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SUMMARY:Towards an integral local theta correspondence: universal Weil mod
 ule and first conjectures - Justin Trias (University of East Anglia)
DTSTART:20210223T143000Z
DTEND:20210223T153000Z
UID:TALK156550@talks.cam.ac.uk
CONTACT:Jessica Fintzen
DESCRIPTION:The theta correspondence is an important and somewhat mysterio
 us tool in number theory\, with arithmetic applications ranging from speci
 al values of L-functions\, epsilon factors\, to the local Langlands corres
 pondence. The local variant of the theta correspondence is described as a 
 bijection between prescribed sets of irreducible smooth complex representa
 tions of groups G_1 and G_2\, where (G_1\,G_2) is a reductive dual pair in
  a symplectic p-adic group. The basic setup in the theory (Stone-von Neuma
 nn theorem\, the metaplectic group and the Weil representation) can be ext
 ended beyond complex representations to representations with coefficients 
 in any algebraically closed field R as long as the characteristic of R doe
 s not divide p. However\, the correspondence defined in this way may no lo
 nger be a bijection depending on the characteristic of R compared to the p
 ro-orders of the pair (G_1\,G_2). In the recent years\, there has been a g
 rowing interest in studying representations with coefficients in as genera
 l a ring as possible. In this talk\, I will explain how the basic setup ma
 kes sense over an A-algebra B\, where A is the ring obtained from the inte
 gers by inverting p and adding enough p-power roots of unity. Eventually\,
  I will discuss some conjectures towards an integral local theta correspon
 dence. In particular\, one expects that the failure of this correspondence
  for fields having bad characteristic does appear in terms of some torsion
  submodule in integral isotypic families of the Weil representation with c
 oefficients in B.\n\nIf you like to attend the talk\, please register here
  using your full professional name: https://maths-cam-ac-uk.zoom.us/meetin
 g/register/tJ0rduqvqDkoHNVfiCUn5f9IYxlhZKyCD3-S
LOCATION:Online
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