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."

**Read or Download Abelian l-adic representations and elliptic curves PDF**

**Similar algebraic geometry books**

During this e-book, Professor Novikov describes fresh advancements in soliton thought and their family members to so-called Poisson geometry. This formalism, that's concerning symplectic geometry, is intensely worthwhile for the research of integrable structures which are defined by way of differential equations (ordinary or partial) and quantum box theories.

**Algebraic Geometry Iv Linear Algebraic Groups Invariant Theory**

Contributions on heavily similar matters: the speculation of linear algebraic teams and invariant thought, by way of recognized specialists within the fields. The booklet should be very necessary as a reference and examine consultant to graduate scholars and researchers in arithmetic and theoretical physics.

**Vector fields on singular varieties**

Vector fields on manifolds play a massive function in arithmetic and different sciences. particularly, the Poincaré-Hopf index theorem provides upward thrust to the speculation of Chern sessions, key manifold-invariants in geometry and topology. it truly is average to invite what's the ‘good’ inspiration of the index of a vector box, and of Chern sessions, if the underlying area turns into singular.

E-book by means of

- Positive polynomials and sums of squares
- Functional differential equations / 2. C*-applications. Pt. 2, Equations with discontinuous coefficients and boundary value problems
- Lectures on Theta II Birkhaeuser
- Spaces of Homotopy Self-Equivalences: A Survey
- Constructible Sets in Real Geometry
- Introduction to Modern Number Theory: Fundamental Problems, Ideas and Theories

**Additional info for Abelian l-adic representations and elliptic curves**

**Sample text**

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.