This paper deals with the Langlands' classification for discrete series of unitary quasi-split p-adic groups. We show that such a classification follows from Arthur's work on the simple trace formula which we can use now thanks to Laumon-Ngo's proof of the fundamental lemma and Waldspurger's work on the consequences of it on the transfer.

Classification et changement de bases pour les s\'{e}ries discr\`{e}tes des groupes unitaires p-adiques

Thank you for reading the geometry and cohomology of some simple shimura varieties. As you may know, people have look numerous times for their chosen readings like this the geometry and cohomology of some simple shimura varieties, but end up in harmful downloads. Rather than reading a good book with a cup of coffee in the afternoon, instead they are facing with some infectious bugs inside their computer.

The Geometry And Cohomology Of Some Simple Shimura Varieties

In this article, we study the Beilinson-Bloch-Kato conjecture for motives corresponding to the Rankin-Selberg product of conjugate self-dual automorphic representations, within the framework of the Gan-Gross-Prasad conjecture. We show that if the central critical value of the Rankin-Selberg $L$-function does not vanish, then the Bloch-Kato Selmer group with coefficients in a favorable field of the corresponding motive vanishes. We also show that if the class in the Bloch-Kato Selmer group constructed from certain diagonal cycle does not vanish, which is conjecturally equivalent to the nonvanishing of the central critical first derivative of the Rankin-Selberg $L$-function, then the Bloch-Kato Selmer group is of rank one.

On the Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives

We prove the local Kudla--Rapoport conjecture, which is a precise identity between the arithmetic intersection numbers of special cycles on unitary Rapoport--Zink spaces and the derivatives of local representation densities of hermitian forms. As a first application, we prove the global Kudla--Rapoport conjecture, which relates the arithmetic intersection numbers of special cycles on unitary Shimura varieties and the central derivatives of the Fourier coefficients of incoherent Eisenstein series. Combining previous results of Liu and Garcia--Sankaran, we also prove cases of the arithmetic Siegel--Weil formula in any dimension.

Kudla--Rapoport cycles and derivatives of local densities

Abstract Zhang proved the global Gan–Gross–Prasad conjecture for U ⁡ ( n + 1 ) × U ⁡ ( n ) \\operatorname{U}(n+1)\\times\\operatorname{U}(n) under some local conditions [19]. One of the conditions is that the unitary groups are split at the archimedean places. We remove this assumption at the archimedean places in this paper.

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On the global Gan–Gross–Prasad conjecture for unitary groups: Approximating smooth transfer of Jacquet–Rallis

We prove an integral formula computing multiplicities of square-integrable representations relative to Galois pairs over p-adic fields and we apply this formula to verify two consequences of a conjecture of Dipendra Prasad. One concerns the exact computation of the multiplicity of the Steinberg representation and the other the invariance of multiplicities by transfer among inner forms.

https://hal.archives-ouvertes.fr/hal-01941337/document

On distinguished square-integrable representations for Galois pairs and a conjecture of Prasad

In this paper, we prove a conjecture of Wei Zhang on comparison of certain local spherical characters from which we draw some consequences for the Ichino-Ikeda conjecture for unitary groups.

Comparison of local spherical characters and the Ichino-Ikeda conjecture for unitary groups

We introduce a new technique for isolating components on the spectral side of the trace formula. By applying it to the Jacquet--Rallis relative trace formula, we complete the proof of the global Gan--Gross--Prasad conjecture and its refinement Ichino--Ikeda conjecture for $\mathrm{U}(n)\times\mathrm{U}(n+1)$ in the stable case.

Isolation of cuspidal spectrum, with application to the Gan--Gross--Prasad conjecture

We prove an integral formula computing multiplicities of square-integrable representations relative to Galois pairs over $p$-adic fields and we apply this formula to verify two consequences of a conjecture of Dipendra Prasad. One concerns the exact computation of the multiplicity of the Steinberg representation and the other the invariance of multiplicities by transfer among inner forms.

https://hal.archives-ouvertes.fr/hal-01941343/document

Raphaël Beuzart-Plessis

Papers

On distinguished square-integrable representations for Galois pairs and a conjecture of Prasad

Comparison of local spherical characters and the Ichino-Ikeda conjecture for unitary groups

Isolation of cuspidal spectrum, with application to the Gan--Gross--Prasad conjecture

On distinguished square-integrable representations for Galois pairs and a conjecture of Prasad

Comparison of local spherical characters and the Ichino-Ikeda conjecture for unitary groups