicolas Bergeron
(École normale supérieure\, Paris France) as part of CRM workshop: Arith
metic quotients of locally symmetric spaces and their cohomology\n\nLectur
e held in Virtual.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/Cohomology/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Johannes Sprang (Universität Regensburg)
DTSTART:20201021T144500Z
DTEND:20201021T154500Z
DTSTAMP:20260422T212932Z
UID:Cohomology/7
DESCRIPTION:Title: The Equivariant Polylogarithm and Eisenstein classes\nby Johannes S
prang (Universität Regensburg) as part of CRM workshop: Arithmetic quotie
nts of locally symmetric spaces and their cohomology\n\nLecture held in Vi
rtual.\n\nAbstract\nIn this lecture\, I will report on recent results\, jo
int with Guido Kings\, on the construction of equivariant\nEisenstein clas
ses. The equivariant polylogarithm is a very general tool for constructing
motivic cohomology\nclasses of arithmetic groups. A certain refinement of
the de Rham realization of these classes gives interesting\nalgebraic Eis
enstein classes. As an application of our construction\, we prove algebrai
city results for critical\nHecke L-values of totally imaginary fields. Thi
s generalizes previous results of Damerell\, Shimura and Katz in\nthe CM c
ase. The integrality of our construction allows us to construct p-adic L-f
unctions for totally imaginary\nfields at ordinary primes.\n
LOCATION:https://researchseminars.org/talk/Cohomology/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luis Garcia Martinez (University College London)
DTSTART:20201021T160000Z
DTEND:20201021T170000Z
DTSTAMP:20260422T212932Z
UID:Cohomology/8
DESCRIPTION:Title: An Eisenstein cocycle for imaginary quadratic fields\nby Luis Garci
a Martinez (University College London) as part of CRM workshop: Arithmetic
quotients of locally symmetric spaces and their cohomology\n\nLecture hel
d in Virtual.\n\nAbstract\nI will give details of the general picture disc
ussed by Nicolas Bergeron in the case of arithmetic subgroups of\nSL_N(k)\
, where k is an imaginary quadratic field. I will introduce a cocycle for
such groups whose values are\npolynomials in classical Kronecker—Eisenst
ein series. We will then see how this cocycle leads to explicit\nformulas
for critical values of Hecke L—functions of degree N extensions of k\, g
eneralising work of Colmez.\nThis is based on a work-in-progress with Pier
re Charollois\, Luis Garcia and Akshay Venkatesh.\n
LOCATION:https://researchseminars.org/talk/Cohomology/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicolas Bergeron (École normale supérieure\, Paris France)
DTSTART:20201022T133000Z
DTEND:20201022T143000Z
DTSTAMP:20260422T212932Z
UID:Cohomology/9
DESCRIPTION:Title: Sczech cocycles and hyperplane arrangements 4\nby Nicolas Bergeron
(École normale supérieure\, Paris France) as part of CRM workshop: Arith
metic quotients of locally symmetric spaces and their cohomology\n\nLectur
e held in Virtual.\n\nAbstract\nMany authors\, among which Nori\, Sczech\,
Solomon\, Stevens\, or more recently Beilinson—Kings—Levin and\nCharo
llois—Dasgupta—Greenberg\, have constructed different\, but related\,
linear groups cocycles that are\nusually referred to as « Eisenstein cocy
cles. » In these series of lectures I will explain a topological\nconstru
ction that is a common source for all these cocycles. One interesting feat
ure of this construction is that\nstarting from a purely topological class
it leads to the algebraic world of meromorphic forms on hyperplane\ncompl
ements in n-fold products of either the (complex) additive group\, the mul
tiplicative group or a (family of)\nelliptic curve(s). We will see that ev
entually our construction reveals hidden relations between products of\nel
ementary (rational\, trigonometric or elliptic) functions) governed by rel
ations between classes in the\nhomology of linear groups.\nThis is based o
n a work-in-progress with Pierre Charollois\, Luis Garcia and Akshay Venka
tesh.\n
LOCATION:https://researchseminars.org/talk/Cohomology/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Preston Wake (Michigan State University)
DTSTART:20201022T150000Z
DTEND:20201022T160000Z
DTSTAMP:20260422T212932Z
UID:Cohomology/10
DESCRIPTION:Title: Torsion in the cohomology of Hilbert modular surfaces\nby Preston
Wake (Michigan State University) as part of CRM workshop: Arithmetic quoti
ents of locally symmetric spaces and their cohomology\n\nLecture held in V
irtual.\n\nAbstract\nWe investigate the analogue of Mazur's Eisenstein ide
al for Hilbert modular forms over a real quadratic field.\nUnlike in the c
ase of modular forms\, we show that\, even in weight two\, there are mod-p
modular forms that\ndon't lift to characteristic zero. We explain this by
computing the torsion in the cohomology of the Hilbert\nmodular surface.
This is joint work with Akshay Venkatesh.\n
LOCATION:https://researchseminars.org/talk/Cohomology/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicolas Bergeron (École normale supérieure\, Paris France)
DTSTART:20201023T133000Z
DTEND:20201023T143000Z
DTSTAMP:20260422T212932Z
UID:Cohomology/11
DESCRIPTION:Title: Sczech cocycles and hyperplane arrangements 5\nby Nicolas Bergeron
(École normale supérieure\, Paris France) as part of CRM workshop: Arit
hmetic quotients of locally symmetric spaces and their cohomology\n\nLectu
re held in Virtual.\n\nAbstract\nMany authors\, among which Nori\, Sczech\
, Solomon\, Stevens\, or more recently Beilinson—Kings—Levin and\nChar
ollois—Dasgupta—Greenberg\, have constructed different\, but related\,
linear groups cocycles that are\nusually referred to as « Eisenstein coc
ycles. » In these series of lectures I will explain a topological\nconstr
uction that is a common source for all these cocycles. One interesting fea
ture of this construction is that\nstarting from a purely topological clas
s it leads to the algebraic world of meromorphic forms on hyperplane\ncomp
lements in n-fold products of either the (complex) additive group\, the mu
ltiplicative group or a (family of)\nelliptic curve(s). We will see that e
ventually our construction reveals hidden relations between products of\ne
lementary (rational\, trigonometric or elliptic) functions) governed by re
lations between classes in the\nhomology of linear groups.\nThis is based
on a work-in-progress with Pierre Charollois\, Luis Garcia and Akshay Venk
atesh.\n
LOCATION:https://researchseminars.org/talk/Cohomology/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pierre Charollois (Université Paris 6)
DTSTART:20201023T150000Z
DTEND:20201023T160000Z
DTSTAMP:20260422T212932Z
UID:Cohomology/12
DESCRIPTION:Title: Trigonometric and elliptic cocycles overGLN(Z)\nby Pierre Charollo
is (Université Paris 6) as part of CRM workshop: Arithmetic quotients of
locally symmetric spaces and their cohomology\n\nLecture held in Virtual.\
n\nAbstract\nI will first recall a joint work with Dasgupta and Greenberg
(2016)\, where we elaborate on Shintani's method\nto construct an Eisenste
in cocycle over GLN(<Z) taking values in a ring o
f\nrational generating series that can be expressed in terms of basic trig
onometric functions. We establish that it\nis cohomologous to a former coc
ycle of Sczech. After smoothing it enjoys nice integral properties. Combin
ed\nwith evaluation on a tori\, it allows us to recover the basic properti
es of the Cassou-Noguès p-adic zeta\nfunctions attached to totally real f
ields.\nThen I'll give an overview of the elliptic generalization I have l
ater obtained\, where trigonometric functions are\nnow replaced by the Kro
necker-Eisenstein function\, which is a generating series for modular form
s. The action\nof Hecke operators over GLN on that new c
ocycle has been studied by Hao Zhang in\nhis thesis (2020)\, and the end o
f talk present some of his results too.\n
LOCATION:https://researchseminars.org/talk/Cohomology/12/
END:VEVENT
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