Download PDF by Wu X.: 8-ranks of Class Groups of Some Imaginary Quadratic Number

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By Wu X.

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Consider the open subset obtained by removing all hyperplanes {Xi = X j } R" \ (U{Xi =R = Zj}). The connected components of R" are called chambers. We shall point out a particular chamber given by c+= {. E R I 2 1 > . . > x,}. Next, note that the entries of any x in R" are pair-wise different, so there exists a unique permutation which puts them in a decreasing order. This means that R" is a disjoint union of w(C+) as w runs thru S,. In particular, every chamber C is equal to c = {x E R I Xil > ...

Lyl, = lxyJp, and the triangle inequality. In fact we have an even stronger property: 1% +YIP 5 max{l4,1 IYlP). The ring of p a d i c integers Z,is the completion of Zwith respect to this metric. To understand this ring we can proceed as follows. Let B ( z , r ) be the (closed) ball of radius T 2 0, centered at x. Note that Z can be written as a union of p disjoint balls of radius l/p, centered a t 0,1,. . ,p - 1: 1 1 1 P P P Z = B(0,-) u B(1, -) u . . u B(p - 1, -). Note that B ( i ,$) is simply i+pZ, a coset of the maximal ideal (p).

2: Let obtained by conjugating Then x x be a character of T , and x" be the character by w, which simply means permuting x1 and x 2 . 0 +. 6 1 / 2 x w --f IndE(X)N -+ d1/2x-+ 0. Lectures on Representations of p-adic Groups 31 Proof: Let I n d g ( & be the B-submodule of functions in I n d g ( X ) supported on the big cell B U N . We shall use the filtration IndE(x)w c I&(X) and the exactness of V -+ V ~toJ calculate I n d g ( X ) . Define a functional Q on I n d g ( X ) by 4 f )= f (1). Since a(7r(nt)f) = 6 1 ' 2 X ( t ) .

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8-ranks of Class Groups of Some Imaginary Quadratic Number Fields by Wu X.


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