An application of norm fields.
(Une application de corps des normes.)

*(French)*Zbl 0933.11055This paper shows that a certain restriction functor on \(p\)-adic Galois representations of absolutely unramified \(p\)-adic fields \(K_0\) is fully faithful, and its image is closed under taking subobjects and quotients. More precisely, one restricts representations of \(\text{Gal}(\overline K_0/K_0)\) to \(\text{Gal}(\overline K_0/K_\infty)\) where for once \(K_\infty\) is not the cyclotomic extension but \(K_\infty=K_0(\root {p^\infty} \of \pi)\) with \(\pi\) any uniformizer of \(K_0\). (As the author points out, the theorem would fail in a rather obvious way if one were to take the cyclotomic extension.)

There are two more conditions in the theorem: firstly the representations considered are all supposed crystalline, and secondly the Hodge-Tate weights are supposed to lie inside an interval \([g,h]\) of length \(p-2\). The author surmises that the second condition might be unnecessary, but goes on to show by a counterexample that some condition is certainly needed: one can exhibit an explicit nontrivial extension \[ 0\to {\mathbb{Z}}_p(1) \to V \to {\mathbb{Z}}_p \to 0 \] whose restriction to \(\text{Gal}(\overline K_0/K_\infty)\) becomes the split extension. Thus the restriction functor does sometimes map nonisomorphic objects to isomorphic ones, so it cannot be fully faithful on all representations.

The proof of the main result goes by reduction to representations mod \(p\). It draws on several auxiliary categories and functors, considered previously by Breuil, and Fontaine and Lafaille. In the proof that one of these functors is exact and fully faithful (Cor. 4.2.2), the author makes essential use of Wintenberger’s norm field \(X\) of \(K_\infty\); the Galois group of \(X^{\text{sep}}\) over \(X\) is canonically isomorphic to \(\text{Gal}(\overline K_0/K_\infty)\), and mod \(p\) representations of \(\text{Gal}(X^{\text{sep}}/X)\) can be classified, as shown by Fontaine, by one of the auxiliary categories. For the precise descriptions, and the interrelations between all these categories, we have to refer to the paper.

Reviewer’s remarks: The so-called dĂ©vissage, that is, the reduction to representations mod \(p\) on p. 202 is explained a bit perfunctorily, and there are misprints; but this does not really impair the readability. The author even states that the article is elementary; perhaps some readers won’t agree and some others will.

There are two more conditions in the theorem: firstly the representations considered are all supposed crystalline, and secondly the Hodge-Tate weights are supposed to lie inside an interval \([g,h]\) of length \(p-2\). The author surmises that the second condition might be unnecessary, but goes on to show by a counterexample that some condition is certainly needed: one can exhibit an explicit nontrivial extension \[ 0\to {\mathbb{Z}}_p(1) \to V \to {\mathbb{Z}}_p \to 0 \] whose restriction to \(\text{Gal}(\overline K_0/K_\infty)\) becomes the split extension. Thus the restriction functor does sometimes map nonisomorphic objects to isomorphic ones, so it cannot be fully faithful on all representations.

The proof of the main result goes by reduction to representations mod \(p\). It draws on several auxiliary categories and functors, considered previously by Breuil, and Fontaine and Lafaille. In the proof that one of these functors is exact and fully faithful (Cor. 4.2.2), the author makes essential use of Wintenberger’s norm field \(X\) of \(K_\infty\); the Galois group of \(X^{\text{sep}}\) over \(X\) is canonically isomorphic to \(\text{Gal}(\overline K_0/K_\infty)\), and mod \(p\) representations of \(\text{Gal}(X^{\text{sep}}/X)\) can be classified, as shown by Fontaine, by one of the auxiliary categories. For the precise descriptions, and the interrelations between all these categories, we have to refer to the paper.

Reviewer’s remarks: The so-called dĂ©vissage, that is, the reduction to representations mod \(p\) on p. 202 is explained a bit perfunctorily, and there are misprints; but this does not really impair the readability. The author even states that the article is elementary; perhaps some readers won’t agree and some others will.

Reviewer: Cornelius Greither (Neubiberg)