## Number Theory and its Applications by C. Yildrim

By C. Yildrim

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61 In the application, we let q=ll L = [2,3, g>= 71, . 2 gives us the structure of G L for the Shimura curve. We then have to see how it mixes with G5, which amounts to determining the Goursat subgroups. 3. e. giving the identification of G 5 and GL on the common subfield K s I1 KL, which is a finite extension of Q. We shall determine explicitly what this subfield: is, and what the above isomorphism is. - Z(4) 9 1, where U s = 1 + 5M5. The map H 5 -, Z(4) is surjective because Q(~5) is disjoint from K L (the field of fifth roots of unity 0(#5) is ramified only at 5, and K L is unramified at 5).

We c a n therefore d i a g o n a l i z e o13 over our c o o r d i n a t e s s o c h o s e n that 6 + 25a GI3 = 0 Z 5, a n d w e a s s u m e ~ - 2 + 25b We then o b t a i n In particular, 043 is not s c a l a r rood 25. Again from t a b l e s , for the prime p = 653 we have t653 = - 4 1 , and g6s 3 has c h a r a c t e r i s t i c polynomial X 2 - 41X + 653 -= (X +11) (X - 2) (rood 25) . Since 1 ! ~ 2 mod 5, it follows that G653 is d i a g o n a l i z a b l e over Z5, and s i n c e 114 --- ( - 2 ) 4 - 16 ( m o d 2 5 ) 57 we obtain a~53 -= 1 6 I - I + 5 .

Lim Up(S) = 1 FM(t0) p where S is the congruence class o[ t o rood M. e. that our probability assumption is compatible with the Tchebotarev density property for the sequence of Frobenius elements to be viewed as a random sequence in our model. Given a subinterval I of [ - 1 , 1], let SI, p be the set of integers yp such that ~ ( y p , p ) 9 I. By definition, gp(SI,p) = Lemma 2. E fM(t' p) " ~=(t,p) EI limpgp(Si,p)= f g(~:)d~:. I Proof. Entirely analogous to the proof of Lemma 1. We start with approximat- ing Riemann sums for the integral f g(~) d~, I and use the analogue of (1) and (2) which have this number as a factor instead of the number 1.