By Jean-Pierre Serre
This vintage booklet includes an advent to platforms of l-adic representations, a subject matter of significant significance in quantity idea and algebraic geometry, as mirrored by means of the superb contemporary advancements at the Taniyama-Weil conjecture and Fermat's final Theorem. The preliminary chapters are dedicated to the Abelian case (complex multiplication), the place one unearths a pleasant correspondence among the l-adic representations and the linear representations of a few algebraic teams (now known as Taniyama groups). The final bankruptcy handles the case of elliptic curves without advanced multiplication, the most results of that is that a twin of the Galois team (in the corresponding l-adic illustration) is "large."
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Additional info for Abelian l-adic representations and elliptic curves
For k = 1 the generic splitting type is aE = (1, 0), for on any line L which does not meet Y the restriction E|L is given by 0 → OL → E|L → OL (1) → 0, and this sequence splits because H 1 (L, OL (−1)) = 0. For k = 2 we have for any line L ⊂ P2 with L ∩ Y = ∅ that E|L is given by the extension 0 → OL → E|L → OL (2) → 0. This however does not mean that the splitting type of E on these lines is (2, 0), for not all extensions of OL (2) by OL split. In fact the generic splitting type for k = 2 is aE = (1, 1), for (1, 1) is the (lexicographically) smallest possible pair (a, b) with a ≥ b and a + b = 2 and on a line L which meets exactly one point xi 54 1.
The examples of Tango give simple (n − 1)-bundles over Pn . Indecomposable (n − 2)-bundles over Pn are diﬃcult to construct and are known only for n = 4 (examples of Horrocks and Mumford , cf. , example 6) and n = 5 (cf. Horrocks ). The following theorem of Barth and Van de Ven  also sheds some light on this problem: a holomorphic 2-bundle over Pn which can be holomorphically extended over arbitrary PN ⊃ Pn is necessarily of the form OPn (a) ⊕ OPn (b). This theorem has played an important psychological rˆole in the development and has been written by Tjurin  and Sato  for bundles of arbitrary rank.
Uniform bundles In this paragraph we explain the “standard construction”, which systematizes the study of a vector bundle over Pn by considering its restrictions to lines. As a ﬁrst application we ﬁnd that a bundle whose restriction to every line through some given point is trivial must itself be trivial. Then we show that uniform r-bundles over Pn always split if r < n. This is no longer true for r ≥ n (see the remarks at the end of this section). Finally we give an example of a uniform bundle which is not homogeneous.