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CATEGORIES:Number Theory Seminar
SUMMARY:Towards an integral local theta correspondence: un
iversal Weil module and first conjectures - Justin
Trias (University of East Anglia)
DTSTART;TZID=Europe/London:20210223T143000
DTEND;TZID=Europe/London:20210223T153000
UID:TALK156550AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/156550
DESCRIPTION:The theta correspondence is an important and somew
hat mysterious tool in number theory\, with arithm
etic applications ranging from special values of L
-functions\, epsilon factors\, to the local Langla
nds correspondence. The local variant of the theta
correspondence is described as a bijection betwee
n prescribed sets of irreducible smooth complex re
presentations of groups G_1 and G_2\, where (G_1\,
G_2) is a reductive dual pair in a symplectic p-ad
ic group. The basic setup in the theory (Stone-von
Neumann theorem\, the metaplectic group and the W
eil representation) can be extended beyond complex
representations to representations with coefficie
nts in any algebraically closed field R as long as
the characteristic of R does not divide p. Howeve
r\, the correspondence defined in this way may no
longer be a bijection depending on the characteris
tic of R compared to the pro-orders of the pair (G
_1\,G_2). In the recent years\, there has been a g
rowing interest in studying representations with c
oefficients in as general a ring as possible. In t
his talk\, I will explain how the basic setup make
s sense over an A-algebra B\, where A is the ring
obtained from the integers by inverting p and addi
ng enough p-power roots of unity. Eventually\, I w
ill discuss some conjectures towards an integral l
ocal theta correspondence. In particular\, one exp
ects that the failure of this correspondence for f
ields having bad characteristic does appear in ter
ms of some torsion submodule in integral isotypic
families of the Weil representation with coefficie
nts in B.\n\nIf you like to attend the talk\, plea
se register here using your full professional name
: https://maths-cam-ac-uk.zoom.us/meeting/register
/tJ0rduqvqDkoHNVfiCUn5f9IYxlhZKyCD3-S
LOCATION:Online
CONTACT:Jessica Fintzen
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