By Robert B. Ash, W. P. Novinger

ISBN-10: 0486462501

ISBN-13: 9780486462509

Aimed toward complicated undergraduates and graduate scholars, this considerably revised and up to date version of a well-liked textual content bargains a concise therapy that gives cautious and whole reasons in addition to a number of difficulties and suggestions. subject matters contain simple concept, basic Cauchy theorem and functions, analytic capabilities, and leading quantity theorem. 2004 version.

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**Sample text**

The contribution of elements in _ Sp 12, j)_ which are conjugate in fo il L 2 =| x 0J Sp(2,Z)/(±l) to L 1 = K Q x E 2 ( resp. x E 2 ) is given by K 1 3 = 2 63"1(-i)k[(k-2)-i(k-l)] ( resp. ) K Proof: contains 14 = 2 6s 1 The centralizer E ? x SL^(7) representatives elements of (i) k [ (k-2)+i(k-l)]. E4 SL2(R) CT _ of Ln in Sp(2,Z), as a subgroup of index 2 with coset and L, . The Iwasawa decomposition of is I" cose [-sine sinC"| Ta cos^J [0 b -| a"" J 0. Z12elC] MINKING EIE 28 Hence a fundamental domain for F l : z SL2(R) is t Tzi = x E2 ] [t i I t ' 1 0, and dx 1 2 dy 1 2 = tdtdc.

It follows that (mod 2) J [SL2(Z) : H ] = 3 [SL2(Z) : r\(2)] = 6. ' w , e w ] C 91 r'R h as t h e which i s an e l e m e n t of £ + co = mir/2 , £ - OJ = nTT/2. I t is a subgroup of C of index 9l' Z MINKING E IE 36 8 with coset representatives | [ 1,-1 ] , [1,1 ] , [ l , - i ] , By (1) and K24 ( 2 ) , we 2"3- = 2TT2. [1 , 1 ] } X { E 4 ,1 J}. v get (2TT)""1I2( 9 = TT/2) 2~"7(2k-3) = THEOREM 11. The contribution of elements in are conjugate in Sp(2,Z)/{±l} to M = [ S, U] Sp(2,z) which with S = diag and 011= 1[I -, l\| ,, s a [l oj|>) and [ s, 0 ] ( resp.

E 2 x Sp(2,R) , we can reduce Theorem 3 of CHAPTER I. This MINKING EIE 54 LEMMA 3. Suppose an element of a - bi M € Sp(3,Z) A^R) of the form are eigenvales of Q\ /E P/ VO S E (3) U. Then to P Q is conjugate in , where First, we can choose P+Qi L M b f 0 with Sp(3,R) to and a + bi is conjugate in la+bi 0 0 0 a b 0 -b a and Sp(3,R) 0 s = s 0 s 0 s' 0 0 s as in the proof of the previous Lemma so that E B\ 'P Q\/E * B =( t ° B 1 2 ) , 1 \-Q P 0 E / B 12 B 2 is a 1 x 2 row vector and B 2 is a 2 x 2 L with B, 2 M L= matrix.