Automorphic Forms and Galois Representations: Volume 2


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Mladen Dimitrov

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What do you want to do in September for MO's tenth anniversary? Related Question feed. Divisors and closed ideals of B Definition 2. The proof of this theorem goes as follows. The set Y is the set of closed maximal ideals of B. The first step in the proof of their classification [20] theo. From this one deduces that the line bundle L should be ample. Multiplicative structure of the graded algebra P Definition 2.

Let us note the following important corollary. If F is algebraically closed the graded algebra P is graded factorial with irreducible elements of degree 1. In the preceding theorem, the injectivity is an easy application of Theorem 2. The surjectivity uses Weierstrass products. Weierstrass products and the logarithm of a Lubin—Tate group We use the notation from Section 2.

In fact we have the following period isomorphism. This equivalence class of norms defines the Banach space topology of the preceding theorem. The fact that such a series makes sense in the Witt bivectors is an essential ingredient in the proof of Theorem 2. The curve 5. The fundamental exact sequence Using the results from the preceding section we give a new proof of the fundamental exact sequence. In fact this fundamental exact sequence is a little bit more general than the usual one.

Let t1 ,. Vector bundles on curves and p-adic Hodge theory 55 Proof. We conclude since Cm is algebraically closed. We will use the following corollary. The curve when F is algebraically closed Theorem 2. As a consequence of Corollary 2. To prove it is a P. Using Corollary 2. Vector bundles on curves and p-adic Hodge theory 57 Using the fundamental exact sequence one verifies this embedding of D.

Other assertions of the theorem are easily verified. The following proposition makes clear the difference between X and P1 and will have important consequences on the classification of vector bundles on X.

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Mladen Dimitrov

It is deduced from Corollary 2. Moreover Be , deg is not euclidean. We put subscripts to indicate the dependence on the field F of the preceding constructions. The curve X of the preceding section is equipped with an action of G F. F Theorem 2. It satisfies the following properties. Let us give a few indications on the tools used in the proof.

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F Proof. According to Proposition 2. Applying Theorem 2. The key tool is the following cohomological F ,e computation. F ,d Proof. Let LT be a Lubin—Tate group law. This is deduced from Theorem 2.

Computational Aspects of Modular Forms and Galois Representations

Then, according to Theorem 2. According to Theorem 2. The arguments used in the proof of Theorem 2. Let us remark the preceding theorem implies the following. If the residue field of F is algebraically closed, the choice of another uniformizing element gives a graded algebra that is isomorphic to the preceding, but such an isomorphism is not canonical. We now put a second subscript in our notations to indicate the dependence on E. Suppose we have fixed algebraic closures F and E. We can prove the following. Vector bundles 6.

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Generalities Definition 2. If F is algebraically closed, the ring Be is a P. Line bundles 6. If F is algebraically closed, with the notation of Section 6. Suppose now F is general.


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F F But according to Theorem 2. Vector bundles on curves and p-adic Hodge theory 65 6. Cohomology of line bundles Suppose F is algebraically closed. With the notation of Section 6. The fact that Be , deg is almost euclidean 2. From this one obtains the following proposition. The classification theorem when F is algebraically closed 6. Statement of the theorem Here is the main theorem about vector bundles. It is an analogue of Kedlaya [21],[22] or Hartl—Pink [20] classification theorems.

Automorphic Forms and Galois Representations: Volume 2 Automorphic Forms and Galois Representations: Volume 2
Automorphic Forms and Galois Representations: Volume 2 Automorphic Forms and Galois Representations: Volume 2
Automorphic Forms and Galois Representations: Volume 2 Automorphic Forms and Galois Representations: Volume 2
Automorphic Forms and Galois Representations: Volume 2 Automorphic Forms and Galois Representations: Volume 2
Automorphic Forms and Galois Representations: Volume 2 Automorphic Forms and Galois Representations: Volume 2
Automorphic Forms and Galois Representations: Volume 2 Automorphic Forms and Galois Representations: Volume 2
Automorphic Forms and Galois Representations: Volume 2 Automorphic Forms and Galois Representations: Volume 2
Automorphic Forms and Galois Representations: Volume 2 Automorphic Forms and Galois Representations: Volume 2

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