Number Theory: An Introduction to Mathematics

566 XIII Connections with Number Theory This shows that each prime which dividesm, but notm^2 +Ame^2 +Be^4 , must occur to an ev ...

4 Mordell’s Theorem 567 Moreover, if we define a mapφof the groupEof all rational points ofCA,Binto the groupE′of all rational p ...

568 XIII Connections with Number Theory and similarlyy 2 (x 22 −B)/x 22 =y′. It follows that φ(x 1 ,y 1 )=φ(x 2 ,y 2 )=(x′,y′). ...

5 Further Results and Conjectures 569 Hence forψ◦φ(P)=P∗=(x∗,y∗)we have x∗=(x^2 −B)^2 / 4 y^2 , y∗=(x^2 −B)[(x^2 +Ax+B)^2 −(A^2 ...

570 XIII Connections with Number Theory A deep theorem of Mazur (1977) says that the torsion group must be one of the following: ...

5 Further Results and Conjectures 571 which is now an integer, has minimal absolute value. (It has been proved by Tate that we t ...

572 XIII Connections with Number Theory and|γp|=| ̄γp|=p−^1 /^2. Multiplying out the products, we obtain forRs> 3 / 2 an abso ...

5 Further Results and Conjectures 573 This is sometimes called the ‘weak’ conjecture of Birch and Swinnerton-Dyer, since they al ...

574 XIII Connections with Number Theory HW-conjecture but also, for sufficiently many Dirichlet charactersχ,the‘twisted’ L-funct ...

6 Some Applications 575 In the same paper in which he proved his theorem, Mordell (1922) conjectured that if a non-singular irre ...

576 XIII Connections with Number Theory for some relatively prime positive integersc 1 ,d 1 .Then (c^21 −d 12 )(c 12 +d 12 )=c 1 ...

6 Some Applications 577 Conversely, ifu,v,ware rational numbers such thatuv= 2 nandu^2 +v^2 =w^2 ,then (u+v)^2 =w^2 + 4 n,(u−v)^ ...

578 XIII Connections with Number Theory Moreoverx′,x′+nandx′−nare allnonzerorational squares. Since 2Pis of finite order, the th ...

6 Some Applications 579 The most celebrated application of the arithmetic of elliptic curves has been the recent proof of Fermat ...

580 XIII Connections with Number Theory Moreover, if we putC=−(A+B),thenC=0and(A,C)=(B,C)=1. The linear change of variables X→ ...

7 Further Remarks 581 Elkies (1988) used the arithmetic of ellipticcurves to find infinitely many counterex- amples forn=4, the ...

582 XIII Connections with Number Theory was already studied by Heine (1847). There is indeed a whole world ofq-analysis, which m ...

7 Further Remarks 583 For reference, we give here the formulas for addition on an elliptic curve in the so-called Weierstrass’s ...

584 XIII Connections with Number Theory For congruent numbers, see Volume II, Chapter XVI of Dickson [23], Tunnell [57], and Nod ...

8 Selected References 585 [21] J.E. Cremona,Algorithms for modular elliptic curves, 2nd ed., Cambridge University Press, 1997. [ ...