Download A Course in p-adic Analysis (Graduate Texts in Mathematics) by Alain M. Robert PDF

By Alain M. Robert

ISBN-10: 1441931503

ISBN-13: 9781441931504

Found on the flip of the twentieth century, p-adic numbers are often utilized by mathematicians and physicists. this article is a self-contained presentation of uncomplicated p-adic research with a spotlight on analytic subject matters. It bargains many positive factors not often handled in introductory p-adic texts similar to topological versions of p-adic areas within Euclidian house, a unique case of Hazewinkel’s sensible equation lemma, and a remedy of analytic parts.

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0 '. a . 0 1 2 0- A projection of C x C 3. 1. Topological Groups Definition. A topological group is a group G equipped with a topology such that the map (x, y) H xy-t : G x G -* G is continuous. 18 L _ p-adic Numbers If G is a topological group, the inverse map x H x-1 is continuous (fix x = e in the continuous map (x, y) t-* xy-1) and hence a homeomorphism of order 2 of G. The translations x H ax (resp. , the inverse of x r- ax is x F+ a-tx). A subgroup of a topological group is a topological group for the induced topology.

Proposition, The following properties are equivalent: (1) P = 0 admits a solution in Zp. (ii) For each n > 0, P = 0 admits a solution in Z/ p" Z. (iii) For each n > 0, there are integers a", b" such that P(an,b,,)=- 0modp". PROOF. (iii) is a simple reformulation of (ii). Now for x = Et>o ai p' E Zp, define x" _ >t<" a; p' mod p" E Z/p'Z. Then if (x, y) E Zp x Zp, then P(xn, y,,) = P(x, y) mod p"ZP E ZP/p"ZP (= Z/p"Z), and hence (i) (i i ). Conversely, to prove (i i) (i) let us consider the finite sets X" ={(x,y)EZ/p"ZxZ/p"Z: P(x,y)=0}.

Let )), be an affine linear functional on V such that A. < l on E, A(v) = 1 for some v E E. Choose 1 = b - 1. Then v(0 1 i>o bi+l r>o so that the image F of %P is also contained in the convex hull of E: F C E = Ko. Moreover, by choice of the constant $, v >gE,>ob+l)=1. From the self-similarity representation of F we get a better approximation F=U $U+ b b) I ( 6 bv C VEX VEX b f. b) Iterating this inclusion in the self-similarity representation of F we get an even better approximation: I VE C 0-+b b, alb F= v +1 I VEX b uEE(Ob+ c) Eventually, this leads to a representation of the fractal F as the intersection of a decreasing sequence of compact sets K.

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