Diophantine equations, modular forms, and cycles · 2010-09-13 · Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Diophantine equations, modular forms,

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves

Diophantine equations, modular forms, and cycles

Henri Darmon

McGill University

Ottawa, September 2010


1 Diophantine equations

2 Cubic equations

3 FLT

4 Pell’s equation

5 Elliptic curves


Definition

A Diophantine equation is a system of polynomial equations withinteger 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 ofDiophantine equations.


Definition


f1(x1, . . . , xm) = · · · = fn(x1, . . . , xm) = 0,


Some examples:






Definition


f1(x1, . . . , xm) = · · · = fn(x1, . . . , xm) = 0,


Some examples:






Definition


f1(x1, . . . , xm) = · · · = fn(x1, . . . , xm) = 0,


Some examples:






Definition


f1(x1, . . . , xm) = · · · = fn(x1, . . . , xm) = 0,


Some examples:






Definition


f1(x1, . . . , xm) = · · · = fn(x1, . . . , xm) = 0,


Some examples:






Why Diophantine equations?

What distinguishes the study of Diophantine equations from amerely recreational pursuit?

Claim: Diophantine equations lie beyond the realm of recreationalmathematics, because their study draws on a rich panoply ofmathematical ideas. These ideas, and the new questions they leadto, are just as interesting (perhaps more!) than the equationswhich might have led to their discovery.











2 Cubic equations

3 FLT

4 Pell’s equation

5 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 = x31 , 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.



Theorem


Proof.




2 .


{y , y + 1} ⊂ {. . . ,−27,−8,−1, 0, 1, 8, 27, . . .}.




Theorem


Proof.




2 .


{y , y + 1} ⊂ {. . . ,−27,−8,−1, 0, 1, 8, 27, . . .}.




Theorem


Proof.




2 .


{y , y + 1} ⊂ {. . . ,−27,−8,−1, 0, 1, 8, 27, . . .}.




Theorem


Proof.




2 .


{y , y + 1} ⊂ {. . . ,−27,−8,−1, 0, 1, 8, 27, . . .}.



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.



Theorem (Euler)


Proof.


(y +√−2)(y −

√−2) = x3.


√−2) = 1.


y +√−2 = (a + b

√−2)3 = a(a2 − 6b2) + b(3a2 − 2b2)

√−2.




Theorem (Euler)


Proof.


(y +√−2)(y −

√−2) = x3.


√−2) = 1.


y +√−2 = (a + b

√−2)3 = a(a2 − 6b2) + b(3a2 − 2b2)

√−2.




Theorem (Euler)


Proof.


(y +√−2)(y −

√−2) = x3.


√−2) = 1.


y +√−2 = (a + b

√−2)3 = a(a2 − 6b2) + b(3a2 − 2b2)

√−2.




Theorem (Euler)


Proof.


(y +√−2)(y −

√−2) = x3.


√−2) = 1.


y +√−2 = (a + b

√−2)3 = a(a2 − 6b2) + b(3a2 − 2b2)

√−2.




Theorem (Euler)


Proof.


(y +√−2)(y −

√−2) = x3.


√−2) = 1.


y +√−2 = (a + b

√−2)3 = a(a2 − 6b2) + b(3a2 − 2b2)

√−2.



The gap in Euler’s proof

Euler’s proof is interesting because it invokes a non-trivialstructural property – unique factorisation – of the the ring Z[

√−2].


Third example: the equation y 2 + 118 = x3

Theorem

The equation y2 + 118 = x3 has no integer solutions.

Proof.


(y +√−118)(y −

√−118) = x3.

Proceed exactly as before, using unique factorisation inZ[√−118].

But... 152 + 118 = 73, so the theorem is wrong!



Theorem


Proof.


(y +√−118)(y −

√−118) = x3.





Theorem


Proof.


(y +√−118)(y −

√−118) = x3.





Theorem


Proof.


(y +√−118)(y −

√−118) = x3.





Theorem


Proof.


(y +√−118)(y −

√−118) = x3.




Unique factorisation

Conclusion: Unique factorisation fails in Z[√−118].

The possible failure of unique factorisation which often arises as anobstruction to analysing diophantine equations, is a highlyinteresting phenomenon.

It can be measured in terms of a class group of an appropriate ring.

Number theorists have devoted a lot of efforts to betterunderstanding and controlling class groups, spurring thedevelopment of algebraic number theory and commutative algebra.





















2 Cubic equations

3 FLT

4 Pell’s equation

5 Elliptic curves


Fermat’s Last Theorem

Theorem (Fermat, 1635?)

If n ≥ 3, then the equation xn + yn = zn has no integer solutionwith xyz 6= 0.

Natural opening gambit:

(x + y)(x + ζny) · · · (x + ζn−1n y) = zn,

where ζn = e2πi/n is an nth root of unity.

Theorem (Lame)

Suppose p > 2 is prime. If Z[ζp] has unique factorisation, thenxp + yp = zp has no non-trivial solution.






(x + y)(x + ζny) · · · (x + ζn−1n y) = zn,


Theorem (Lame)







(x + y)(x + ζny) · · · (x + ζn−1n y) = zn,


Theorem (Lame)



Kummer’s theorem

Theorem (Kummer)

Suppose p > 2 is prime. If p does not divide the class number ofZ[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 cyclotomicrings (rings of the form Z[ζn]). Many of these are still open!

As we all know, Fermat’s Last Theorem was eventually proved in1995, by Andrew Wiles, relying on a very different circle of ideas.


Kummer’s theorem

Theorem (Kummer)





Kummer’s theorem

Theorem (Kummer)






2 Cubic equations

3 FLT

4 Pell’s equation

5 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 anon-trivial fundamental solution (x0, y0) such that all othersolutions are of the form

(±x ,±y) = (x0, y0)∗n.


Pell’s equation


x2 − dy2 = 1,


The group law.

(x1, y1) ∗ (x2, y2) = (x1x2 + dy1y2, x1y2 + y1x2).

Theorem (Fermat)


(±x ,±y) = (x0, y0)∗n.


Pell’s equation


x2 − dy2 = 1,


The group law.

(x1, y1) ∗ (x2, y2) = (x1x2 + dy1y2, x1y2 + y1x2).

Theorem (Fermat)


(±x ,±y) = (x0, y0)∗n.


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 fundamentalsolution is the method based on continued fractions. It wasdiscovered by the Indian mathematicians of the 12th century, andrediscovered by Fermat in the 17th century.



If d = 2, then (x0, y0) = (3, 2).

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

If d = 313, then

(x0, y0) = (32188120829134849, 1819380158564160).




If d = 2, then (x0, y0) = (3, 2).

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

If d = 313, then

(x0, y0) = (32188120829134849, 1819380158564160).




If d = 2, then (x0, y0) = (3, 2).

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

If d = 313, then

(x0, y0) = (32188120829134849, 1819380158564160).



Explanation of the group law

Key remark: If (x , y) is a solution to Pell’s equation, thenx + 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 ].





√d ].

One can rewrite

(x1, y1) ∗ (x2, y2) = (x3, y3)

as(x1 + y1

√d)(x2 + y2

√d) = (x3 + y3

√d).



d ].





√d ].

One can rewrite

(x1, y1) ∗ (x2, y2) = (x3, y3)

as(x1 + y1

√d)(x2 + y2

√d) = (x3 + y3

√d).



d ].


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

)ζ jd .

Direct calculation:g2 = d .



Theorem (Gauss)



Proof.

Gauss sums:

g =d−1∑j=0

(j

d

)ζ jd .




Theorem (Gauss)



Proof.

Gauss sums:

g =d−1∑j=0

(j

d

)ζ jd .



The cyclotomic approach to Pell’s equation, cont’d

The usefulness of Gauss’s theorem for Pell’s equation arises fromthe fact that Z[ζd ] contains some obvious units: the circular units.

u = ζd + 1 =ζ2d − 1

ζd − 1.

Now letx + y

√d := norm

Z[ζd ]

Z[√

d ](u).

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




u = ζd + 1 =ζ2d − 1

ζd − 1.

Now letx + y

√d := norm

Z[ζd ]

Z[√

d ](u).





u = ζd + 1 =ζ2d − 1

ζd − 1.

Now letx + y

√d := norm

Z[ζd ]

Z[√

d ](u).




2 Cubic equations

3 FLT

4 Pell’s equation

5 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 tosuch an equation.

Elliptic curve equations exhibit many of the features of Pell’sequation:

1 The set of (rational) solutions to an elliptic curve equation isequipped with a natural group law;

2 The cyclotomic approach to solving Pell’s equation has aninteresting (and quite deep) counterpart for elliptic curves.


Elliptic Curves


y2 = x3 + ax + b, with a, b ∈ Q.






Elliptic Curves


y2 = x3 + ax + b, with a, b ∈ Q.






Elliptic Curves


y2 = x3 + ax + b, with a, b ∈ Q.






The group law for elliptic curves

x

y y = x + a x + b2 3

P

Q

R

P+Q


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 arein natural bijection with homomorphisms from QE to Q: given asolution (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 toQ) in a non-trivial way.




QE := Q[x , y ]/(y2 − (x3 + ax + b)).


ϕ(x) = r , ϕ(y) = s.





QE := Q[x , y ]/(y2 − (x3 + ax + b)).


ϕ(x) = r , ϕ(y) = s.





QE := Q[x , y ]/(y2 − (x3 + ax + b)).


ϕ(x) = r , ϕ(y) = s.



Modular functions

Let H be the Poincare 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 suchobjects.


Modular functions


Theorem


j

(az + b

cz + d

)= j(z), for all

a b

c d

∈ SL2(Z),




Modular functions


Theorem


j

(az + b

cz + d

)= j(z), for all

a b

c d

∈ SL2(Z),




Modular functions


Theorem


j

(az + b

cz + d

)= j(z), for all

a b

c d

∈ SL2(Z),




Why number theorists like the j-function

1 Moonshine: Its q expansion, or Fourier expansion, hasinteger coefficients:

j(q) = q−1 + 196884q + 21493760q2 + · · ·

The coefficients in this expansion encode information aboutfinite-dimensional representations of certain sporadic simplegroups. (John McKay’s “monstrous moonshine”).

2 Modular polynomials: Let N be an integer. The functionsj(z) and j(Nz) satisfy a polynomial equation ΦN(x , y) in twovariables with integer coefficients. The polynomial ΦN(x , y)is called the N-th modular polynomial.

3 Complex multiplication: If z ∈ H satisfies a quadraticequation with rational coefficients, then j(z) is an algebraicnumber.




j(q) = q−1 + 196884q + 21493760q2 + · · ·







j(q) = q−1 + 196884q + 21493760q2 + · · ·







j(q) = q−1 + 196884q + 21493760q2 + · · ·





Modular rings

Using the modular polynomial ΦN(x , y), we can associate to eachN 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. Modularrings play the same role in the study of elliptic curves ascyclotomic rings in the study of Pell’s equation.


Modular rings





Modular rings





Modular rings





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) themodular 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 LastTheorem. Annals of Mathematics 141: 443–551.

Taylor R, Wiles A. Ring theoretic properties of certain Heckealgebras. Annals of Mathematics 141: 553–572.


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) themodular 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 LastTheorem. Annals of Mathematics 141: 443–551.

Taylor R, Wiles A. Ring theoretic properties of certain Heckealgebras. Annals of Mathematics 141: 553–572.


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 ahomomorphism

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 .





evτ : QN −→ Q,




ϕτ : QE −→ Q;






evτ : QN −→ Q,




ϕτ : QE −→ Q;






evτ : QN −→ Q,




ϕτ : QE −→ Q;



Heegner points on modular curves, and elliptic curves

Some terminology:

The curve X0(N) whose function field is QN is called the modularcurve of level N.

The morphism X0(N) −→ E attached to the inclusion QE ⊂ QN iscalled a modular parametrisation for E .

The imaginary quadratic irrationalities correspond to a canonicalcollection of algebraic points on X0(N), known as Heegner points.

Question: Can the method of finding points on E based onmodular parametrisations and Heegner points be generalised?



Some terminology:







Some terminology:







Some terminology:






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 ofdimension r + 1.

Such a Π induces a map

{ r -dimensional, null-homologous subvarieties of V} −→ E

by the ruleΠ(∆) = πE (π−1

V (∆) · Π).

The resulting map

Π : CHr+1(V )0 −→ E

is a generalisation of a modular parametrisation.


Algebraic cycles






V (∆) · Π).

The resulting map

Π : CHr+1(V )0 −→ E



Algebraic cycles






V (∆) · Π).

The resulting map

Π : CHr+1(V )0 −→ E



Algebraic cycles






V (∆) · Π).

The resulting map

Π : CHr+1(V )0 −→ E



Chow-Heegner points

If V contains a natural, systematic supply of r -dimensional cycleswhich are null-homologous, their images under Π give rise tonatural algebraic points on E , generalising Heegner points.

Key examples:

(Bertolini, Prasanna, D): V = Wr × E r , where W − r is the r -foldfiber product of the universal elliptic curve over a modular curve,and E is a CM elliptic curve.

(Rotger, D): V = Wr1 ×Wr2 ×Wr3 .


Chow-Heegner points


Key examples:




Chow-Heegner points


Key examples:




Stark-Heegner points

In some cases, one can conjecturally construct canonical points onelliptic curves as the images of certain non-algebraic cycles oncertain modular varieties.

These mysterious points are called Stark-Heegner points; they are,at present, very poorly understood.

Gaining a better understanding of the phenomena underlyingStark-Heegner points has been one of the goals of my research inthe last 10 years or so.

A vague question: Can ideas like these, which lead to efficientalgorithms for studying elliptic curves over global fields, eventuallyfind practical applications similar to the theory of elliptic curvesover finite fields?




















Thank you for your attention.

Documents

Diophantine equations, modular forms, and cycles · 2010-09-13 · Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Diophantine equations, modular forms,