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报告人:丁一文 博士(Imperial college London)

I. 时间: 2016-03-31, 10h00-11h30, 地点:晨兴610 
We recall some theory of p-adic modular forms and recall the result of Breuil-Emerton on the existence of overconvergent companion forms using p-adic comparison theorems. We then explain Bergdall's reproof of this result via the geometry of Coleman-Mazur eigencurves. 
 
II. 时间: 2016-04-01, 10h00-11h30, 地点:晨兴610 
(2) We give a quick introduction of p-adic Langlands program and reformulate the companion forms problem (and some of the proof)  in terms of locally analytic  representation theory (which is referred to as Breuil's locally analytic socle conjecture in general case). We then discuss locally analytic representation theory  in more details (in particular for GL2(L) with L finite over Qp).

III. 时间:2016-04-05, 10h00-11h30, 地点:晨兴610 
(3) We construct (patched) eigenvariety for rank 2 definite unitary group (splitting at p). Using locally analytic representation theory, we construct "partially classical" stratifications of the (patched) eigenvariety. And we explain how to deduce Breuil's locally analytic socle conjecture for GL2(L) from the geometry of these stratifications. We leave the study of these stratifications to the next lecture.

IV. 时间:2016-04-06, 10h00-11h30, 地点:晨兴610 
We recall trianguline variety (for Galois representations), and construct "partially de Rham" stratifications of the trianguline variety, which can be studied by (partially de Rham) Galois cohomology. Based on a "R=T" result due to Breuil-Helleman-Schraen (which relates the patched eigenvariety and the trianguline variety) , we explain how to study the "partically classical" stratifications of the patched eigenvariety by studying "partially de Rham" stratifications of the trianguline variety. If time permits, we discuss a little about general GLn case.
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