Let \(l\) and \(p\) be two prime numbers. The Langlands program aims to establish a bijection with good properties between \(l\)-adic representations of absolute Galois groups of number fields (respectively, \(p\)-adic local fields; this is the local case) and \(l\)-adic representations attached to automorphic representations of reductive groups over number fields (respectively, certain smooth \(l\)-adic representations of reductive groups over \(p\)-adic fields). This domain of research has seen a spectacular progress during last two decades culminating in establishing the existence of the desired correspondence for \(GL_n\), when \(l \not = p\) (work of Harris and Taylor).
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The \(p\)-adic local Langlands corresponcence focuses on the local case when \(l = p\). This turns out to be much harder than the \(l \not = p\) case and demands different technical tools. As for now, the \(p\)-adic correspondence is known only in the case of \(GL_2({\bf Q}_p)\) (by works of Berger, Breuil, Colmez, Emerton, Kisin, Paskunas and many others). There is a growing interest in generalizing this correspondence to other groups, especially because of many potential number-theoretic applications.
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In our talk, we shall review recent progress in this theory, together with the latest technical input: completed cohomology of Emerton and perfectoid spaces of Scholze.
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