BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Justin Trias (University of East Anglia) DTSTART:20210223T143000Z DTEND:20210223T153000Z DTSTAMP:20260421T153238Z UID:CamNT/10 DESCRIPTION:Title: T owards an integral local theta correspondence: universal Weil module and f irst conjectures\nby Justin Trias (University of East Anglia) as part of Cambridge Number Theory Seminar\n\n\nAbstract\nThe theta correspondence is an important and somewhat mysterious tool in number theory\, with arit hmetic applications ranging from special values of L-functions\, epsilon f actors\, to the local Langlands correspondence. The local variant of the t heta correspondence is described as a bijection between prescribed sets of irreducible smooth complex representations of groups $G_1$ and $G_2$\, wh ere $(G_1\,G_2)$ is a reductive dual pair in a symplectic p-adic group. Th e basic setup in the theory (Stone-von Neumann theorem\, the metaplectic g roup and the Weil representation) can be extended beyond complex represent ations to representations with coefficients in any algebraically closed fi eld R as long as the characteristic of R does not divide p. However\, the correspondence defined in this way may no longer be a bijection depending on the characteristic of R compared to the pro-orders of the pair $(G_1\,G _2)$. In the recent years\, there has been a growing interest in studying representations with coefficients in as general a ring as possible. In thi s talk\, I will explain how the basic setup makes 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 will discuss some conject ures towards an integral local theta correspondence. In particular\, one e xpects that the failure of this correspondence for fields having bad chara cteristic does appear in terms of some torsion submodule in integral isoty pic families of the Weil representation with coefficients in B.\n LOCATION:https://researchseminars.org/talk/CamNT/10/ END:VEVENT END:VCALENDAR