Algebraic Geometry

Download Arithmetic of elliptic curves with complex multiplication by B.H. Gross, B. Mazur PDF

By B.H. Gross, B. Mazur

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Example text

Suppose x' E (R')

4) and the fact that F is non-principal. t = 0. t \ otherwise. t := { s ala] E A[a) n B~] } E :F. t ; moreover, for each A ~ u; U>. t (since a~] E Bt] implies a~l E B~))j therefore n>'E(\/ U>. t = U (U>. t E F, as claimed. t, for all p: This contradicts the hypothesis that the 811(R*) cover ¢(R*). D. 12: R* is uncountable. Proof: Suppose R* = {all I v EN}; we seek a contradiction. Let 811 be the expression X - all = O. Then 811(R*) = {all}' These sets, for v E N, would be a cover of R* possessing no finite sub cover.

This celebrated generalization of Euler's four-square theorem for Q [1754/5, pp . 368-9] had been announced repeatedly by Hilbert ([1899, §38] and [1904, p. 696]), but without proof. Landau [1919] proved Hilbert's claim in the case where K is a quadratic number field; finally, Siegel proved Hilbert's claim completely, in [1921]. 2. 12) to Hilbert's 17th problem as explained in the Introduction. 10). In order to prove the latter, we briefly introduce ultraproducts, and use some arguments that originated in Model Theory.

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