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Raphaël Beuzart-Plessis, On the spectral decomposition of the Jacquet-Rallis trace formula and the Gan-Gross-Prasad conjecture for unitary groups.
Reference: Raphaël Beuzart-Plessis, On the spectral decomposition of the Jacquet-Rallis trace formula and the Gan-Gross-Prasad conjecture for unitary groups. BC NT/AG Seminar.
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Look at automorphic cuspidal representations to \(\operatorname{PGL}_2({ \mathbf{F} })\) for \({ \mathbf{F} }\) a number field.
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This talk: generalizing some of these formulas:
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See split torus, quaternion algebra.
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See cusp forms, period, trace formulas
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Gan-Gross-Prasad and Ichino-Ikeda conjecture for unitary groups
- \(E/{ \mathbf{F} }\) a quadratic extension of number fields, so \({ \mathsf{Gal}} (E/{ \mathbf{F} }) \cong {\mathbf{Z}}/2 = \left\{{ 1, c }\right\}\).
- Take a nondegenerate Hermitian form \(h\) on \(E^n\), define \(U(h)\) as the unitary group of \(h\) and set \(U_h \coloneqq U(h) \times U(h \oplus h_0)\) and \(h_0: E^2 \to E\) where \((x,y) \mapsto xy^c\).
- Define an L function :
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Theorem: from the Langlands philosophy. There is a Hermitian form and a cuspidal representation in the same L packet where \(P\) is nonvanishing .
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Theorem/conjecture: if \(\sigma\) is tempered everywhere, there is a formula:
- See special values of L functions.
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Proved in special cases.
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Recent result: \(P_h\neq - \implies L(1/2, \Pi)\neq 0\) proved using twisted automorphic descent.
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See regularization, Langlands decomposition, Eisenstein series, Levi, GIT quotients, orbital integrals.
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Global distributions break into an Euler product of local distributions.
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Part of proof: use zeta integrals of kernel functions, fix one variable to get a function in a Schwartz space.
- See unipotent and parabolic subgroups
- Uses a “standard unfolding” process?
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See Whittaker function, Petersson inner product, Phragmen-Lindelof principle to control one zeta integral in terms of another.
See Serre’s conjecture.