Download PDF by Minking Eie: Dimension Formulae for the Vector Spaces of Siegel Cusp

By Minking Eie

ISBN-10: 0821824368

ISBN-13: 9780821824368

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Additional resources for Dimension Formulae for the Vector Spaces of Siegel Cusp Forms of Degree Three

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M' = 0 Here we only prove (5). Others follow with Proof. , U, ] with MINKING EIE 58 Vl^V 1 a b 0 -1 1 0 -1 0 a = ±(u-2p-q), Thus we can choose suitable integers or 1 . If a = 0 and element of the form p, q b = ±(v+p-q) so that a = 0, b = 0 b = 0 , then with a conjugation by [T, E] e r~ , we are done. b = 1 , with a conjugation with may assume that the entries of [T, E] e r~ S^ at If a = 0 and if necessary, we 1, 12, 13 positions are 0 . Then we have the decomposition [S2, VUjV"1] = [S2, E][0, Vj^lIO, U] with A 1 1 -1 0 1 0 0 0 1 1 0 and L.

Suppose M L = [M; p, q; s] be an element of has an isolated fixed point, and let characteristic polynomial; then M x. n si 0 1 , s G Z Proof. if L r« (x) be its is conjugate in Sp(3, Z) tp_

Now it suffices to find conjugacy classes of elements with property (2) or (4) or (5). Let 2 1 r«, r~ and 0 r« be subgroups of r^ defined as follows: r2 = \ M e Sp(3, Z ) | M = A 0 B *" * 1 * * 'A B] C 0 D Vc C Dj 0 0 0 1. e Sp(2, Z) SIEGEL CUSP FORMS OF DEGREE THREE i-< M e Sp(3, Z)| M = a 0 b * * u * * c 0 d * 0 0 0 35 "a b' c d * V1 e SL 2 (Z) r° = {M e sp(3, z)| M = [ s, u] , s = t s e M3(Z), U, iT1 € GL3(Z)} . 2 Note that Sp(3, Z) r3, 1 r3 r~ and are the stabilizers in of the three-dimensional cusp, one-dimensional cusp and zero-dimensional cusp (or boundary component as in [24]) which we shall denote by Z Z F F z 2 = l 7 l Z 12 z = < 12 * z Im 2 z l Z 12 12 z 2 > oi Im z, > 0 L oo loo and F Q = {[i»]} respectively.

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Dimension Formulae for the Vector Spaces of Siegel Cusp Forms of Degree Three by Minking Eie


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