Algebraic Solutions of Differential Equations
115
of E. Since ~
extends to ~, take a finite covering of E by affine open
sets V~ which trivialize Lg, and suppose ~
is given by transition func
tions f u. Then V' is given by 1forms ~oi holomorphic on V/c~E~ = Vi[1/m].
By the quasicoherence of
f2~/r,
there is an power M of m such that M co~
is holomorphic on all of V~. As there are only finitely many V~, a common
M works for all. Now (~,
17,)
is given by transition
functions
(M~,
M
f/j ), which
do
extend.
Q.E.D.
Combining (7.5.9.3) with the reduction steps (7.5.4)(7.5.6), we find
(7.5.10)
Tautology. Let E be an elliptic curve over
T=Z[1/n],
with a
(nontrivial) rational point of order two. Then the truth of the analogue
of (7.4.4) for the function field
K/Q
of EQ is equivalent to the following
conjecture:
(7.5.11)
Conjecture. The
kerne; of the homomorphism
(7.5.9.1)
FI rcE ,
EQ(Q)=E(T) ~
p>
7,pg,
r(E, n~/r)
is contained in the torsion subgroup of
EQ(Q).
Appendix
Riemann's Existence Theorem
An extremely useful form of the theorem is an easy consequence of
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 Fall '11
 NormanKatz
 Differential Equations, Manifold, locally free sheaf

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