By Michael Artin

Those notes are according to lectures given at Yale college within the spring of 1969. Their item is to teach how algebraic services can be utilized systematically to improve definite notions of algebraic geometry,which are typically taken care of by means of rational services through the use of projective equipment. the worldwide constitution that's traditional during this context is that of an algebraic space—a area got by means of gluing jointly sheets of affine schemes through algebraic functions.I attempted to imagine no past wisdom of algebraic geometry on thepart of the reader yet was once not able to be constant approximately this. The test purely 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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E. The p-adic dilogarithm and the dianalog. The next dilogarithm variant we mention is the p-adic dilogarithm, studied by R. Coleman and other authors. We fix a prime number p and define (p) Li2 (x) = n>0, p∤n xn . n2 (14) This function can be written as Li2 (x)−p−2 Li2 (xp ), so in the complex domain it is simply a combination of ordinary dilogarithms and of no independent interest, but because we have omitted the terms in (14) with p’s in the denominator, the power series converges p-adically for all p-adic numbers x with (p) valuation |x|p < 1.

Z n =x Next, there is the two-variable, five-term relation Li2 (x) + Li2 (y) + Li2 = 1−x 1 − xy 1−y 1 − xy 1−x log 1 − xy + Li2 (1 − xy) + Li2 π2 − log(x) log(1 − x) − log(y) log(1 − y) + log 6 1−y 1 − xy which (in this or one of the many equivalent forms obtained by applying the symmetry properties given above) was discovered and rediscovered by Spence (1809), Abel (1827), Hill (1828), Kummer (1840), Schaeffer (1846), and doubtless others. ) There is also the six-term relation − 1 zn+1 1 − zn = zn−1 1 1 1 + + =1 x y z ⇒ Li2 (x) + Li2 (y) + Li2 (z) = xy yz zx 1 Li2 − + Li2 − + Li2 − 2 z x y discovered by Kummer (1840) and Newman (1892).

Z n =x Next, there is the two-variable, five-term relation Li2 (x) + Li2 (y) + Li2 = 1−x 1 − xy 1−y 1 − xy 1−x log 1 − xy + Li2 (1 − xy) + Li2 π2 − log(x) log(1 − x) − log(y) log(1 − y) + log 6 1−y 1 − xy which (in this or one of the many equivalent forms obtained by applying the symmetry properties given above) was discovered and rediscovered by Spence (1809), Abel (1827), Hill (1828), Kummer (1840), Schaeffer (1846), and doubtless others. ) There is also the six-term relation − 1 zn+1 1 − zn = zn−1 1 1 1 + + =1 x y z ⇒ Li2 (x) + Li2 (y) + Li2 (z) = xy yz zx 1 Li2 − + Li2 − + Li2 − 2 z x y discovered by Kummer (1840) and Newman (1892).

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