Non-abelian Lubin-Tate theory.

*(English)*Zbl 0704.11049
Automorphic forms, Shimura varieties, and L-functions. Vol. II, Proc. Conf., Ann Arbor/MI (USA) 1988, Perspect. Math. 11, 15-39 (1990).

[For the entire collection see Zbl 0684.00004.]

The paper gives, in an extremely condensed but nevertheless highly valuable form, an overview on a field that vaguely may be circumscribed as “local Langlands conjectures”. We cannot further abbreviate the yet brief explanations of the article, so we limit ourselves to shortly indicate the contents.

In the introduction, a short discussion of the motivation and history of Langlands’ conjectures is given. Let F be a local non-archimedean field with residual characteristic p and Weil group \(W_ F\). Let further \(B=B_{h,F}\) be the central division algebra over F with invariant h.

In section 1 and 2, representations \({\mathcal A}^ v\) and \({\mathcal A}^ r\) of the group \(GL(h,F)\times B^*xW_ F\) are constructed. The first, \({\mathcal A}^ v\), acts essentially on the \(\ell\)-adic (\(\ell \neq p)\) cohomology of the moduli scheme for certain formal \(O_ F\)-modules, whereas \({\mathcal A}^ r\) acts on the rigid \(\ell\)-adic cohomology of some p-adic symmetric domain. (No formal proof of the existence of such a cohomology theory has been given by now!)

Now in section 3, assuming the local Langlands conjecture for F and h, precise conjectures for the decompositions of \({\mathcal A}^ v\) and \({\mathcal A}^ r\) are proposed. The author notes that “...these conjectures are not specially mine. They seem to have been known for a certain time by some people, mostly by Deligne and Drinfel’d...”. If \(h=1\), these conjectures collapse to Lubin-Tate theory, which explains the title.

In section 4, the author sketches a proof for the conjecture on \({\mathcal A}^ r\), in the case \(F={\mathbb{Q}}_ p\) and \(h=2\). He concludes, in section 5, with a discussion to what extent that proof might be generalized to higher dimensions.

The paper gives, in an extremely condensed but nevertheless highly valuable form, an overview on a field that vaguely may be circumscribed as “local Langlands conjectures”. We cannot further abbreviate the yet brief explanations of the article, so we limit ourselves to shortly indicate the contents.

In the introduction, a short discussion of the motivation and history of Langlands’ conjectures is given. Let F be a local non-archimedean field with residual characteristic p and Weil group \(W_ F\). Let further \(B=B_{h,F}\) be the central division algebra over F with invariant h.

In section 1 and 2, representations \({\mathcal A}^ v\) and \({\mathcal A}^ r\) of the group \(GL(h,F)\times B^*xW_ F\) are constructed. The first, \({\mathcal A}^ v\), acts essentially on the \(\ell\)-adic (\(\ell \neq p)\) cohomology of the moduli scheme for certain formal \(O_ F\)-modules, whereas \({\mathcal A}^ r\) acts on the rigid \(\ell\)-adic cohomology of some p-adic symmetric domain. (No formal proof of the existence of such a cohomology theory has been given by now!)

Now in section 3, assuming the local Langlands conjecture for F and h, precise conjectures for the decompositions of \({\mathcal A}^ v\) and \({\mathcal A}^ r\) are proposed. The author notes that “...these conjectures are not specially mine. They seem to have been known for a certain time by some people, mostly by Deligne and Drinfel’d...”. If \(h=1\), these conjectures collapse to Lubin-Tate theory, which explains the title.

In section 4, the author sketches a proof for the conjecture on \({\mathcal A}^ r\), in the case \(F={\mathbb{Q}}_ p\) and \(h=2\). He concludes, in section 5, with a discussion to what extent that proof might be generalized to higher dimensions.

Reviewer: E.U.Gekeler

##### MSC:

11R39 | Langlands-Weil conjectures, nonabelian class field theory |

11S37 | Langlands-Weil conjectures, nonabelian class field theory |

14L05 | Formal groups, \(p\)-divisible groups |

11G05 | Elliptic curves over global fields |

11F70 | Representation-theoretic methods; automorphic representations over local and global fields |