What do we mean by local ? To get an answer to this question let us start from the following two problems.First problem is from number theory. When does the diophantine equationhave a non-trivial solution in rational numbers ? In order to solve the problem, let us consider the quadratic norm residue symbol (−, −) p where p runs through all primes p and also ∞. This symbol is a bi-multiplicative map (−, −) p : Q * × Q * → {±1} and it is easily computed in terms of the Legendre symbol. Then, a non-trivial solution exists if and only if, for any p, (a, b) p = 1. However, these conditions are not independent:This is essentially the Gauss reciprocity law in the Hilbert form. The "points" p correspond to all possible completions of the field Q of rational numbers, namely to the p-adic fields Q p and the field R of real numbers. One can show that the equation f = 0 has a non-trivial solution in Q p if and only if (a, b) p = 1.The second problem comes from complex analysis. Let X be a compact Riemann surface (= complete smooth algebraic curve defined over C). For a point P ∈ X, denote by K P = C((t P )) the field of Laurent formal power series in a local coordinate t P at the point P . The field K P contains the ring O P = C[[t P ]] of Taylor formal power series. These have an invariant meaning and are called the local field and the local ring at P respectively. Let us now fix finitely many points P 1 , . . . , P n ∈ X and assign to every P in X some elements f P such that f P 1 ∈ K P 1 , . . . , f Pn ∈ K Pn and f P = 0 for all other points. *

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