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Dr. Katz DEq Homework Solutions 115

# Dr. Katz DEq Homework Solutions 115 - A lgebraic Solutions...

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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 1-forms ~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 (non-trivial) 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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