Divisors - Elliptic Curves: Number Theory and Cryptography - page 366

PROOF Letting D =0and D = K in the Riemann-Roch theorem, then

using (3) in Proposition 11.14, yields

( K )= g, and ( K )=deg( K ) − g +2 .

Therefore,

deg( K )=2 g − 2 ,

as desired.

COROLLARY 11.17
If deg( D ) > 2 g − 2 ,then ( D )=deg( D ) − g +1 .

PROOF Since deg( K−D ) < 0, Proposition 11.14 (1) says that ( K−D )=
0. The Riemann-Roch theorem therefore yields the result.

COROLLARY 11.18
Let P, Q be pointson C .If g

≥ 1 and [ P ] − [ Q ] ∼ 0 ,then P = Q .

PROOF =[ Q ].

Since f n has a pole of order n at Q , and since functions with different orders
of poles at Q are linearly independent, the set

By assumption, [ P ] − [ Q ]=div( f )forsome f . Assume [ P ]

{ 1 ,f,f 2 ,...,f 2 g− 1

}

spans a subspace of L ((2 g − 1)[ Q ]) of dimension 2 g . Therefore,

2 g ≤ ((2 g − 1)[ Q ]) = (2 g − 1) − g +1= g,

by Corollary 11.17. Since g ≥ 1, this is a contradiction. Therefore, P = Q .

Our goal is to show that a curve C of genus one is isomorphic over K to
an elliptic curve given by a generalized Weierstrass equation. The following
will be used to construct the functions needed to map from C to the elliptic
curve.

COROLLARY 11.19
If C has genus g =1 and deg( D ) > 0 ,then

( D )=deg( D ) .

PROOF This is simply a restatement of Corollary 11.17 in the case g =1.