International Mathematical Conference 1982: Proceedings by Louis H. Chen, Louis H.T. Chen, etc.

By Louis H. Chen, Louis H.T. Chen, etc.

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X Cc 1 (g) i . I = IC,(g) x(g) - r S ( g ) 5 0 (mod v ( IC,(g) Next suppose t h a t by any element o f :C (g) 'i v ( ICG(g)I ) = 0 Thus w e have t h a t i f defect Pi E S y l p ( H ) NH(Pi) = Irr(Ci) 0 V(5(9)) 2 v(ICc :1 1. (mod then 1) > 0. Then since (9) i I 3). 1) Then I), v ( ICc (9) 1 ) Then I) 1 x, 5 V(lCC i v(rl(g)) 2 > 0. Let r) i and a r e a l l of rl (g) I), V(ICC i (9) I) ' 0. 2) - r 5 ( h ) )m +l(x(g) - r c ( g ) ) W r (mod From ( 4 . & 1. and ( 4 . 2 ) w e have t h a t ( x l nG 1 and hence o 5 (xc (rc,rl) = r 6 ,r)) (mod 50 i i s a unique element of I r r ( C i I o = 1 1, such t h a t +5 0 ~ (mod ) PI.

1941): Some idiosyncratic numbers of Ramanujan. Proc. Sci. A 12, 519-520. Gupta, H. (1977): Ramanujan's ternary quadratic form x 2+y 2+1Oz 2 . Punjab Univ. 124 (1973), No 1-2,57 (1977). Hasse, H. (1923): fiber die Darstellbarkeit von Zahlen durch quadratische Formen im Kdrper der rationalen Zahlen. reine angew. Abh. bd. 1, 3-22). Rational quadratic forms 23 Hasse, H. l, 23-42). S. (1976): Representation by spinor genera. Math. 63, 147-152. S. (1979): Arithmetic theory of integral quadratic forms.

His investigations were continued by his pupil Malygev (cf. Malygev (1962)). They consider only forms f of a special kind: in particular its genus contains only a single spinor genus. Their arguments require an auxiliary prime g , which may be chosen arbitrarily subject to mild restrictions, and they consider only the b which satisfy (-db/q) = 1, Here d is the discriminant of Legendre symbol. 4). 4). e. the result is ineffective). In fact they use Siegel's estimate for the class number of binary quadratic forms, which is well known to be ineffective.

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