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Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Diophantine equations, modular forms, and cycles

Henri Darmon

McGill University

Ottawa, September 2010

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

1

Diophantine equations

2

Cubic equations

3

FLT

4

Pell’s equation

5

Elliptic curves

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Definition A Diophantine equation is a system of polynomial equations with integer coefficients: f1(x1, . . . , xm) = · · · = fn(x1, . . . , xm) = 0, in which one is solely interested in the integer solutions. Some examples:

1 Cubic equations, like y2 = x3 + 1; 2 The Fermat-Pell equation: x2 − Dy2 = 1; 3 Fermat’s equation: xn + yn = zn;

A large part of number theory is concerned with the study of Diophantine equations.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Definition A Diophantine equation is a system of polynomial equations with integer coefficients: f1(x1, . . . , xm) = · · · = fn(x1, . . . , xm) = 0, in which one is solely interested in the integer solutions. Some examples:

1 Cubic equations, like y2 = x3 + 1; 2 The Fermat-Pell equation: x2 − Dy2 = 1; 3 Fermat’s equation: xn + yn = zn;

A large part of number theory is concerned with the study of Diophantine equations.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Definition A Diophantine equation is a system of polynomial equations with integer coefficients: f1(x1, . . . , xm) = · · · = fn(x1, . . . , xm) = 0, in which one is solely interested in the integer solutions. Some examples:

1 Cubic equations, like y2 = x3 + 1; 2 The Fermat-Pell equation: x2 − Dy2 = 1; 3 Fermat’s equation: xn + yn = zn;

A large part of number theory is concerned with the study of Diophantine equations.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Definition A Diophantine equation is a system of polynomial equations with integer coefficients: f1(x1, . . . , xm) = · · · = fn(x1, . . . , xm) = 0, in which one is solely interested in the integer solutions. Some examples:

1 Cubic equations, like y2 = x3 + 1; 2 The Fermat-Pell equation: x2 − Dy2 = 1; 3 Fermat’s equation: xn + yn = zn;

A large part of number theory is concerned with the study of Diophantine equations.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Definition A Diophantine equation is a system of polynomial equations with integer coefficients: f1(x1, . . . , xm) = · · · = fn(x1, . . . , xm) = 0, in which one is solely interested in the integer solutions. Some examples:

1 Cubic equations, like y2 = x3 + 1; 2 The Fermat-Pell equation: x2 − Dy2 = 1; 3 Fermat’s equation: xn + yn = zn;

A large part of number theory is concerned with the study of Diophantine equations.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Definition A Diophantine equation is a system of polynomial equations with integer coefficients: f1(x1, . . . , xm) = · · · = fn(x1, . . . , xm) = 0, in which one is solely interested in the integer solutions. Some examples:

1 Cubic equations, like y2 = x3 + 1; 2 The Fermat-Pell equation: x2 − Dy2 = 1; 3 Fermat’s equation: xn + yn = zn;

A large part of number theory is concerned with the study of Diophantine equations.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Why Diophantine equations?

What distinguishes the study of Diophantine equations from a merely recreational pursuit? Claim: Diophantine equations lie beyond the realm of recreational mathematics, because their study draws on a rich panoply of mathematical ideas. These ideas, and the new questions they lead to, are just as interesting (perhaps more!) than the equations which might have led to their discovery.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Why Diophantine equations?

What distinguishes the study of Diophantine equations from a merely recreational pursuit? Claim: Diophantine equations lie beyond the realm of recreational mathematics, because their study draws on a rich panoply of mathematical ideas. These ideas, and the new questions they lead to, are just as interesting (perhaps more!) than the equations which might have led to their discovery.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Why Diophantine equations?

What distinguishes the study of Diophantine equations from a merely recreational pursuit? Claim: Diophantine equations lie beyond the realm of recreational mathematics, because their study draws on a rich panoply of mathematical ideas. These ideas, and the new questions they lead to, are just as interesting (perhaps more!) than the equations which might have led to their discovery.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

1

Diophantine equations

2

Cubic equations

3

FLT

4

Pell’s equation

5

Elliptic curves

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

First example: the equation y 2 + y = x3

Theorem The equation y2 + y = x3 has only two solutions, namely (x, y) = (0, 0) and (0, −1). Proof. Factor the left-hand side: y(y + 1) = x3. Unique factorisation in Z: If gcd(a, b) = 1 and ab = x3, then a = x3

1,

b = x3

2.

Hence y and y + 1 are perfect cubes, {y, y + 1} ⊂ {. . . , −27, −8, −1, 0, 1, 8, 27, . . .}. It follows that y = −1 or 0.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

First example: the equation y 2 + y = x3

Theorem The equation y2 + y = x3 has only two solutions, namely (x, y) = (0, 0) and (0, −1). Proof. Factor the left-hand side: y(y + 1) = x3. Unique factorisation in Z: If gcd(a, b) = 1 and ab = x3, then a = x3

1,

b = x3

2.

Hence y and y + 1 are perfect cubes, {y, y + 1} ⊂ {. . . , −27, −8, −1, 0, 1, 8, 27, . . .}. It follows that y = −1 or 0.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

First example: the equation y 2 + y = x3

Theorem The equation y2 + y = x3 has only two solutions, namely (x, y) = (0, 0) and (0, −1). Proof. Factor the left-hand side: y(y + 1) = x3. Unique factorisation in Z: If gcd(a, b) = 1 and ab = x3, then a = x3

1,

b = x3

2.

Hence y and y + 1 are perfect cubes, {y, y + 1} ⊂ {. . . , −27, −8, −1, 0, 1, 8, 27, . . .}. It follows that y = −1 or 0.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

First example: the equation y 2 + y = x3

Theorem The equation y2 + y = x3 has only two solutions, namely (x, y) = (0, 0) and (0, −1). Proof. Factor the left-hand side: y(y + 1) = x3. Unique factorisation in Z: If gcd(a, b) = 1 and ab = x3, then a = x3

1,

b = x3

2.

Hence y and y + 1 are perfect cubes, {y, y + 1} ⊂ {. . . , −27, −8, −1, 0, 1, 8, 27, . . .}. It follows that y = −1 or 0.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

First example: the equation y 2 + y = x3

Theorem The equation y2 + y = x3 has only two solutions, namely (x, y) = (0, 0) and (0, −1). Proof. Factor the left-hand side: y(y + 1) = x3. Unique factorisation in Z: If gcd(a, b) = 1 and ab = x3, then a = x3

1,

b = x3

2.

Hence y and y + 1 are perfect cubes, {y, y + 1} ⊂ {. . . , −27, −8, −1, 0, 1, 8, 27, . . .}. It follows that y = −1 or 0.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Second example: the equation y 2 + 2 = x3

Theorem (Euler) The equation y2 + 2 = x3 has only two solutions, namely (x, y) = (3, ±5). Proof. Factor the left hand side in the larger ring Z[√−2]: (y + √ −2)(y − √ −2) = x3. Observe that y is odd, so gcd(y + √−2, y − √−2) = 1. Unique factorisation in Z[√−2] = ⇒ y + √ −2 = (a + b √ −2)3 = a(a2 − 6b2) + b(3a2 − 2b2) √ −2. Elementary manipulations = ⇒ b = 1, a = ±1, so y = ±5.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Second example: the equation y 2 + 2 = x3

Theorem (Euler) The equation y2 + 2 = x3 has only two solutions, namely (x, y) = (3, ±5). Proof. Factor the left hand side in the larger ring Z[√−2]: (y + √ −2)(y − √ −2) = x3. Observe that y is odd, so gcd(y + √−2, y − √−2) = 1. Unique factorisation in Z[√−2] = ⇒ y + √ −2 = (a + b √ −2)3 = a(a2 − 6b2) + b(3a2 − 2b2) √ −2. Elementary manipulations = ⇒ b = 1, a = ±1, so y = ±5.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Second example: the equation y 2 + 2 = x3

Theorem (Euler) The equation y2 + 2 = x3 has only two solutions, namely (x, y) = (3, ±5). Proof. Factor the left hand side in the larger ring Z[√−2]: (y + √ −2)(y − √ −2) = x3. Observe that y is odd, so gcd(y + √−2, y − √−2) = 1. Unique factorisation in Z[√−2] = ⇒ y + √ −2 = (a + b √ −2)3 = a(a2 − 6b2) + b(3a2 − 2b2) √ −2. Elementary manipulations = ⇒ b = 1, a = ±1, so y = ±5.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Second example: the equation y 2 + 2 = x3

Theorem (Euler) The equation y2 + 2 = x3 has only two solutions, namely (x, y) = (3, ±5). Proof. Factor the left hand side in the larger ring Z[√−2]: (y + √ −2)(y − √ −2) = x3. Observe that y is odd, so gcd(y + √−2, y − √−2) = 1. Unique factorisation in Z[√−2] = ⇒ y + √ −2 = (a + b √ −2)3 = a(a2 − 6b2) + b(3a2 − 2b2) √ −2. Elementary manipulations = ⇒ b = 1, a = ±1, so y = ±5.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Second example: the equation y 2 + 2 = x3

Theorem (Euler) The equation y2 + 2 = x3 has only two solutions, namely (x, y) = (3, ±5). Proof. Factor the left hand side in the larger ring Z[√−2]: (y + √ −2)(y − √ −2) = x3. Observe that y is odd, so gcd(y + √−2, y − √−2) = 1. Unique factorisation in Z[√−2] = ⇒ y + √ −2 = (a + b √ −2)3 = a(a2 − 6b2) + b(3a2 − 2b2) √ −2. Elementary manipulations = ⇒ b = 1, a = ±1, so y = ±5.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Second example: the equation y 2 + 2 = x3

Theorem (Euler) The equation y2 + 2 = x3 has only two solutions, namely (x, y) = (3, ±5). Proof. Factor the left hand side in the larger ring Z[√−2]: (y + √ −2)(y − √ −2) = x3. Observe that y is odd, so gcd(y + √−2, y − √−2) = 1. Unique factorisation in Z[√−2] = ⇒ y + √ −2 = (a + b √ −2)3 = a(a2 − 6b2) + b(3a2 − 2b2) √ −2. Elementary manipulations = ⇒ b = 1, a = ±1, so y = ±5.

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].

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Third example: the equation y 2 + 118 = x3

Theorem The equation y2 + 118 = x3 has no integer solutions. Proof. Factor the left hand side in the larger ring Z[√−118]: (y + √ −118)(y − √ −118) = x3. Proceed exactly as before, using unique factorisation in Z[√−118]. But... 152 + 118 = 73, so the theorem is wrong!

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Third example: the equation y 2 + 118 = x3

Theorem The equation y2 + 118 = x3 has no integer solutions. Proof. Factor the left hand side in the larger ring Z[√−118]: (y + √ −118)(y − √ −118) = x3. Proceed exactly as before, using unique factorisation in Z[√−118]. But... 152 + 118 = 73, so the theorem is wrong!

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Third example: the equation y 2 + 118 = x3

Theorem The equation y2 + 118 = x3 has no integer solutions. Proof. Factor the left hand side in the larger ring Z[√−118]: (y + √ −118)(y − √ −118) = x3. Proceed exactly as before, using unique factorisation in Z[√−118]. But... 152 + 118 = 73, so the theorem is wrong!

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Third example: the equation y 2 + 118 = x3

Theorem The equation y2 + 118 = x3 has no integer solutions. Proof. Factor the left hand side in the larger ring Z[√−118]: (y + √ −118)(y − √ −118) = x3. Proceed exactly as before, using unique factorisation in Z[√−118]. But... 152 + 118 = 73, so the theorem is wrong!

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Third example: the equation y 2 + 118 = x3

Theorem The equation y2 + 118 = x3 has no integer solutions. Proof. Factor the left hand side in the larger ring Z[√−118]: (y + √ −118)(y − √ −118) = x3. Proceed exactly as before, using unique factorisation in Z[√−118]. But... 152 + 118 = 73, so the theorem is wrong!

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

• bstruction 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.

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

• bstruction 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.

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

• bstruction 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.

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

• bstruction 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.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

1

Diophantine equations

2

Cubic equations

3

FLT

4

Pell’s equation

5

Elliptic curves

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Fermat’s Last Theorem

Theorem (Fermat, 1635?) If n ≥ 3, then the equation xn + yn = zn has no integer solution with xyz = 0. Natural opening gambit: (x + y)(x + ζny) · · · (x + ζn−1

n

y) = zn, where ζn = e2πi/n is an nth root of unity. Theorem (Lam´ e) Suppose p > 2 is prime. If Z[ζp] has unique factorisation, then xp + yp = zp has no non-trivial solution.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Fermat’s Last Theorem

Theorem (Fermat, 1635?) If n ≥ 3, then the equation xn + yn = zn has no integer solution with xyz = 0. Natural opening gambit: (x + y)(x + ζny) · · · (x + ζn−1

n

y) = zn, where ζn = e2πi/n is an nth root of unity. Theorem (Lam´ e) Suppose p > 2 is prime. If Z[ζp] has unique factorisation, then xp + yp = zp has no non-trivial solution.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Fermat’s Last Theorem

Theorem (Fermat, 1635?) If n ≥ 3, then the equation xn + yn = zn has no integer solution with xyz = 0. Natural opening gambit: (x + y)(x + ζny) · · · (x + ζn−1

n

y) = zn, where ζn = e2πi/n is an nth root of unity. Theorem (Lam´ e) Suppose p > 2 is prime. If Z[ζp] has unique factorisation, then xp + yp = zp has no non-trivial solution.

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[ζp], then xp + yp = zp 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[ζn]). 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.

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[ζp], then xp + yp = zp 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[ζn]). 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.

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[ζp], then xp + yp = zp 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[ζn]). 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.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

1

Diophantine equations

2

Cubic equations

3

FLT

4

Pell’s equation

5

Elliptic curves

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Pell’s equation

The Fermat-Pell equation is the equation x2 − dy2 = 1, where d > 0 is a non-square integer. The group law. (x1, y1) ∗ (x2, y2) = (x1x2 + dy1y2, x1y2 + y1x2). Theorem (Fermat) For any non-square d > 0, the Pell equation x2 − dy2 has a non-trivial fundamental solution (x0, y0) such that all other solutions are of the form (±x, ±y) = (x0, y0)∗n.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Pell’s equation

The Fermat-Pell equation is the equation x2 − dy2 = 1, where d > 0 is a non-square integer. The group law. (x1, y1) ∗ (x2, y2) = (x1x2 + dy1y2, x1y2 + y1x2). Theorem (Fermat) For any non-square d > 0, the Pell equation x2 − dy2 has a non-trivial fundamental solution (x0, y0) such that all other solutions are of the form (±x, ±y) = (x0, y0)∗n.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Pell’s equation

The Fermat-Pell equation is the equation x2 − dy2 = 1, where d > 0 is a non-square integer. The group law. (x1, y1) ∗ (x2, y2) = (x1x2 + dy1y2, x1y2 + y1x2). Theorem (Fermat) For any non-square d > 0, the Pell equation x2 − dy2 has a non-trivial fundamental solution (x0, y0) such that all other solutions are of the form (±x, ±y) = (x0, y0)∗n.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Some examples of fundamental solutions

If d = 2, then (x0, y0) = (3, 2). If d = 61, then (x0, y0) = (1766319049, 226153980). If d = 313, then (x0, y0) = (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.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Some examples of fundamental solutions

If d = 2, then (x0, y0) = (3, 2). If d = 61, then (x0, y0) = (1766319049, 226153980). If d = 313, then (x0, y0) = (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.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Some examples of fundamental solutions

If d = 2, then (x0, y0) = (3, 2). If d = 61, then (x0, y0) = (1766319049, 226153980). If d = 313, then (x0, y0) = (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.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Some examples of fundamental solutions

If d = 2, then (x0, y0) = (3, 2). If d = 61, then (x0, y0) = (1766319049, 226153980). If d = 313, then (x0, y0) = (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.

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 (x1, y1) ∗ (x2, y2) = (x3, y3) as (x1 + y1 √ d)(x2 + y2 √ d) = (x3 + y3 √ d). Solving Pell’s equation can now be recast as: Problem: Calculate the group of units in the ring Z[ √ d].

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 (x1, y1) ∗ (x2, y2) = (x3, y3) as (x1 + y1 √ d)(x2 + y2 √ d) = (x3 + y3 √ d). Solving Pell’s equation can now be recast as: Problem: Calculate the group of units in the ring Z[ √ d].

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 (x1, y1) ∗ (x2, y2) = (x3, y3) as (x1 + y1 √ d)(x2 + y2 √ d) = (x3 + y3 √ d). Solving Pell’s equation can now be recast as: Problem: Calculate the group of units in the ring Z[ √ d].

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[ζd], where ζd = e2πi/d. Proof. Gauss sums: g =

d−1

• j=0

j d

• ζj

d.

Direct calculation: g2 = d.

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[ζd], where ζd = e2πi/d. Proof. Gauss sums: g =

d−1

• j=0

j d

• ζj

d.

Direct calculation: g2 = d.

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[ζd], where ζd = e2πi/d. Proof. Gauss sums: g =

d−1

• j=0

j d

• ζj

d.

Direct calculation: g2 = d.

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[ζd] contains some obvious units: the circular units. u = ζd + 1 = ζ2

d − 1

ζd − 1. Now let x + y √ d := normZ[ζd]

Z[ √ d](u).

Then (x, y) is a (not necessarily fundamental!) solution to Pell’s equation.

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[ζd] contains some obvious units: the circular units. u = ζd + 1 = ζ2

d − 1

ζd − 1. Now let x + y √ d := normZ[ζd]

Z[ √ d](u).

Then (x, y) is a (not necessarily fundamental!) solution to Pell’s equation.

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[ζd] contains some obvious units: the circular units. u = ζd + 1 = ζ2

d − 1

ζd − 1. Now let x + y √ d := normZ[ζd]

Z[ √ d](u).

Then (x, y) is a (not necessarily fundamental!) solution to Pell’s equation.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

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Diophantine equations

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Cubic equations

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FLT

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Pell’s equation

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Elliptic curves

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 y2 = x3 + 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.

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 y2 = x3 + 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.

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 y2 = x3 + 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.

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 y2 = x3 + 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.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

The group law for elliptic curves

x y y = x + a x + b

2 3

P Q R P+Q

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Ring theoretic formulation of the problem

To the elliptic curve E : y2 = x3 + ax + b, we attach the ring QE := Q[x, y]/(y2 − (x3 + ax + b)). Elementary (but important) remark: Rational solutions of E are in natural bijection with homomorphisms from QE to Q: given a solution (x, y) = (r, s) , let ϕ : QE − → Q be given by ϕ(x) = r, ϕ(y) = s. Problem: Construct homomorphisms from QE to Q (or at least to ¯ Q) in a non-trivial way.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Ring theoretic formulation of the problem

To the elliptic curve E : y2 = x3 + ax + b, we attach the ring QE := Q[x, y]/(y2 − (x3 + ax + b)). Elementary (but important) remark: Rational solutions of E are in natural bijection with homomorphisms from QE to Q: given a solution (x, y) = (r, s) , let ϕ : QE − → Q be given by ϕ(x) = r, ϕ(y) = s. Problem: Construct homomorphisms from QE to Q (or at least to ¯ Q) in a non-trivial way.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Ring theoretic formulation of the problem

To the elliptic curve E : y2 = x3 + ax + b, we attach the ring QE := Q[x, y]/(y2 − (x3 + ax + b)). Elementary (but important) remark: Rational solutions of E are in natural bijection with homomorphisms from QE to Q: given a solution (x, y) = (r, s) , let ϕ : QE − → Q be given by ϕ(x) = r, ϕ(y) = s. Problem: Construct homomorphisms from QE to Q (or at least to ¯ Q) in a non-trivial way.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Ring theoretic formulation of the problem

To the elliptic curve E : y2 = x3 + ax + b, we attach the ring QE := Q[x, y]/(y2 − (x3 + ax + b)). Elementary (but important) remark: Rational solutions of E are in natural bijection with homomorphisms from QE to Q: given a solution (x, y) = (r, s) , let ϕ : QE − → Q be given by ϕ(x) = r, ϕ(y) = s. Problem: Construct homomorphisms from QE to Q (or at least to ¯ Q) in a non-trivial way.

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 j az + b cz + d

• = j(z),

for all    a b c d    ∈ SL2(Z), j(z) = q−1 + O(q), where q = e2πiz. The j-function is the prototypical example of a modular function. It has been said that number theory is largely the study of such

• bjects.

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 j az + b cz + d

• = j(z),

for all    a b c d    ∈ SL2(Z), j(z) = q−1 + O(q), where q = e2πiz. The j-function is the prototypical example of a modular function. It has been said that number theory is largely the study of such

• bjects.

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 j az + b cz + d

• = j(z),

for all    a b c d    ∈ SL2(Z), j(z) = q−1 + O(q), where q = e2πiz. The j-function is the prototypical example of a modular function. It has been said that number theory is largely the study of such

• bjects.

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 j az + b cz + d

• = j(z),

for all    a b c d    ∈ SL2(Z), j(z) = q−1 + O(q), where q = e2πiz. The j-function is the prototypical example of a modular function. It has been said that number theory is largely the study of such

• bjects.

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 + 196884q + 21493760q2 + · · · 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 ΦN(x, y) in two variables with integer coefficients. The polynomial ΦN(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.

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 + 196884q + 21493760q2 + · · · 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 ΦN(x, y) in two variables with integer coefficients. The polynomial ΦN(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.

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 + 196884q + 21493760q2 + · · · 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 ΦN(x, y) in two variables with integer coefficients. The polynomial ΦN(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.

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 + 196884q + 21493760q2 + · · · 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 ΦN(x, y) in two variables with integer coefficients. The polynomial ΦN(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.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Modular rings

Using the modular polynomial ΦN(x, y), we can associate to each N a ring of modular functions QN := Q[x, y]/(ΦN(x, y)) = Q(j(z), j(Nz)). The ring QN 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.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Modular rings

Using the modular polynomial ΦN(x, y), we can associate to each N a ring of modular functions QN := Q[x, y]/(ΦN(x, y)) = Q(j(z), j(Nz)). The ring QN 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.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Modular rings

Using the modular polynomial ΦN(x, y), we can associate to each N a ring of modular functions QN := Q[x, y]/(ΦN(x, y)) = Q(j(z), j(Nz)). The ring QN 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.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Modular rings

Using the modular polynomial ΦN(x, y), we can associate to each N a ring of modular functions QN := Q[x, y]/(ΦN(x, y)) = Q(j(z), j(Nz)). The ring QN 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.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Wiles’ Theorem

Theorem (Wiles, Breuil, Conrad, Diamond, Taylor) Let E : y2 = x3 + ax + b be an elliptic curve (with a, b ∈ Q). Then the ring QE is contained in (the fraction field of) the modular ring QN, 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.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Wiles’ Theorem

Theorem (Wiles, Breuil, Conrad, Diamond, Taylor) Let E : y2 = x3 + ax + b be an elliptic curve (with a, b ∈ Q). Then the ring QE is contained in (the fraction field of) the modular ring QN, 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.

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 evτ : QN − → ¯ Q, sending j(z) to j(τ) and j(Nz) to j(Nτ).

2 By Wiles’ theorem, QE is a subring of the modular ring QN. 3 Restricting evτ to QE gives a homomorphism

ϕτ : QE − → ¯ Q; this homomorphism corresponds to an algebraic solution of E.

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 evτ : QN − → ¯ Q, sending j(z) to j(τ) and j(Nz) to j(Nτ).

2 By Wiles’ theorem, QE is a subring of the modular ring QN. 3 Restricting evτ to QE gives a homomorphism

ϕτ : QE − → ¯ Q; this homomorphism corresponds to an algebraic solution of E.

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 evτ : QN − → ¯ Q, sending j(z) to j(τ) and j(Nz) to j(Nτ).

2 By Wiles’ theorem, QE is a subring of the modular ring QN. 3 Restricting evτ to QE gives a homomorphism

ϕτ : QE − → ¯ Q; this homomorphism corresponds to an algebraic solution of E.

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 evτ : QN − → ¯ Q, sending j(z) to j(τ) and j(Nz) to j(Nτ).

2 By Wiles’ theorem, QE is a subring of the modular ring QN. 3 Restricting evτ to QE gives a homomorphism

ϕτ : QE − → ¯ Q; this homomorphism corresponds to an algebraic solution of E.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Heegner points on modular curves, and elliptic curves

Some terminology: The curve X0(N) whose function field is QN is called the modular curve of level N. The morphism X0(N) − → E attached to the inclusion QE ⊂ QN is called a modular parametrisation for E. The imaginary quadratic irrationalities correspond to a canonical collection of algebraic points on X0(N), known as Heegner points. Question: Can the method of finding points on E based on modular parametrisations and Heegner points be generalised?

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Heegner points on modular curves, and elliptic curves

Some terminology: The curve X0(N) whose function field is QN is called the modular curve of level N. The morphism X0(N) − → E attached to the inclusion QE ⊂ QN is called a modular parametrisation for E. The imaginary quadratic irrationalities correspond to a canonical collection of algebraic points on X0(N), known as Heegner points. Question: Can the method of finding points on E based on modular parametrisations and Heegner points be generalised?

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Heegner points on modular curves, and elliptic curves

Some terminology: The curve X0(N) whose function field is QN is called the modular curve of level N. The morphism X0(N) − → E attached to the inclusion QE ⊂ QN is called a modular parametrisation for E. The imaginary quadratic irrationalities correspond to a canonical collection of algebraic points on X0(N), known as Heegner points. Question: Can the method of finding points on E based on modular parametrisations and Heegner points be generalised?

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Heegner points on modular curves, and elliptic curves

Some terminology: The curve X0(N) whose function field is QN is called the modular curve of level N. The morphism X0(N) − → E attached to the inclusion QE ⊂ QN is called a modular parametrisation for E. The imaginary quadratic irrationalities correspond to a canonical collection of algebraic points on X0(N), known as Heegner points. Question: Can the method of finding points on E based on modular parametrisations and Heegner points be generalised?

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Algebraic cycles

Let V be a variety (of some dimension d = 2r + 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(∆) = πE(π−1

V (∆) · Π).

The resulting map Π : CHr+1(V )0 − → E is a generalisation of a modular parametrisation.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Algebraic cycles

Let V be a variety (of some dimension d = 2r + 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(∆) = πE(π−1

V (∆) · Π).

The resulting map Π : CHr+1(V )0 − → E is a generalisation of a modular parametrisation.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Algebraic cycles

Let V be a variety (of some dimension d = 2r + 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(∆) = πE(π−1

V (∆) · Π).

The resulting map Π : CHr+1(V )0 − → E is a generalisation of a modular parametrisation.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Algebraic cycles

Let V be a variety (of some dimension d = 2r + 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(∆) = πE(π−1

V (∆) · Π).

The resulting map Π : CHr+1(V )0 − → E is a generalisation of a modular parametrisation.

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 = Wr × 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 = Wr1 × Wr2 × Wr3.

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 = Wr × 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 = Wr1 × Wr2 × Wr3.

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 = Wr × 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 = Wr1 × Wr2 × Wr3.

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

• ver finite fields?

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

• ver finite fields?

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

• ver finite fields?

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

• ver finite fields?

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Thank you for your attention.