By Michael Artin

Those notes are in line with lectures given at Yale college within the spring of 1969. Their item is to teach how algebraic features can be utilized systematically to strengthen yes notions of algebraic geometry,which are typically handled through rational services through the use of projective equipment. the worldwide constitution that's ordinary during this context is that of an algebraic space—a house received through gluing jointly sheets of affine schemes via algebraic functions.I attempted to imagine no past wisdom of algebraic geometry on thepart of the reader yet used to be not able to be constant approximately this. The try out in simple terms avoided me from constructing any subject systematically. Thus,at most sensible, the notes can function a naive creation to the topic.

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The resulting space, if one blows up further the irrelevant points (0, ∞), (∞, 0) is also the Deligne-Mumford compactification for the moduli space of curves of genus 0 with 5 marked points. One gets the picture in Fig. 1 xy=1 y=1 Li=Li(1,1) x= 0 x=1 Li=Li(y) y=0 Li= 0 Fig. 1 Analytic continuation to (1, 1) We now try to compute Li(a,b) (1, 1). First we should interpret it as the value of Li(a,b) on a tangent vector (x, y¯) = (1, 1) at the point (1, 1). The first step in the computation of this value is to restrict to the divisor y = 0.

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. This is not surprising based on its expansion and the identity principle. We now repeat the same considerations but this time restricting to the divisor x = 1. In the same way as before the diﬀerential equations are going to be ⎧ ⎪ ⎪ d ⎨Li(a,b−1) (1, y) b > 1 y Li(a,b) (1, y) = ⎪ y ⎪ ⎩ 1−y dy Lia (y) b=1 1 Heidelberg Lectures on Coleman Integration 39 which are of course the same diﬀerential equations satisfied by the single variable Li(a,b) .

Namely, one easily checks that if D = div( f ), then D ω is the global index of Fω and log( f ), and is thus the cup product in H1dR (X/K) of the projections of ω and d f / f , but the projection of d f / f is 0. This method extends to the bad reduction case [Bes05] but it then needs Coleman integration in the bad reduction case, which either uses the methods of Colmez and Zarhin or that of Vologodsky which is far more complicated. (2) Again in the good reduction case it follows from [Bes00a] that the integral factors via the syntomic regulator on the Chow group of zero cycles on X, see below, thus killing principal divisors.