Number Theory: An Introduction to Mathematics

546 XIII Connections with Number Theory In the same way that we proved Proposition 3 we may show that, if|x|<1, then 1 /( 1 − ...

2 Partitions 547 We h av e F(x)−G(x) = ∑ n≥ 0 (− 1 )nx^2 nq^5 n(n+^1 )/^2 {q−^2 n−x^2 q^2 (n+^1 )−q−n+xqn+^1 }/(q)n(xqn+^1 )∞ = ...

548 XIII Connections with Number Theory The Rogers–Ramanujan identities may now be easily derived: Proposition 6If|q|< 1 ,the ...

3 Cubic Curves 549 A remarkable application of the Rogers–Ramanujan identities to the hard hexagon model of statistical mechanic ...

550 XIII Connections with Number Theory where all unwritten terms have degree>1. Sincea,bare uniquely determined byf, we can ...

3 Cubic Curves 551 to be the projective line defined byaX+bY+cZ. It follows from Euler’s theorem on homogeneous functions that ( ...

552 XIII Connections with Number Theory with coefficientsaij∈Ksuch that F(a 11 X′+···,a 21 X′+···,a 31 X′+···)=G(X′,Y′,Z′). It i ...

3 Cubic Curves 553 If K has characteristic= 2 , 3 , then it is projectively equivalent to the projective completionC=Ca,bof an ...

554 XIII Connections with Number Theory Singular cases Node:d=0,a (^0) Cusp:d = a = 0 x y y P x P Non-singular cases y y x x d0 ...

3 Cubic Curves 555 X=T^2 ,Y=T^3. In the node case, if we putT=Y/(X+ 3 b/ 2 a),thenithasthe parametrization X=T^2 + 3 b/a,Y=T^3 + ...

556 XIII Connections with Number Theory where gλ(X)= 4 λX^3 − 4 ( 1 +λ)X^2 + 4 X is Riemann’s normal form andλ = 0 ,1. IfS(u)is ...

3 Cubic Curves 557 y x P P P*=P +P 0 1 2 3 3 12 P Fig. 2.Addition onCa,b. LetCbe a projective cubic curve over the fieldK, defin ...

558 XIII Connections with Number Theory Birational equivalence may be defined in the following way. Arational transfor- mationof ...

4 Mordell’s Theorem 559 we write simplyE:=E(Q). This section is devoted to the basic theorem of Mordell (1922), which says thatt ...

560 XIII Connections with Number Theory The Euclidean algorithm may be used to derive the polynomial identity ( 3 X^2 + 4 a)(X^4 ...

4 Mordell’s Theorem 561 Proof Supposehˆhas the properties (i),(ii). Then, by (ii), 4nhˆ(P) =hˆ( 2 nP)and hence, by (i), 4nhˆ(P)− ...

562 XIII Connections with Number Theory To prove (∗) we may evidently assume thatP 1 =(x 1 ,y 1 )andP 2 =(x 2 ,y 2 )are both fin ...

4 Mordell’s Theorem 563 By Proposition II.16,q 3 andq 4 each divideA, and so their product dividesA^2. Hence, for some integerD ...

564 XIII Connections with Number Theory then evidently (P,Q)=(Q,P),(P,P)= 2 hˆ(P)≥ 0. It remains to show that (P,Q+R)=(P,Q)+(P,R ...

4 Mordell’s Theorem 565 and hence hˆ(P 1 )≤2[hˆ(P)+C]/m^2 ≤[hˆ(P)+C]/ 2. ButP 1 ∈/E′,sinceP∈/E′, and hencehˆ(P 1 )≥hˆ(P). It fol ...