By Jean-Pierre Serre

ISBN-10: 0201093847

ISBN-13: 9780201093841

This vintage publication comprises an creation to structures of l-adic representations, a subject of serious significance in quantity conception and algebraic geometry, as mirrored by means of the marvelous 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 reveals 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's that similar to the Galois crew (in the corresponding l-adic illustration) is "large."

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In [BdJ03] we have shown that the same result holds under some integrality assumptions for the map ˜ • (L)) → Q ¯p, H1 ( M (n) ¯ p , with the single induced by the syntomic regulator and an embedding τ : L → Q valued version of the complex polylogarithm replaced by the function n−1 Ln (z) + Ln−1(z) log(z) (−1)m with Ln (z) = Lin−m (z) logm (z) . n m! m=0 1 Heidelberg Lectures on Coleman Integration 35 We end this subsection by recalling another relation between syntomic cohomology and Coleman integration.

The first step in the computation of this value is to restrict to the divisor y = 0. By this we mean that we analytically continue to the normal bundle of y = 0 minus the 0 section and then restrict to the section y¯ = 1. 14), together with the restriction to y¯ = 1 boils down to removing the part multiplying dy and then setting y = 0 in the formulas. The equations are therefore going to become ⎧ ⎪ ⎪ d ⎨Li(a−1,b) (x, 0) a > 1 x Li(a,b) (x, 0) = ⎪ ⎪ ⎩0 dx a=1. Since the boundary conditions on these functions are always set so that the constant term at 0 is 0 it follows immediately that the function Li(a,b) (x, 0) is identically 0.

Start from a horizontal section in M(U f¯(x) ), pull back by f to obtain a horizontal section of f¯∗ (M) on U x , apply the rule α to obtain a horizontal section on Uz and finally apply the inverse of pullback by f . 10). In particular, when x = z, f¯ : G x → G f¯(x) is a group homomorphism and in general it is compatible with the structure of P x,z as a principal homogeneous space for G x . ¯ →A ¯ and f¯ fixes both x and z. Then we can check what Suppose now that f¯ : A it means for a path α ∈ P x,z to be fixed by f¯.

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

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