Dr. Katz DEq Homework Solutions 112

Dr. Katz DEq Homework Solutions 112 - 112 N.M Katz n...

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112 N.M. Katz: non-trivial rational point of order two, for reasons which will become clear. For each prime p which does not divide n, we put Ep=E x r(SpecFp)), the elliptic curve over Fp obtained by reduction modulo p. The fact that Z [i/n] is a principal ideal domain, and the projectivity of E over it, gives (7.5.1.0) E(T) ~, Eo(Q ) and permits the definition of the homomorphism of '~ reduction modulo p" for each p not dividing n (7.5.1.1) EQ(Q) ~, E(T) , Ep(Fp). This homomorphism is injective on the subgroup of EQ(Q) consisting of torsion elements of order prime to p. If p 4:2 and p is prime to n, the finite group Ep(Fp) contains a non-trivial element of order two (this being true of EQ (Q) by hypothesis), hence has an even number of elements. (7.5.2) Lemma. Hypotheses as in (7.5.1), for all primes p~ 7 prime to n, the finite group Ep(Fp) has order prime to p. Proof If not, it has even order divisible by p, hence has > 2p elements. For p > 7, this contradicts the Reimann hypothesis t ~Ep(Fp)-p-1[ <2]/p. Q.E.D.
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