112
N.M. Katz:
nontrivial 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 nontrivial 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.
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This note was uploaded on 12/21/2011 for the course MAP 4341 taught by Professor Normankatz during the Fall '11 term at UNF.
 Fall '11
 NormanKatz

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