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Diophantine equations Cubic equations FLT Pells equation Elliptic curves Diophantine equations, modular forms, and cycles Henri Darmon McGill University Ottawa, September 2010 Diophantine equations Cubic equations FLT Pells

  1. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Second example: the equation y 2 + 2 = x 3 Theorem (Euler) The equation y 2 + 2 = x 3 has only two solutions, namely ( x , y ) = (3 , ± 5) . Proof. Factor the left hand side in the larger ring Z [ √− 2]: √ √ − 2) = x 3 . ( y + − 2)( y − Observe that y is odd, so gcd( y + √− 2 , y − √− 2) = 1. Unique factorisation in Z [ √− 2] = ⇒ √ √ √ − 2) 3 = a ( a 2 − 6 b 2 ) + b (3 a 2 − 2 b 2 ) y + − 2 = ( a + b − 2 . Elementary manipulations = ⇒ b = 1 , a = ± 1, so y = ± 5.

  2. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Second example: the equation y 2 + 2 = x 3 Theorem (Euler) The equation y 2 + 2 = x 3 has only two solutions, namely ( x , y ) = (3 , ± 5) . Proof. Factor the left hand side in the larger ring Z [ √− 2]: √ √ − 2) = x 3 . ( y + − 2)( y − Observe that y is odd, so gcd( y + √− 2 , y − √− 2) = 1. Unique factorisation in Z [ √− 2] = ⇒ √ √ √ − 2) 3 = a ( a 2 − 6 b 2 ) + b (3 a 2 − 2 b 2 ) y + − 2 = ( a + b − 2 . Elementary manipulations = ⇒ b = 1 , a = ± 1, so y = ± 5.

  3. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Second example: the equation y 2 + 2 = x 3 Theorem (Euler) The equation y 2 + 2 = x 3 has only two solutions, namely ( x , y ) = (3 , ± 5) . Proof. Factor the left hand side in the larger ring Z [ √− 2]: √ √ − 2) = x 3 . ( y + − 2)( y − Observe that y is odd, so gcd( y + √− 2 , y − √− 2) = 1. Unique factorisation in Z [ √− 2] = ⇒ √ √ √ − 2) 3 = a ( a 2 − 6 b 2 ) + b (3 a 2 − 2 b 2 ) y + − 2 = ( a + b − 2 . Elementary manipulations = ⇒ b = 1 , a = ± 1, so y = ± 5.

  4. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves The gap in Euler’s proof Euler’s proof is interesting because it invokes a non-trivial structural property – unique factorisation – of the the ring Z [ √− 2].

  5. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Third example: the equation y 2 + 118 = x 3 Theorem The equation y 2 + 118 = x 3 has no integer solutions. Proof. Factor the left hand side in the larger ring Z [ √− 118]: √ √ − 118) = x 3 . ( y + − 118)( y − Proceed exactly as before, using unique factorisation in Z [ √− 118]. But... 15 2 + 118 = 7 3 , so the theorem is wrong!

  6. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Third example: the equation y 2 + 118 = x 3 Theorem The equation y 2 + 118 = x 3 has no integer solutions. Proof. Factor the left hand side in the larger ring Z [ √− 118]: √ √ − 118) = x 3 . ( y + − 118)( y − Proceed exactly as before, using unique factorisation in Z [ √− 118]. But... 15 2 + 118 = 7 3 , so the theorem is wrong!

  7. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Third example: the equation y 2 + 118 = x 3 Theorem The equation y 2 + 118 = x 3 has no integer solutions. Proof. Factor the left hand side in the larger ring Z [ √− 118]: √ √ − 118) = x 3 . ( y + − 118)( y − Proceed exactly as before, using unique factorisation in Z [ √− 118]. But... 15 2 + 118 = 7 3 , so the theorem is wrong!

  8. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Third example: the equation y 2 + 118 = x 3 Theorem The equation y 2 + 118 = x 3 has no integer solutions. Proof. Factor the left hand side in the larger ring Z [ √− 118]: √ √ − 118) = x 3 . ( y + − 118)( y − Proceed exactly as before, using unique factorisation in Z [ √− 118]. But... 15 2 + 118 = 7 3 , so the theorem is wrong!

  9. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Third example: the equation y 2 + 118 = x 3 Theorem The equation y 2 + 118 = x 3 has no integer solutions. Proof. Factor the left hand side in the larger ring Z [ √− 118]: √ √ − 118) = x 3 . ( y + − 118)( y − Proceed exactly as before, using unique factorisation in Z [ √− 118]. But... 15 2 + 118 = 7 3 , so the theorem is wrong!

  10. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Unique factorisation Conclusion : Unique factorisation fails in Z [ √− 118]. The possible failure of unique factorisation which often arises as an obstruction to analysing diophantine equations, is a highly interesting phenomenon. It can be measured in terms of a class group of an appropriate ring. Number theorists have devoted a lot of efforts to better understanding and controlling class groups, spurring the development of algebraic number theory and commutative algebra.

  11. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Unique factorisation Conclusion : Unique factorisation fails in Z [ √− 118]. The possible failure of unique factorisation which often arises as an obstruction to analysing diophantine equations, is a highly interesting phenomenon. It can be measured in terms of a class group of an appropriate ring. Number theorists have devoted a lot of efforts to better understanding and controlling class groups, spurring the development of algebraic number theory and commutative algebra.

  12. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Unique factorisation Conclusion : Unique factorisation fails in Z [ √− 118]. The possible failure of unique factorisation which often arises as an obstruction to analysing diophantine equations, is a highly interesting phenomenon. It can be measured in terms of a class group of an appropriate ring. Number theorists have devoted a lot of efforts to better understanding and controlling class groups, spurring the development of algebraic number theory and commutative algebra.

  13. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Unique factorisation Conclusion : Unique factorisation fails in Z [ √− 118]. The possible failure of unique factorisation which often arises as an obstruction to analysing diophantine equations, is a highly interesting phenomenon. It can be measured in terms of a class group of an appropriate ring. Number theorists have devoted a lot of efforts to better understanding and controlling class groups, spurring the development of algebraic number theory and commutative algebra.

  14. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Diophantine equations 1 Cubic equations 2 FLT 3 Pell’s equation 4 Elliptic curves 5

  15. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Fermat’s Last Theorem Theorem (Fermat, 1635?) If n ≥ 3 , then the equation x n + y n = z n has no integer solution with xyz � = 0 . Natural opening gambit: ( x + y )( x + ζ n y ) · · · ( x + ζ n − 1 y ) = z n , n where ζ n = e 2 π i / n is an n th root of unity. Theorem (Lam´ e) Suppose p > 2 is prime. If Z [ ζ p ] has unique factorisation, then x p + y p = z p has no non-trivial solution.

  16. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Fermat’s Last Theorem Theorem (Fermat, 1635?) If n ≥ 3 , then the equation x n + y n = z n has no integer solution with xyz � = 0 . Natural opening gambit: ( x + y )( x + ζ n y ) · · · ( x + ζ n − 1 y ) = z n , n where ζ n = e 2 π i / n is an n th root of unity. Theorem (Lam´ e) Suppose p > 2 is prime. If Z [ ζ p ] has unique factorisation, then x p + y p = z p has no non-trivial solution.

  17. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Fermat’s Last Theorem Theorem (Fermat, 1635?) If n ≥ 3 , then the equation x n + y n = z n has no integer solution with xyz � = 0 . Natural opening gambit: ( x + y )( x + ζ n y ) · · · ( x + ζ n − 1 y ) = z n , n where ζ n = e 2 π i / n is an n th root of unity. Theorem (Lam´ e) Suppose p > 2 is prime. If Z [ ζ p ] has unique factorisation, then x p + y p = z p has no non-trivial solution.

  18. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Kummer’s theorem Theorem (Kummer) Suppose p > 2 is prime. If p does not divide the class number of Z [ ζ p ] , then x p + y p = z p has no non-trivial solution. In particular, Fermat’s Last theorem is true for p < 100 . Kummer’s theorem leads to fascinating questions about cyclotomic rings (rings of the form Z [ ζ n ]). Many of these are still open! As we all know, Fermat’s Last Theorem was eventually proved in 1995, by Andrew Wiles, relying on a very different circle of ideas.

  19. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Kummer’s theorem Theorem (Kummer) Suppose p > 2 is prime. If p does not divide the class number of Z [ ζ p ] , then x p + y p = z p has no non-trivial solution. In particular, Fermat’s Last theorem is true for p < 100 . Kummer’s theorem leads to fascinating questions about cyclotomic rings (rings of the form Z [ ζ n ]). Many of these are still open! As we all know, Fermat’s Last Theorem was eventually proved in 1995, by Andrew Wiles, relying on a very different circle of ideas.

  20. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Kummer’s theorem Theorem (Kummer) Suppose p > 2 is prime. If p does not divide the class number of Z [ ζ p ] , then x p + y p = z p has no non-trivial solution. In particular, Fermat’s Last theorem is true for p < 100 . Kummer’s theorem leads to fascinating questions about cyclotomic rings (rings of the form Z [ ζ n ]). Many of these are still open! As we all know, Fermat’s Last Theorem was eventually proved in 1995, by Andrew Wiles, relying on a very different circle of ideas.

  21. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Diophantine equations 1 Cubic equations 2 FLT 3 Pell’s equation 4 Elliptic curves 5

  22. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Pell’s equation The Fermat-Pell equation is the equation x 2 − dy 2 = 1 , where d > 0 is a non-square integer. The group law . ( x 1 , y 1 ) ∗ ( x 2 , y 2 ) = ( x 1 x 2 + dy 1 y 2 , x 1 y 2 + y 1 x 2 ) . Theorem (Fermat) For any non-square d > 0 , the Pell equation x 2 − dy 2 has a non-trivial fundamental solution ( x 0 , y 0 ) such that all other solutions are of the form ( ± x , ± y ) = ( x 0 , y 0 ) ∗ n .

  23. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Pell’s equation The Fermat-Pell equation is the equation x 2 − dy 2 = 1 , where d > 0 is a non-square integer. The group law . ( x 1 , y 1 ) ∗ ( x 2 , y 2 ) = ( x 1 x 2 + dy 1 y 2 , x 1 y 2 + y 1 x 2 ) . Theorem (Fermat) For any non-square d > 0 , the Pell equation x 2 − dy 2 has a non-trivial fundamental solution ( x 0 , y 0 ) such that all other solutions are of the form ( ± x , ± y ) = ( x 0 , y 0 ) ∗ n .

  24. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Pell’s equation The Fermat-Pell equation is the equation x 2 − dy 2 = 1 , where d > 0 is a non-square integer. The group law . ( x 1 , y 1 ) ∗ ( x 2 , y 2 ) = ( x 1 x 2 + dy 1 y 2 , x 1 y 2 + y 1 x 2 ) . Theorem (Fermat) For any non-square d > 0 , the Pell equation x 2 − dy 2 has a non-trivial fundamental solution ( x 0 , y 0 ) such that all other solutions are of the form ( ± x , ± y ) = ( x 0 , y 0 ) ∗ n .

  25. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Some examples of fundamental solutions If d = 2, then ( x 0 , y 0 ) = (3 , 2). If d = 61, then ( x 0 , y 0 ) = (1766319049 , 226153980) . If d = 313, then ( x 0 , y 0 ) = (32188120829134849 , 1819380158564160). The standard (and still the best) method to find the fundamental solution is the method based on continued fractions . It was discovered by the Indian mathematicians of the 12th century, and rediscovered by Fermat in the 17th century.

  26. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Some examples of fundamental solutions If d = 2, then ( x 0 , y 0 ) = (3 , 2). If d = 61, then ( x 0 , y 0 ) = (1766319049 , 226153980) . If d = 313, then ( x 0 , y 0 ) = (32188120829134849 , 1819380158564160). The standard (and still the best) method to find the fundamental solution is the method based on continued fractions . It was discovered by the Indian mathematicians of the 12th century, and rediscovered by Fermat in the 17th century.

  27. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Some examples of fundamental solutions If d = 2, then ( x 0 , y 0 ) = (3 , 2). If d = 61, then ( x 0 , y 0 ) = (1766319049 , 226153980) . If d = 313, then ( x 0 , y 0 ) = (32188120829134849 , 1819380158564160). The standard (and still the best) method to find the fundamental solution is the method based on continued fractions . It was discovered by the Indian mathematicians of the 12th century, and rediscovered by Fermat in the 17th century.

  28. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Some examples of fundamental solutions If d = 2, then ( x 0 , y 0 ) = (3 , 2). If d = 61, then ( x 0 , y 0 ) = (1766319049 , 226153980) . If d = 313, then ( x 0 , y 0 ) = (32188120829134849 , 1819380158564160). The standard (and still the best) method to find the fundamental solution is the method based on continued fractions . It was discovered by the Indian mathematicians of the 12th century, and rediscovered by Fermat in the 17th century.

  29. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Explanation of the group law Key remark : If ( x , y ) is a solution to Pell’s equation, then √ √ x + y d is a unit (invertible element) of the ring Z [ d ]. One can rewrite ( x 1 , y 1 ) ∗ ( x 2 , y 2 ) = ( x 3 , y 3 ) as √ √ √ ( x 1 + y 1 d )( x 2 + y 2 d ) = ( x 3 + y 3 d ) . Solving Pell’s equation can now be recast as: √ Problem : Calculate the group of units in the ring Z [ d ].

  30. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Explanation of the group law Key remark : If ( x , y ) is a solution to Pell’s equation, then √ √ x + y d is a unit (invertible element) of the ring Z [ d ]. One can rewrite ( x 1 , y 1 ) ∗ ( x 2 , y 2 ) = ( x 3 , y 3 ) as √ √ √ ( x 1 + y 1 d )( x 2 + y 2 d ) = ( x 3 + y 3 d ) . Solving Pell’s equation can now be recast as: √ Problem : Calculate the group of units in the ring Z [ d ].

  31. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Explanation of the group law Key remark : If ( x , y ) is a solution to Pell’s equation, then √ √ x + y d is a unit (invertible element) of the ring Z [ d ]. One can rewrite ( x 1 , y 1 ) ∗ ( x 2 , y 2 ) = ( x 3 , y 3 ) as √ √ √ ( x 1 + y 1 d )( x 2 + y 2 d ) = ( x 3 + y 3 d ) . Solving Pell’s equation can now be recast as: √ Problem : Calculate the group of units in the ring Z [ d ].

  32. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves A cyclotomic approach to Pell’s equation Theorem (Gauss) √ Suppose (for simplicity) that d ≡ 1 (mod 4) . Then the ring Z [ d ] is contained in the cyclotomic ring Z [ ζ d ] , where ζ d = e 2 π i / d . Proof. Gauss sums: � j d − 1 � � ζ j g = d . d j =0 Direct calculation: g 2 = d .

  33. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves A cyclotomic approach to Pell’s equation Theorem (Gauss) √ Suppose (for simplicity) that d ≡ 1 (mod 4) . Then the ring Z [ d ] is contained in the cyclotomic ring Z [ ζ d ] , where ζ d = e 2 π i / d . Proof. Gauss sums: � j d − 1 � � ζ j g = d . d j =0 Direct calculation: g 2 = d .

  34. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves A cyclotomic approach to Pell’s equation Theorem (Gauss) √ Suppose (for simplicity) that d ≡ 1 (mod 4) . Then the ring Z [ d ] is contained in the cyclotomic ring Z [ ζ d ] , where ζ d = e 2 π i / d . Proof. Gauss sums: � j d − 1 � � ζ j g = d . d j =0 Direct calculation: g 2 = d .

  35. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves The cyclotomic approach to Pell’s equation, cont’d The usefulness of Gauss’s theorem for Pell’s equation arises from the fact that Z [ ζ d ] contains some obvious units: the circular units . u = ζ d + 1 = ζ 2 d − 1 ζ d − 1 . Now let √ d := norm Z [ ζ d ] x + y d ] ( u ) . √ Z [ Then ( x , y ) is a (not necessarily fundamental!) solution to Pell’s equation.

  36. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves The cyclotomic approach to Pell’s equation, cont’d The usefulness of Gauss’s theorem for Pell’s equation arises from the fact that Z [ ζ d ] contains some obvious units: the circular units . u = ζ d + 1 = ζ 2 d − 1 ζ d − 1 . Now let √ d := norm Z [ ζ d ] x + y d ] ( u ) . √ Z [ Then ( x , y ) is a (not necessarily fundamental!) solution to Pell’s equation.

  37. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves The cyclotomic approach to Pell’s equation, cont’d The usefulness of Gauss’s theorem for Pell’s equation arises from the fact that Z [ ζ d ] contains some obvious units: the circular units . u = ζ d + 1 = ζ 2 d − 1 ζ d − 1 . Now let √ d := norm Z [ ζ d ] x + y d ] ( u ) . √ Z [ Then ( x , y ) is a (not necessarily fundamental!) solution to Pell’s equation.

  38. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Diophantine equations 1 Cubic equations 2 FLT 3 Pell’s equation 4 Elliptic curves 5

  39. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Elliptic Curves An elliptic curve is an equation in two variables x , y of the form y 2 = x 3 + ax + b , with a , b ∈ Q . We are interested in the rational rather than integer solutions to such an equation. Elliptic curve equations exhibit many of the features of Pell’s equation: 1 The set of (rational) solutions to an elliptic curve equation is equipped with a natural group law; 2 The cyclotomic approach to solving Pell’s equation has an interesting (and quite deep) counterpart for elliptic curves.

  40. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Elliptic Curves An elliptic curve is an equation in two variables x , y of the form y 2 = x 3 + ax + b , with a , b ∈ Q . We are interested in the rational rather than integer solutions to such an equation. Elliptic curve equations exhibit many of the features of Pell’s equation: 1 The set of (rational) solutions to an elliptic curve equation is equipped with a natural group law; 2 The cyclotomic approach to solving Pell’s equation has an interesting (and quite deep) counterpart for elliptic curves.

  41. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Elliptic Curves An elliptic curve is an equation in two variables x , y of the form y 2 = x 3 + ax + b , with a , b ∈ Q . We are interested in the rational rather than integer solutions to such an equation. Elliptic curve equations exhibit many of the features of Pell’s equation: 1 The set of (rational) solutions to an elliptic curve equation is equipped with a natural group law; 2 The cyclotomic approach to solving Pell’s equation has an interesting (and quite deep) counterpart for elliptic curves.

  42. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Elliptic Curves An elliptic curve is an equation in two variables x , y of the form y 2 = x 3 + ax + b , with a , b ∈ Q . We are interested in the rational rather than integer solutions to such an equation. Elliptic curve equations exhibit many of the features of Pell’s equation: 1 The set of (rational) solutions to an elliptic curve equation is equipped with a natural group law; 2 The cyclotomic approach to solving Pell’s equation has an interesting (and quite deep) counterpart for elliptic curves.

  43. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves The group law for elliptic curves y 2 3 y = x + a x + b R Q x P P+Q

  44. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Ring theoretic formulation of the problem To the elliptic curve E : y 2 = x 3 + ax + b , we attach the ring Q E := Q [ x , y ] / ( y 2 − ( x 3 + ax + b )) . Elementary (but important) remark : Rational solutions of E are in natural bijection with homomorphisms from Q E to Q : given a solution ( x , y ) = ( r , s ) , let ϕ : Q E − → Q be given by ϕ ( x ) = r , ϕ ( y ) = s . Problem : Construct homomorphisms from Q E to Q (or at least to ¯ Q ) in a non-trivial way.

  45. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Ring theoretic formulation of the problem To the elliptic curve E : y 2 = x 3 + ax + b , we attach the ring Q E := Q [ x , y ] / ( y 2 − ( x 3 + ax + b )) . Elementary (but important) remark : Rational solutions of E are in natural bijection with homomorphisms from Q E to Q : given a solution ( x , y ) = ( r , s ) , let ϕ : Q E − → Q be given by ϕ ( x ) = r , ϕ ( y ) = s . Problem : Construct homomorphisms from Q E to Q (or at least to ¯ Q ) in a non-trivial way.

  46. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Ring theoretic formulation of the problem To the elliptic curve E : y 2 = x 3 + ax + b , we attach the ring Q E := Q [ x , y ] / ( y 2 − ( x 3 + ax + b )) . Elementary (but important) remark : Rational solutions of E are in natural bijection with homomorphisms from Q E to Q : given a solution ( x , y ) = ( r , s ) , let ϕ : Q E − → Q be given by ϕ ( x ) = r , ϕ ( y ) = s . Problem : Construct homomorphisms from Q E to Q (or at least to ¯ Q ) in a non-trivial way.

  47. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Ring theoretic formulation of the problem To the elliptic curve E : y 2 = x 3 + ax + b , we attach the ring Q E := Q [ x , y ] / ( y 2 − ( x 3 + ax + b )) . Elementary (but important) remark : Rational solutions of E are in natural bijection with homomorphisms from Q E to Q : given a solution ( x , y ) = ( r , s ) , let ϕ : Q E − → Q be given by ϕ ( x ) = r , ϕ ( y ) = s . Problem : Construct homomorphisms from Q E to Q (or at least to ¯ Q ) in a non-trivial way.

  48. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Modular functions Let H be the Poincar´ e upper half plane. Theorem There is a unique holomorphic function j : H − → C satisfying   � az + b �  a b j = j ( z ) , for all  ∈ SL 2 ( Z ) ,   cz + d c d j ( z ) = q − 1 + O ( q ) , where q = e 2 π iz . The j -function is the prototypical example of a modular function . It has been said that number theory is largely the study of such objects.

  49. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Modular functions Let H be the Poincar´ e upper half plane. Theorem There is a unique holomorphic function j : H − → C satisfying   � az + b �  a b j = j ( z ) , for all  ∈ SL 2 ( Z ) ,   cz + d c d j ( z ) = q − 1 + O ( q ) , where q = e 2 π iz . The j -function is the prototypical example of a modular function . It has been said that number theory is largely the study of such objects.

  50. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Modular functions Let H be the Poincar´ e upper half plane. Theorem There is a unique holomorphic function j : H − → C satisfying   � az + b �  a b j = j ( z ) , for all  ∈ SL 2 ( Z ) ,   cz + d c d j ( z ) = q − 1 + O ( q ) , where q = e 2 π iz . The j -function is the prototypical example of a modular function . It has been said that number theory is largely the study of such objects.

  51. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Modular functions Let H be the Poincar´ e upper half plane. Theorem There is a unique holomorphic function j : H − → C satisfying   � az + b �  a b j = j ( z ) , for all  ∈ SL 2 ( Z ) ,   cz + d c d j ( z ) = q − 1 + O ( q ) , where q = e 2 π iz . The j -function is the prototypical example of a modular function . It has been said that number theory is largely the study of such objects.

  52. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Why number theorists like the j -function 1 Moonshine : Its q expansion, or Fourier expansion, has integer coefficients: j ( q ) = q − 1 + 196884 q + 21493760 q 2 + · · · The coefficients in this expansion encode information about finite-dimensional representations of certain sporadic simple groups. (John McKay’s “monstrous moonshine”). 2 Modular polynomials : Let N be an integer. The functions j ( z ) and j ( Nz ) satisfy a polynomial equation Φ N ( x , y ) in two variables with integer coefficients. The polynomial Φ N ( x , y ) is called the N -th modular polynomial . 3 Complex multiplication : If z ∈ H satisfies a quadratic equation with rational coefficients, then j ( z ) is an algebraic number.

  53. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Why number theorists like the j -function 1 Moonshine : Its q expansion, or Fourier expansion, has integer coefficients: j ( q ) = q − 1 + 196884 q + 21493760 q 2 + · · · The coefficients in this expansion encode information about finite-dimensional representations of certain sporadic simple groups. (John McKay’s “monstrous moonshine”). 2 Modular polynomials : Let N be an integer. The functions j ( z ) and j ( Nz ) satisfy a polynomial equation Φ N ( x , y ) in two variables with integer coefficients. The polynomial Φ N ( x , y ) is called the N -th modular polynomial . 3 Complex multiplication : If z ∈ H satisfies a quadratic equation with rational coefficients, then j ( z ) is an algebraic number.

  54. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Why number theorists like the j -function 1 Moonshine : Its q expansion, or Fourier expansion, has integer coefficients: j ( q ) = q − 1 + 196884 q + 21493760 q 2 + · · · The coefficients in this expansion encode information about finite-dimensional representations of certain sporadic simple groups. (John McKay’s “monstrous moonshine”). 2 Modular polynomials : Let N be an integer. The functions j ( z ) and j ( Nz ) satisfy a polynomial equation Φ N ( x , y ) in two variables with integer coefficients. The polynomial Φ N ( x , y ) is called the N -th modular polynomial . 3 Complex multiplication : If z ∈ H satisfies a quadratic equation with rational coefficients, then j ( z ) is an algebraic number.

  55. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Why number theorists like the j -function 1 Moonshine : Its q expansion, or Fourier expansion, has integer coefficients: j ( q ) = q − 1 + 196884 q + 21493760 q 2 + · · · The coefficients in this expansion encode information about finite-dimensional representations of certain sporadic simple groups. (John McKay’s “monstrous moonshine”). 2 Modular polynomials : Let N be an integer. The functions j ( z ) and j ( Nz ) satisfy a polynomial equation Φ N ( x , y ) in two variables with integer coefficients. The polynomial Φ N ( x , y ) is called the N -th modular polynomial . 3 Complex multiplication : If z ∈ H satisfies a quadratic equation with rational coefficients, then j ( z ) is an algebraic number.

  56. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Modular rings Using the modular polynomial Φ N ( x , y ), we can associate to each N a ring of modular functions Q N := Q [ x , y ] / (Φ N ( x , y )) = Q ( j ( z ) , j ( Nz )) . The ring Q N will be called the modular ring of level N . Modular rings play the same role in the study of elliptic curves as cyclotomic rings in the study of Pell’s equation.

  57. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Modular rings Using the modular polynomial Φ N ( x , y ), we can associate to each N a ring of modular functions Q N := Q [ x , y ] / (Φ N ( x , y )) = Q ( j ( z ) , j ( Nz )) . The ring Q N will be called the modular ring of level N . Modular rings play the same role in the study of elliptic curves as cyclotomic rings in the study of Pell’s equation.

  58. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Modular rings Using the modular polynomial Φ N ( x , y ), we can associate to each N a ring of modular functions Q N := Q [ x , y ] / (Φ N ( x , y )) = Q ( j ( z ) , j ( Nz )) . The ring Q N will be called the modular ring of level N . Modular rings play the same role in the study of elliptic curves as cyclotomic rings in the study of Pell’s equation.

  59. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Modular rings Using the modular polynomial Φ N ( x , y ), we can associate to each N a ring of modular functions Q N := Q [ x , y ] / (Φ N ( x , y )) = Q ( j ( z ) , j ( Nz )) . The ring Q N will be called the modular ring of level N . Modular rings play the same role in the study of elliptic curves as cyclotomic rings in the study of Pell’s equation.

  60. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Wiles’ Theorem Theorem ( Wiles , Breuil, Conrad, Diamond, Taylor) Let E : y 2 = x 3 + ax + b be an elliptic curve (with a , b ∈ Q ). Then the ring Q E is contained in (the fraction field of) the modular ring Q N , for some integer N ≥ 1 (the conductor of E, which can be explicitly calculated from an equation). Proof. Wiles, Andrew. Modular elliptic curves and Fermat’s Last Theorem . Annals of Mathematics 141: 443–551. Taylor R, Wiles A. Ring theoretic properties of certain Hecke algebras . Annals of Mathematics 141: 553–572.

  61. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Wiles’ Theorem Theorem ( Wiles , Breuil, Conrad, Diamond, Taylor) Let E : y 2 = x 3 + ax + b be an elliptic curve (with a , b ∈ Q ). Then the ring Q E is contained in (the fraction field of) the modular ring Q N , for some integer N ≥ 1 (the conductor of E, which can be explicitly calculated from an equation). Proof. Wiles, Andrew. Modular elliptic curves and Fermat’s Last Theorem . Annals of Mathematics 141: 443–551. Taylor R, Wiles A. Ring theoretic properties of certain Hecke algebras . Annals of Mathematics 141: 553–572.

  62. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Using Wiles’ theorem to solve elliptic curve equations √ Let τ = a + b − d ∈ H be any quadratic number. 1 By the theory of complex multiplication, we have a homomorphism → ¯ ev τ : Q N − Q , sending j ( z ) to j ( τ ) and j ( Nz ) to j ( N τ ). 2 By Wiles’ theorem, Q E is a subring of the modular ring Q N . 3 Restricting ev τ to Q E gives a homomorphism → ¯ ϕ τ : Q E − Q ; this homomorphism corresponds to an algebraic solution of E .

  63. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Using Wiles’ theorem to solve elliptic curve equations √ Let τ = a + b − d ∈ H be any quadratic number. 1 By the theory of complex multiplication, we have a homomorphism → ¯ ev τ : Q N − Q , sending j ( z ) to j ( τ ) and j ( Nz ) to j ( N τ ). 2 By Wiles’ theorem, Q E is a subring of the modular ring Q N . 3 Restricting ev τ to Q E gives a homomorphism → ¯ ϕ τ : Q E − Q ; this homomorphism corresponds to an algebraic solution of E .

  64. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Using Wiles’ theorem to solve elliptic curve equations √ Let τ = a + b − d ∈ H be any quadratic number. 1 By the theory of complex multiplication, we have a homomorphism → ¯ ev τ : Q N − Q , sending j ( z ) to j ( τ ) and j ( Nz ) to j ( N τ ). 2 By Wiles’ theorem, Q E is a subring of the modular ring Q N . 3 Restricting ev τ to Q E gives a homomorphism → ¯ ϕ τ : Q E − Q ; this homomorphism corresponds to an algebraic solution of E .

  65. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Using Wiles’ theorem to solve elliptic curve equations √ Let τ = a + b − d ∈ H be any quadratic number. 1 By the theory of complex multiplication, we have a homomorphism → ¯ ev τ : Q N − Q , sending j ( z ) to j ( τ ) and j ( Nz ) to j ( N τ ). 2 By Wiles’ theorem, Q E is a subring of the modular ring Q N . 3 Restricting ev τ to Q E gives a homomorphism → ¯ ϕ τ : Q E − Q ; this homomorphism corresponds to an algebraic solution of E .

  66. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Heegner points on modular curves, and elliptic curves Some terminology : The curve X 0 ( N ) whose function field is Q N is called the modular curve of level N . The morphism X 0 ( N ) − → E attached to the inclusion Q E ⊂ Q N is called a modular parametrisation for E . The imaginary quadratic irrationalities correspond to a canonical collection of algebraic points on X 0 ( N ), known as Heegner points. Question : Can the method of finding points on E based on modular parametrisations and Heegner points be generalised?

  67. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Heegner points on modular curves, and elliptic curves Some terminology : The curve X 0 ( N ) whose function field is Q N is called the modular curve of level N . The morphism X 0 ( N ) − → E attached to the inclusion Q E ⊂ Q N is called a modular parametrisation for E . The imaginary quadratic irrationalities correspond to a canonical collection of algebraic points on X 0 ( N ), known as Heegner points. Question : Can the method of finding points on E based on modular parametrisations and Heegner points be generalised?

  68. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Heegner points on modular curves, and elliptic curves Some terminology : The curve X 0 ( N ) whose function field is Q N is called the modular curve of level N . The morphism X 0 ( N ) − → E attached to the inclusion Q E ⊂ Q N is called a modular parametrisation for E . The imaginary quadratic irrationalities correspond to a canonical collection of algebraic points on X 0 ( N ), known as Heegner points. Question : Can the method of finding points on E based on modular parametrisations and Heegner points be generalised?

  69. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Heegner points on modular curves, and elliptic curves Some terminology : The curve X 0 ( N ) whose function field is Q N is called the modular curve of level N . The morphism X 0 ( N ) − → E attached to the inclusion Q E ⊂ Q N is called a modular parametrisation for E . The imaginary quadratic irrationalities correspond to a canonical collection of algebraic points on X 0 ( N ), known as Heegner points. Question : Can the method of finding points on E based on modular parametrisations and Heegner points be generalised?

  70. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Algebraic cycles Let V be a variety (of some dimension d = 2 r + 1). A correspondence from V to E is a subvariety Π ⊂ V × E of dimension r + 1. Such a Π induces a map { r -dimensional, null-homologous subvarieties of V } − → E by the rule Π(∆) = π E ( π − 1 V (∆) · Π) . The resulting map Π : CH r +1 ( V ) 0 − → E is a generalisation of a modular parametrisation.

  71. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Algebraic cycles Let V be a variety (of some dimension d = 2 r + 1). A correspondence from V to E is a subvariety Π ⊂ V × E of dimension r + 1. Such a Π induces a map { r -dimensional, null-homologous subvarieties of V } − → E by the rule Π(∆) = π E ( π − 1 V (∆) · Π) . The resulting map Π : CH r +1 ( V ) 0 − → E is a generalisation of a modular parametrisation.

  72. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Algebraic cycles Let V be a variety (of some dimension d = 2 r + 1). A correspondence from V to E is a subvariety Π ⊂ V × E of dimension r + 1. Such a Π induces a map { r -dimensional, null-homologous subvarieties of V } − → E by the rule Π(∆) = π E ( π − 1 V (∆) · Π) . The resulting map Π : CH r +1 ( V ) 0 − → E is a generalisation of a modular parametrisation.

  73. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Algebraic cycles Let V be a variety (of some dimension d = 2 r + 1). A correspondence from V to E is a subvariety Π ⊂ V × E of dimension r + 1. Such a Π induces a map { r -dimensional, null-homologous subvarieties of V } − → E by the rule Π(∆) = π E ( π − 1 V (∆) · Π) . The resulting map Π : CH r +1 ( V ) 0 − → E is a generalisation of a modular parametrisation.

  74. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Chow-Heegner points If V contains a natural, systematic supply of r -dimensional cycles which are null-homologous, their images under Π give rise to natural algebraic points on E , generalising Heegner points. Key examples : (Bertolini, Prasanna, D): V = W r × E r , where W − r is the r -fold fiber product of the universal elliptic curve over a modular curve, and E is a CM elliptic curve. (Rotger, D): V = W r 1 × W r 2 × W r 3 .

  75. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Chow-Heegner points If V contains a natural, systematic supply of r -dimensional cycles which are null-homologous, their images under Π give rise to natural algebraic points on E , generalising Heegner points. Key examples : (Bertolini, Prasanna, D): V = W r × E r , where W − r is the r -fold fiber product of the universal elliptic curve over a modular curve, and E is a CM elliptic curve. (Rotger, D): V = W r 1 × W r 2 × W r 3 .

  76. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Chow-Heegner points If V contains a natural, systematic supply of r -dimensional cycles which are null-homologous, their images under Π give rise to natural algebraic points on E , generalising Heegner points. Key examples : (Bertolini, Prasanna, D): V = W r × E r , where W − r is the r -fold fiber product of the universal elliptic curve over a modular curve, and E is a CM elliptic curve. (Rotger, D): V = W r 1 × W r 2 × W r 3 .

  77. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Stark-Heegner points In some cases, one can conjecturally construct canonical points on elliptic curves as the images of certain non-algebraic cycles on certain modular varieties. These mysterious points are called Stark-Heegner points ; they are, at present, very poorly understood. Gaining a better understanding of the phenomena underlying Stark-Heegner points has been one of the goals of my research in the last 10 years or so. A vague question : Can ideas like these, which lead to efficient algorithms for studying elliptic curves over global fields , eventually find practical applications similar to the theory of elliptic curves over finite fields?

  78. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Stark-Heegner points In some cases, one can conjecturally construct canonical points on elliptic curves as the images of certain non-algebraic cycles on certain modular varieties. These mysterious points are called Stark-Heegner points ; they are, at present, very poorly understood. Gaining a better understanding of the phenomena underlying Stark-Heegner points has been one of the goals of my research in the last 10 years or so. A vague question : Can ideas like these, which lead to efficient algorithms for studying elliptic curves over global fields , eventually find practical applications similar to the theory of elliptic curves over finite fields?

  79. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Stark-Heegner points In some cases, one can conjecturally construct canonical points on elliptic curves as the images of certain non-algebraic cycles on certain modular varieties. These mysterious points are called Stark-Heegner points ; they are, at present, very poorly understood. Gaining a better understanding of the phenomena underlying Stark-Heegner points has been one of the goals of my research in the last 10 years or so. A vague question : Can ideas like these, which lead to efficient algorithms for studying elliptic curves over global fields , eventually find practical applications similar to the theory of elliptic curves over finite fields?

  80. Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Stark-Heegner points In some cases, one can conjecturally construct canonical points on elliptic curves as the images of certain non-algebraic cycles on certain modular varieties. These mysterious points are called Stark-Heegner points ; they are, at present, very poorly understood. Gaining a better understanding of the phenomena underlying Stark-Heegner points has been one of the goals of my research in the last 10 years or so. A vague question : Can ideas like these, which lead to efficient algorithms for studying elliptic curves over global fields , eventually find practical applications similar to the theory of elliptic curves over finite fields?

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