Quadratic Base Change for p-adic SL(2) as a Theta Correspondence III: Pairing
✍ Scribed by David Manderscheid
- Book ID
- 102605086
- Publisher
- Elsevier Science
- Year
- 2000
- Tongue
- English
- Weight
- 173 KB
- Volume
- 85
- Category
- Article
- ISSN
- 0022-314X
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✦ Synopsis
This paper is the third in a series of papers examining in detail the local theta correspondences attached to the reductive dual pairs (SL 2 (F ), O(F )) where F is a p-adic field of characteristic zero and O is the orthogonal group attached to a quaternary quadratic form with coefficients in F and of Witt rank one over F. In this paper we make a conjecture giving a functorial interpretation of the correspondences and we provide substantial evidence for the conjecture. The conjecture generalizes and refines a result of Cognet [C] and it is consistent with the generally held expectation that theta correspondences attached to groups close in split rank should have functorial interpretations; see, e.g., [A], [Au] and [MS] for examples. When theta correspondences do not have functorial interpretations (in the strict sense of Langlands as opposed to Arthur's formulation), the examples can have profound arithmetic significance, as witnessed by the counterexample to the generalized Ramanujan conjecture in [HPS]. The results that we prove here that provide evidence for the conjecture complement the results of [C] and [R2] on these correspondences. They are also consistent with Kudla's conjectures (see, e.g., [R1]) and those of Prasad [P]. In future papers we plan to provide further evidence for our conjecture, in particular, for p odd, using the lattice model of the oscillator representation and the results of [M3].
To explain our results and methods, we first recall the general setting of theta correspondences for symplectic and orthogonal groups (see, e.g., [MVW], [H]). For i=1, 2, let V i be a finite-dimensional vector space over F equipped with a nondegenerate bilinear form ( , ) i ; assume that ( , ) 1 is skew-symmetric while ( , ) 2 is symmetric. Equip W=V 1 V 2 with the skew-symmetric form ( , ) coming from tensoring the ( , ) i . Let