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Read e-book online Number Theory, Carbondale 1979: Proceedings of the Southern PDF

By M.B. Nathanson

ISBN-10: 0387095594

ISBN-13: 9780387095592

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Additional info for Number Theory, Carbondale 1979: Proceedings of the Southern Illinois Number Theory Conference Carbondale

Example text

Left translation by 7rp and by ( both leave C invariant; 7rp corresponds to (x, y, B) f---+ (x + 1, y, B), and ( to (x, y, B) f---+ (x, y, B+n). If CJ is large enough, then pr- 1 Up(CJ) is isomorphic to C modulo the group generated by the actions of 7rp and (. 5 Choice of the Ap. We choose Äp E (0, oo) suitable in the sense just mentioned, such that in addition all Up (Äp) for P E P are pairwise disjoint. To see that this is possible, note that the set P is finite, and that the Up ( CJ) are a basis for the neighborhoods of P in the Hausdorff space X.

We call v the order of the elliptic point z. The order is the same for all points in the f' -orbit of an elliptic point. 2. 10. 7 Cuspidal points of f' are elements of the boundary IR. fj which are fixed by parabolic transformations in f'. An element 'Y E f' is parabolic if ltrace(i')l = 2 and 'Y ~ Z. Parabolic elements are of the form 1r = gn(x)g- 1 k(lrr) with x =f 0, l E Z, and g E G such that g · oo is the cuspidal point that 1r leaves fixed. The subgroup of f' fixing a cuspidal point g · oo may be written in the form { gn(r)g- 1 k(lrr) : r, l E Z }.

The Lie algebra 9r of Lie algebra of G. It has the JR-basis X= (0001) ) w = ( 01) -10 ) H = G is the same as the (10-10) ) with n(x) = exp(xX), k(B) = exp(BW) and a(y) = exp(~ log(y)H). , [29], Ch. VI, §1. For f E C 00 ( G) and Z E 9r we have the right differentiation Zf(g) = . dd f(gexp(tZ))I t=O t This can be extended C-linearly to give an action of the complexified Lie algebra g = c ®JR 9r in C 00 (G) by left invariant differential Operators of first order. All left invariant differential Operators are obtained from these first order operators.

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Number Theory, Carbondale 1979: Proceedings of the Southern Illinois Number Theory Conference Carbondale by M.B. Nathanson


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