Download Abelian l-adic representations and elliptic curves by Jean-Pierre Serre PDF

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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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 [68], cf. , example 6) and n = 5 (cf. Horrocks [69]). The following theorem of Barth and Van de Ven [9] 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 [128] and Sato [103] 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.

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