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We determine local test vectors for Waldspurger functionals for , in the case where both the representation of and the character of the degree two extension are ramified, with certain restrictions. We use this to obtain an explicit version of Waldspurger’s formula relating twisted central -values of automorphic representations on with certain toric period integrals. As a consequence, we generalize an average value formula of Feigon and Whitehouse, and obtain some nonvanishing results.
We generalize Kato’s (commutative) -adic local -conjecture for families of -modules over the Robba rings. In particular, we prove the essential parts of the generalized local -conjecture for families of trianguline -modules. The key ingredients are the author’s previous work on the Bloch–Kato exponential map for -modules and the recent results of Kedlaya, Pottharst and Xiao on the finiteness of cohomology of -modules.
For an odd prime , we construct some admissible Banach representations of that conjecturally should correspond to some -dimensional tamely ramified, potentially Barsotti–Tate representations of via the -adic local Langlands correspondence. To achieve this, we generalize Breuil’s work in the semistable case and work on the first covering of the Drinfel’d upper half-plane. Our main tool is an explicit semistable model of the first covering.
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