Invitation to the Proof of
Fermat's Last “Theorem”

William F. Hammond

University at Albany Math Club
March, 2011


Invitation to the Proof of
Fermat's Last “Theorem”

Fermat's Last “Theorem” (ca. 1637) was finally proved in the mid-1990s by using the study of plane cubic curves of the form y2=xAxBxC where A, B, and C are distinct integers.

This talk will provide an overview of the main ingredients.

1.  The Statement

There is no solution in positive integers x,y,z of the equation xn+yn=zn for n3.

Note: There are infinitely many essentially different solutions when n=1,2.

2.  Old History

  • The statement is equivalent to the statement that there are no non-zero integers x,y,z satisfying xn+yn=zn for n3.

  • The statement is equivalent to the statement that there are no rational points off the coordinate axes on the plane curve xn+yn=1 for n3.

  • For odd exponents n3 the statement is equivalent to the statement that there are no non-zero integers such that xn+yn+zn=0.

3.  Old History (continued)

  • If the theorem is true when the exponent is a given n, then it is certainly true when the exponent is a multiple of that value of n.

  • The case where n is 3 or 4 can be handled within the realm of “elementary” number theory. (See, for example, the classic text of Hardy & Wright.)

  • Any integer n3 not divisible by 4 must be divisible by an odd prime.

  • It remains to prove the theorem when the exponent n is a prime p5.

4.  A Solution leads to a Cubic Curve

Let p5 be prime.

Suppose there are non-zero integers a,b,c such that ap+bp=cp.

Then the plane cubic curve y2=xxapx+bp. is an elliptic curve “defined over” Q — the Frey-Hellegouarch curve.

5.  Cubic Curves

Over any field K, e.g., Q, C, or Fp=ZpZ, after a (projective) change of coordinates in K a non-singular cubic curve with at least one K-valued point may be brought into “generalized Weierstrass form” y2+a1xy+a3y=x3+a2x2+a4x+a6, and over an algebraically closed field of characteristic 2,3 into an equation of the form y2=xλ1xλ2xλ3.

The latter is a non-singular cubic when Δ=i<jλiλj20.

Example: For the Frey-Hellegouarch curve Δ=abc2p.

6.  Cubics over the Complex Numbers

Given a non-singular cubic curve C, y2+a1xy+a3y=x3+a2x2+a4x+a6, with coefficients in C and Δ0, the set of all solutions x,y in C2 together with the “distinguished point at infinity” forms a compact Riemann surface of genus one — a torus.

7.  The Projective Plane

For a given field K P2K=K2line at infinity where  line at infinity=classes of parallel linesinK2 =lines through0,0inK2 =slopes of lines =K

Each line contains one and only one point (its parallel class) on the line at infinity. The “distinguished point at infinity” is the parallel class of vertical lines.

8.  A Line Meets a Cubic in 3 Points

Given a non-singular cubic curve C, y2+a1xy+a3y=x3+a2x2+a4x+a6, with coefficients in K, every line in K2 passing through 2 points of C meets C in a third point, allowing for multiplicities.

Proof. Parameterize the line and get a cubic equation in the parameter with two known roots in K.

9.  The Distinguished Point at Infinity

Given a non-singular cubic curve C, y2+a1xy+a3y=x3+a2x2+a4x+a6, with coefficients in K, the distinguished point at infinity in P2K lies on C.

Proof. Introduce homogeneous coordinates x,y,z0,0,0 in P2 where:

In homogeneous coordinates the curve C has the equation y2z+a1xyz+a3yz2=x3+a2x2z+a4xz2+a6z3.

In homogeneous coordinates the line at infinity has the equation z=0.

The intersection of the line at infinity with C has the equation x3=0. Thus, C meets the line at infinity “triply” in the distinguished point at infinity.

10.  The Group Law

Given a non-singular cubic curve C, y2+a1xy+a3y=x3+a2x2+a4x+a6, with coefficients in K, there is a unique “algebraic” group law on the points of C in P2K characterized by the two conditions

  1. The group origin 0 is the distinguished point at infinity.

  2. For three points P,Q,R of C one has P+Q+R=0 if and only if P,Q,R lie on a line.

Note: Although the commutative law is obviously automatic here, it is not easy to check the associative law.

11.  The Group Negative

For a given point c,d on the cubic curve y2+a1xy+a3y=x3+a2x2+a4x+a6, its negative in the group law is the point c,d where d,d are the two roots of y2+a1c+a3y=c3+a2c2+a4c+a6, as a quadratic equation in y.

12.  Elliptic Curves

  • The non-singular cubic curves defined over K with at least one K-valued point are the “group objects” in the category of algebraic curves defined over K.

  • For a curve in generalized Weierstrass form, the required K-valued point may always be taken to be the distinguished point at infinity.

  • These are called elliptic curves.

  • When K=Q, much is known about them.

  • Modular forms — objects associated with hyperbolic geometry — provide a dictionary for elliptic curves defined over Q.

  • The Frey-Hellegouarch curve cannot be in that dictionary.

13.  The mod reduction of an elliptic curve

Let E be an elliptic curve of the form y2=xAxBxC where A,B,C are distinct integers. When is a prime not dividing Δ (the square of the product of the root differences), E determines also a curve E defined over the finite field F=ZZ.

E is non-singular when l is not a factor of Δ.

For our purposes, i.e., in the case of the Frey-Hellegouarch curve, the conductor N of E may be defined to be N=|Δ, the square-free part of Δ.

Let c be defined by c=1EF+ when lN. Here EF denotes the number of points of E in the field F.

c is defined in a slightly more complicated way for each of the finitely many primes dividing N.

14.  The L-series of E

One defines the “L-series” of E by forming the Euler product, indexed by primes as follows: LE,s=|N11csN11cs+12s

Expanding the product, one obtains a Dirichlet series LE,s=k=1ckks, which converges for Res>32

Series of this type have been seen in other contexts.

15.  Isometries of the Upper-Half Plane

Let H be H=τCImτ>0.

The group G=SL2R operates via M·τ=aτ+bcτ+d,M=abcd,a,b,c,dR,adbc=1

G±1 is the group of isometries (distance-preserving analytic maps) of H relative to the Poincaré metric ds2=dx2+dy2y2,forτ=x+iyH.

(This is the connection with “hyperbolic geometry”.)

16.  Family of Elliptic Curves over C

Let Gw denote the Eisenstein series Gwτ=const·m,nZ20,01mτ+nw, which converges normally for all τH,w4.

Gwτ is not identically 0 for even w4, while it is self-cancelling for odd w.

For given τ with g4τ=60G4τ, g6τ=140G6τ the equation y2=4x3g4τxg6τ gives rise to a cubic curve Cτ in classical Weierstrass form.

Every elliptic curve defined over C occurs this way, and one has CτCττ=M·τforMSL2Z.

Thus, over C isomorphism classes of elliptic curvesHSL2Z

17.  Modular Forms

The Eisenstein series are examples of modular forms: complex-valued holomorphic functions f in H satisfying fM·τ=cτ+dwfτ for M=abcdΓ,τH. where Γ is SL2Z or a subgroup of finite index in SL2Z.

  • The integer w is the weight of f.

  • Gw is a modular form of weight k.

  • A modular form is, more or less, a holomorphic section of a “line bundle” on the quotient space HΓ.

  • Modular forms are also required to be “holomorphic at cusps”, i.e., approach a finite limit at a “cusp” (see below).

18.  Action of SL2Z on H

The action of Γ=SL2Z on H is portrayed in this picture:

fundamental domain
(Wikipedia image licensed under GFDL)
  • The gray area is a fundamental domain. It has infinite extent.

  • HΓ is non-compact.

19.  Cusps Compactify the Quotient

Let Γ be a subgroup of finite index in Γ01=SL2Z.

  • HΓ “covers” HΓ01

  • Γ operates on H*=HQ.

  • For Γ=Γ1 the orbit of is Q.

  • For general Γ the number of orbits in Q is finite.

  • H*Γ compactifies HΓ by adjoining the finitely many “cusps”.

20.  Cusp Forms

Let Γ be a subgroup of finite index in Γ01=SL2Z. A modular form for Γ is a cusp form if its limiting value at each cusp of Γ is 0.

Example: For Γ=Γ01=SL2Z the modular form λτ=g4τ327g6τ2 is a cusp form of weight 12 — the smallest weight of a cusp form for Γ01.

21.  The Groups Γ0N

Let N1 be a positive integer. The group Γ0N is given by M=abcdSL2Zc0modN. In particular M1=1101Γ0Nfor allN1. If f is a modular form, then fM1·τ=fτ+1=fτ is periodic, so has a Fourier expansion fτ=kZcke2πikτ. Because f is holomorphic at the cusp one has ck=0 for k<0, and if f is a cusp form c0=0 so that then fτ=k=1cke2πikτ. N is called the level.

22.  The Dirichlet Series

There are certain operators, called Hecke operators Twmm1, that act semi-simply on the space of cusp forms for Γ0N not coming from levels dividing N.

The structure of the algebra of these operators shows that if fτ=k=1cke2πikτ is a cusp form of weight 2 that is a simultaneous eigenform of these operators then the corresponding Dirichlet series φfs=k=1ckks has an Euler product expansion just like the Euler product that is the L-function of an elliptic curve defined over Q: φfs=|N11csN11cs+12s

23.  Cusp Forms of Weight 2 on Γ0N

A cusp form f of weight 2 for Γ0N is essentially a regular differential on the quotient X0N=H*Γ0N. When f, not coming from levels dividing N, is an eigenform of the Hecke operators, it determines in a straightforward way a 1-dimensional quotient variety of the Jacobian variety J0N of X0N, X0NJ0NEf which quotient is an elliptic curve Ef defined over Q with conductor N, and, therefore a regular map – the modular parametrization of Ef – from X0N to Ef with the property that the unique (up to a constant) regular differential on Ef pulls back to the differential on X0N determined by f.

24.  The Dictionary for Elliptic Curves over Q

25.  Dictionary Trivia

  • 11 is the smallest value of N for which there is a non-zero cusp form of weight 2 for the group Γ0N. In this case the dimension of the space of cusp forms is 1. There are 3 non-isomorphic but isogenous elliptic curves with conductor 11:

  • The Cremona database — an encoding of the dictionary — has been built into Sage ( Documentation for its use may be found at

  • The smallest conductor having more than 1 isogeny class is 26, which has 2.

  • The smallest conductor having more than 2 isogeny classes is 57, which has 3.

  • There are 38402 isogeny classes with conductors smaller than 10000.

26.  For More Information

G. Cornell, J. Silverman, & G. Stevens,
Modular Forms and Fermat's Last Theorem,
Springer, 1997
— the record of an instructional conference held at Boston University in August, 1995

27.  Acknowledgement

The XHTML + MathML version of these slides uses W3C's Slidy by Dave Raggett, a JavaScript/CSS package for sizing and flow control of an HTML or XHTML slide show.
(The slides were generated in a non-standard fashion from GELLMU source.)