7 Further Remarks 583For reference, we give here the formulas for addition on an elliptic curve in the
so-called Weierstrass’s normal form. IfP 1 =(x 1 ,y 1 )andP 2 =(x 2 ,y 2 )are points of
the curve
Y^2 +a 1 XY+a 3 Y−(X^3 +a 2 X^2 +a 4 X+a 6 ),then
−P 1 =(x 1 ,−y 1 −a 1 x 1 −a 3 ), P 1 +P 2 =P 3 ∗=(x 3 ,−y 3 ),where
x 3 =λ(λ+a 1 )−a 2 −x 1 −x 2 , y 3 =(λ+a 1 )x 3 +μ+a 3 ,and
λ=(y 2 −y 1 )/(x 2 −x 1 ), μ=(y 1 x 2 −y 2 x 1 )/(x 2 −x 1 ) ifx 1 =x 2 ;
λ=( 3 x 12 + 2 a 2 x 1 +a 4 −a 1 y 1 )/N,μ=(−x 13 +a 4 x 1 + 2 a 6 −a 3 y 1 )/N,withN= 2 y 1 +a 1 x 1 +a 3 ifx 1 =x 2 ,P 2 =−P 1.
An algorithm for obtaining a minimal model of an elliptic curve is described in
Laska [40]. Other algorithms connected with elliptic curves are given in Cremona [21].
The original conjecture of Birch and Swinnerton-Dyer was generalized by Tate [54]
and Bloch [11]. For a first introduction to the theory of modular forms see Serre [49],
and for a second see Lang [39].
Hasse actually showed that, ifEis an elliptic curve over any finite fieldFqcontain-
ingqelements, then the numberNqofFq-points onE(including the point at infinity)
satisfies the inequality
|Nq−(q+ 1 )|≤ 2 q^1 /^2.For an elementary proof, see Chahal [17]. Hasse’s result is the special case, when the
genusg=1, of the Riemann hypothesis for function fields, which was mentioned in
Chapter IX,§5.
It follows from the result of Siegel (1929), mentioned in§9 of Chapter IV, and even
from the earlier work of Thue (1909), that an elliptic curve with integral coefficients
has at most finitely manyintegralpoints. However, their method is not constructive.
Baker [6], using the results on linear forms in the logarithms of algebraic numbers
which he developed for the theory of transcendental numbers, obtained an explicit up-
per bound for the magnitude of any integral point in terms of an upper bound for the
absolute values of all coefficients. Sharper bounds have since been obtained, e.g. by
Bugeaud [15]. (For modern proofs of Baker’s theorem on the linear independence of
logarithms of algebraic numbers, see Waldschmidt [59]. The history of Baker’s method
is described in Baker [7].)
For information about the proof of Mordell’s conjecture we refer to Bloch [12],
Szpiro [53], and Cornell and Silverman [19]. The last includes an English translation
of Faltings’ original article. As mentioned in§9 of Chapter IV, Vojta (1991) has given
a proof of the Mordell conjecture which is completely different from that of Faltings.
There is an exposition of this proof, with simplifications due to Bombieri (1990), in
Hindry and Silverman [32].