Towards an integral local theta correspondence: universal Weil module and first conjectures

Abstract: The theta correspondence is an important and somewhat mysterious tool in number theory, with arithmetic applications ranging from special values of L-functions, epsilon factors, to the local Langlands correspondence. The local variant of the theta correspondence is described as a bijection between prescribed sets of irreducible smooth complex representations 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 Neumann theorem, the metaplectic group and the Weil representation) can be extended beyond complex representations to representations with coefficients in any algebraically closed field 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 this 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 adding enough p-power roots of unity. Eventually, I will discuss some conjectures towards an integral local theta correspondence. 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 coefficients in B.

number theory

Audience: researchers in the topic

Cambridge Number Theory Seminar

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