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

William F. Hammond

http://www.albany.edu/~hammond/

University at Albany Math Club
 
March, 2011

Abstract

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:

    y2+y=x3x210x20
    y2+y=x3x27820x263580
    y2+y=x3x2
  • The Cremona database — an encoding of the dictionary — has been built into Sage (http://www.sagemath.org/). Documentation for its use may be found at http://www.sagemath.org/doc/reference/sage/databases/cremona.html.

  • 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

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