Fermat's Last Theorem
After 356 Years

Comments for the GELLMU Edition

Among the challenges that I have been facing with my GELLMU project are (1) convincing mathematicians that it is possible to use comfortable LaTeX-like markup in a fully rigorous way to prepare our articles so that they can have formal inclusion in the markup category known as XML and (2) then convincing them that high quality typesetting may be obtained from the ensuing XML document instance. Toward this end I have revisited the LaTeX markup for the official notes on my October 1993 Albany seminar presentation and edited what was LaTeX source to convert it to LaTeX-like source markup for the article document type that is part of the GELLMU didactic markup production system. Information about this system and my reasons for developing it may be found at http://www.albany.edu/~hammond/gellmu/.

After the time of the original talk and the subsequent preparation of my original write-up, there was a time -- fortunately not long and also not to have been unexpected in the aftermath of so large a new development -- when Andrew Wiles's argument underwent some revision in collaboration with Richard Taylor. Questions about its soundness appeared to have ceased by the fall of 1994, and the work announced by Wiles in June 1993, as revised, was published in the May 1995 issue of the Annals of Mathematics.

There has also been discussion, at times appearing to approach controversy, about the name of the conjecture arising from the 1955 meeting in Japan. What I termed the “Shimura-Taniyama-Weil” conjecture became known as the “modular curve conjecture” and then, from the summer of 1999, as the “modular curve theorem” after the work of Breuil, Conrad, Diamond, and Taylor in the same vein as the work of Wiles and Taylor for the “semi-stable” case.

I list a few references on these matters for the period since my original talk:

A. Wiles, “Modular elliptic curves and Fermat's Last Theorem”, Annals of Mathematics, (second series) vol. 141 (1995), pp. 443-551.

R. Taylor & A. Wiles, “Ring-theoretic properties of certain Hecke algebras”, Annals of Mathematics, (second series) vol. 141 (1995), pp. 553-572.

H. Darmon, F. Diamond, & R. Taylor, “Fermat's Last Theorem”, Current Developments in Mathematics, 1995, International Press, Cambridge, Massachusetts, 1995.

G. Cornell, J. H. Silverman, & G. Stevens, Modular Forms and Fermat's Last Theorem, Springer-Verlag, 1997. This volume is the record of an instructional conference on number theory and arithmetic geometry held August 9-18, 1995 at Boston University.

J. Coates & S.T. Yau, Elliptic curves, modular forms, & Fermat's last theorem, 2nd edition, International Press, Cambridge, MA, 1997. Proceedings of the Conference on Elliptic Curves and Modular Forms held at the Chinese University of Hong Kong, Dec. 1993.

B. Conrad, F. Diamond, & R. Taylor, “Modularity of certain potentially Barsotti-Tate Galois representations”, J. Amer. Math. Soc. 12 (1999), no. 2, 521-567. In this article the modular curve conjecture is proved for any elliptic curve defined over $\mathbf{Q}$ with conductor not divisible by $27$.

C. Breuil, B. Conrad, F. Diamond, & R. Taylor, “On the modularity of elliptic curves over $\mathbf{Q}$: wild $3$-adic exercises”, J. Amer. Math. Soc., to appear.¹

What follows has the same content as the original write-up except that the title of the Appendix has been changed from “Current Status” to “Late 1993/Early 1994 Status”.

1.  Introduction.

The purpose of this expository lecture is to explain the basic ideas underlying the final resolution of “Fermat's Last Theorem” after 356 years as a consequence of the reported establishment by Andrew Wiles of a sufficient portion of the “Shimura-Taniyama-Weil” conjecture. As these notes are being written, the work of Wiles is not available, and the sources of information available to the author are (1) reports by electronic mail, (2) the AMS Notices article [16] of K. Ribet, and (3) a preprint [18] by K. Rubin and A. Silverberg based on the June, 1993 lectures of Wiles at the Newton Institute in Cambridge, England. It should be noted that the fact that “Fermat's Last Theorem” is a consequence of sufficient knowledge of the theory of “elliptic curves” has been fully documented in the publications ([14], [15]) of K. Ribet.

“Fermat's Last Theorem” is the statement, having origin with Pierre de Fermat in 1637, that there are no positive integers $x,y,z$ such that $x^n+y^n=z^n$ for any integer exponent $n>2$. Obviously, if there are no positive integer solutions $x,y,z$ for a particular $n$, then there are certainly none for exponents that are multiples of $n$. Since every integer $n>2$ is divisible either by $4$ or by some odd prime $p$, it follows that “Fermat's Last Theorem” is true if there are no solutions in positive integers of the equation $x^n+y^n=z^n$ when $n=4$ and when $n=p$ for each prime $p>2$. The cases $n=3,4$ are standard fare for textbooks (e.g., see Hardy & Wright [6]) in elementary number theory. Therefore, this discussion will focus on the case $n=p$ where $p>3$ is prime.

Very briefly, the idea is that we now know enough about the classification of non-degenerate plane cubic curves $F\left(x,y\right)=0$ in two variables, also known as “elliptic curves”, with rational coefficients to know how to enumerate them in a logical way so that we may conclude that if there were positive integers $a,b,c$ with $a^p+b^p=c^p$, then the curve $$y^2=x\left(x-a^p\right)\left(x+b^p\right),$$ which is an elliptic curve known as the “Frey curve”, would fall inside of the enumeration. Because the classification is enumerative, when one is presented with a particular elliptic curve with rational coefficients, one knows where to look for the curve in the classification. The curve just written is not to be found within the classification. As a consequence there cannot be positive integers $a,b,c$ with $a^p+b^p=c^p$.

The enumerative classification of non-degenerate plane cubic curves defined by polynomials with rational coefficients has been entirely conjectural (variously known as the “Taniyama Conjecture”, the “Weil Conjecture”, the “Taniyama-Shimura Conjecture”, $\dots$) until June, 1993. This conjecture, even as a conjecture, has served as an important motivating example for the idea of the “Langlands Program”, or perhaps of an extension of that program, that certain kinds of objects in geometry should give rise to certain group representations.

What seems to be believed today² is that the portion of the enumerative classification pertaining to “semi-stable” elliptic curves has been proved by Andrew Wiles. That the existence of positive integers $a,b,c$ with $a^p+b^p=c^p$ would violate the enumerative classification of semi-stable elliptic curves was established by 1987 through the work of G. Frey, J.-P. Serre, and K. Ribet.

The primary purpose of this lecture is to explain the enumerative classification of elliptic curves and to give a brief indication of the mathematics involved in showing that the Frey curve violates that classification.

2.  Elliptic curves

A polynomial $f\left(X,Y\right)$ of degree $d$ in two variables with coefficients in a field $k$ gives rise to what is called an affine plane curve of degree $d$: for each field $K$ containing $k$ (more generally, for each commutative ring that is a $k$-algebra) one has the set $$C_0\left(K\right)=\{\left(x,y\right)\in K^2|f\left(x,y\right)=0\},$$ and for each $k$-linear homomorphism $K\to K^{\prime }$ one has the induced map $C_0\left(K\right)\to C_0\left(K^{\prime }\right)$. From the polynomial $f$ one obtains a homogeneous polynomial of degree d in three variables with coefficients in $k$: $$F\left(X,Y,Z\right)=Z^{d}f\left(X⁄Z,Y⁄Z\right),$$ and the projective plane curve of degree $d$: $$C\left(K\right)=\{\left(\left(x,y,z\right)\right)\in {\mathbf{P}}^2\left(K\right)|F\left(x,y,z\right)=0\},$$ where ${\mathbf{P}}^N\left(K\right)$ denotes $N$-dimensional projective space, which is the quotient set of $K^{N+1}-\left\{0\right\}$ obtained by identifying points lying on the same line through the origin of $K^{N+1}$. Since the projective plane ${\mathbf{P}}^2\left(K\right)$ is the disjoint union of the affine plane $\left\{\left(\left(x,y,1\right)\right)\right|\left(x,y\right)\in K^2\}$ with the “(projective) line at infinity” $$\left\{\left(\left(x,y,0\right)\right)|\left(\left(x,y\right)\right)\in {\mathbf{P}}^1\left(K\right)\right\},$$ it follows that $C\left(K\right)$ is the disjoint union of $C_0\left(K\right)$ with the finite set of its points lying on the projective line at infinity.

An elliptic curve defined over $k$ is the (projective) plane curve $E$ given by a homogeneous polynomial $F$ of degree $3$ in three variables with coefficients in $k$ such that (i) $F$ is irreducible over the algebraic closure $\overline{k}$ of $k$, (ii) the gradient vector $\nabla F$ is a non-vanishing vector at points of ${\overline{k}}^3-\left\{0\right\}$ where $F$ vanishes, and (iii) the set $E\left(k\right)$ is non-empty.

If $k$ is any field, then after an isomorphism (see Silverman [27]) one may obtain a given elliptic curve $E$ with an affine equation of the form $$\begin{array}{cc}\text{(1)}& y^2+a_{1}xy+a_{3}y=x^3+a_2{x}^2+a_{4}x+a_6\text{.}\end{array}$$ Then the homogeneous equation for the intersection of $E\left(K\right)$ with the line at infinity is $$\begin{array}{cc}\text{(2)}& x^3=0\text{.}\end{array}$$ Thus, in this case, $E$ has a unique point on the line at infinity. If the characteristic of $k$ is different from $2$ and $3$³ then one may obtain an equation in “Weierstrass normal form”: $$\begin{array}{cc}\text{(3)}& y^2=4x^3-g_{2}x-g_3,\end{array}$$ which is non-singular if and only if the cubic polynomial in the variable $x$ has distinct roots in $\overline{k}$.

Elliptic curves are the “group objects” in the category of algebraic curves that reside in projective space: for each extension field $K$ of $k$ the set $E\left(K\right)$ of “$K$-valued points” of $E$ is an abelian group. The group law on $E\left(K\right)$ is characterized by two conditions:

1. The origin is a given point of $E\left(k\right)$.

2. The points obtained by intersecting $E\left(K\right)$ with any line in ${\mathbf{P}}^2\left(K\right)$, counted with multiplicities, add up to zero.

When $E$ is given by an equation in the form (1), the origin is usually taken to be the unique point on the line at infinity. If two distinct points of $E\left(K\right)$ are given, they determine a line in ${\mathbf{P}}^2\left(K\right)$; the intersection of that line with $E\left(K\right)$ is given by a cubic polynomial in a parameter for the line which has two roots in $K$ corresponding to the two given points; hence, there is a third root of that cubic polynomial in $K$; this root gives rise to a point of $E\left(K\right)$, which is the negative of the sum of the two given points. The negative of a given point of $E\left(K\right)$ is obtained as the third point in the intersection with $E\left(K\right)$ of the line through the given point and the origin.

For a given field $k$ the set of homogeneous cubic polynomials in three variables is a vector space over $k$ having the set of “monomials” of degree three in three variables as basis. Thus, the dimension of the space of homogeneous cubics is 10. The linear group ${\mathrm{GL}}_3\left(k\right)$ acts on the space of cubics, and two cubic curves in ${\mathbf{P}}^2$ that are related by this action are isomorphic. Since ${\mathrm{GL}}_3\left(k\right)$ is 9-dimensional, one is led to think of the family of isomorphism classes of elliptic curves as 1-dimensional since “non-singularity” is an “open” condition.

3.  Elliptic curves over $\mathbf{C}$

When $k$ is the field $\mathbf{C}$ of complex numbers, one knows (see, e.g., Ahlfors [1]) that for each lattice $\Lambda$ in $\mathbf{C}$ the set of $\Lambda$-periodic meromorphic functions on the complex line $\mathbf{C}$ is the field $\mathbf{C}\left(\wp ,{\wp }^{\prime }\right)$, which is a quadratic extension of the rational function field $\mathbf{C}\left(\wp \right)$, where $\wp$ is the $\wp$-function of Weierstrass. Moreover, $\wp$ satisfies the famous Weierstrass differential equation $$\begin{array}{cc}\text{(4)}& {\wp }^{\prime }{\left(z\right)}^2=4\wp {\left(z\right)}^3-g_2\left(\Lambda \right)\wp \left(z\right)-g_3\left(\Lambda \right);\end{array}$$ thus, the formula $z\mapsto \left(\wp \left(z\right),{\wp }^{\prime }\left(z\right)\right)$ defines a holomorphic map from the punctured complex torus $\mathbf{C}⁄\Lambda -\left\{0\right\}$ to the affine cubic curve $$\begin{array}{cc}\text{(5)}& y^2=4x^3-g_2\left(\Lambda \right)x-g_3\left(\Lambda \right);\end{array}$$ it should hardly be necessary to point out that this map extends to a holomorphic map from the torus $\mathbf{C}⁄\Lambda$ to the corresponding (projective) elliptic curve by sending the origin of the torus to the unique point of the elliptic curve on the line at infinity. The classical theory of theta functions (see, e.g., Igusa [7] or Siegel [26]) leads to a direct demonstration that this map is a homomorphism from the group law on the complex torus to the group law previously described for an elliptic curve. It is not difficult to see that the analytic manifold given by any elliptic curve defined over $\mathbf{C}$ arises from some complex torus. Indeed each non-singular cubic curve $E$ in ${\mathbf{P}}^2\left(\mathbf{C}\right)$ determines a compact connected complex-analytic group. Its universal cover is given by a holomorphic homomorphism $\mathbf{C}\to E$ which has some lattice as kernel.

Any two lattices in $\mathbf{C}$ are related by a change of real basis for $\mathbf{C}$, i.e., by a matrix in ${\mathrm{GL}}_2\left(\mathbf{R}\right)$. Consequently, there is only one real-analytic isomorphism class for the complex torus $\mathbf{C}⁄\Lambda$ as $\Lambda$ varies. The tori corresponding to two lattices are complex-analytically isomorphic if and only if the corresponding real-linear isomorphism of ${\mathbf{R}}^2$ satisfies the Cauchy-Riemann partial differential equations, i.e., if and only if the $\mathbf{R}$-linear isomorphism is $\mathbf{C}$-linear.

A lattice $\Lambda$ may be represented concretely by an ordered basis $\{{\omega }_1,{\omega }_2\}$. If $\tau ={\omega }_2⁄{\omega }_1$, then $\tau$ is not real, and after permuting the basis members, if necessary, one may assume that $\tau$ is in the “upper-half plane”⁴ $H$ of $\mathbf{C}$. Observing that $\Lambda$ is the image under the $\mathbf{C}$-linear map $z\mapsto {\omega }_{1}z$ of the lattice with ordered basis $\{1,\tau \}$, one may assume that $\Lambda$ is this latter lattice. Let $E\left(\tau \right)$ be the complex torus $\mathbf{C}⁄\Lambda$. Allowing for change of basis subject to these assumptions on the basis, one sees that there is an isomorphism of complex-analytic groups $E\left({\tau }^{\prime }\right)\cong E\left(\tau \right)$ if $$\begin{array}{cc}\text{(6)}& {\tau }^{\prime }=\frac{a\tau +b}{c\tau +d},\end{array}$$ for some matrix $$\gamma =\left(\begin{array}{cc}\hfill a& \hfill b\\ \hfill c& \hfill d\end{array}\right)\in {\mathrm{SL}}_2\left(\mathbf{Z}\right)\text{.}$$ Conversely, the monodromy principle may be used to show that every complex-analytic isomorphism among the complex tori $E\left(\tau \right)$ arises in this way.

The coefficients $g_2$ and $g_3$ in the Weierstrass normal form (5) have very explicit constructions as infinite series (see, e.g., Ahlfors [1] or Serre [20]) determined by the given lattice; from this it is straightforward to see that $g_w$ is a modular form of weight $2w$: if $\tau$ and ${\tau }^{\prime }$ are related by (6), then $$g_w\left({\tau }^{\prime }\right)={\left(\lambda \right)}^{2w}{g}_w\left(\tau \right),\lambda =c\tau +d.$$ Consequently, the map $$\left(x,y\right)\mapsto \left({\lambda }^{2}x,{\lambda }^{3}y\right)$$ carries the curve given by (5) for $\tau$ isomorphically to the curve given by (5) for ${\tau }^{\prime }$. The discriminant of the cubic polynomial in the Weierstrass normal form (5) is a modular form of weight $12$, which up to a multiplicative constant, is: $$\Delta \left(\tau \right)={g_2}^3-27{g_3}^2.$$ $\Delta$ is a non-vanishing holomorphic function in $H$. The modular invariant $j\left(\right)$ ([20],[24]) is defined by: $$j\left(\tau \right)=\frac{{\left(12g_2\right)}^3}{\Delta };$$ it is a holomorphic function in the upper-half plane $H$ with the property that $$j\left(\tau \right)=j\left({\tau }^{\prime }\right)$$ if and only if $\tau$ and ${\tau }^{\prime }$ are related by (6). Furthermore, $j$ assumes every value in $\mathbf{C}$ at some point of $H$. Consequently, the complex-analytic isomorphism classes of complex tori or, equivalently, the isomorphism classes of elliptic curves defined over $\mathbf{C}$, are parameterized via $j$ in a one-to-one manner by the complex numbers.

Since this is an expository discourse, it is hoped that the reader will not feel patronized by having noted the fact that the coincidence of (1) the category of elliptic curves over $\mathbf{C}$ and (2) the category of complex tori is the “genus one” case of the coincidence (see Weyl [33]) of (i) the category of “complete” non-singular algebraic curves over $\mathbf{C}$ and (ii) the category of compact Riemann surfaces (one-dimensional connected complex-analytic manifolds).

Although the classification of elliptic curves over $\mathbf{C}$ via the $j$-function is a result that is both beautiful and useful, and although two elliptic curves defined over $\mathbf{Q}$ that are isomorphic as curves defined over $\mathbf{Q}$ give rise to elliptic curves defined over $\mathbf{C}$ that have the same $j$-invariant, it is not true that any two elliptic curves defined over $\mathbf{Q}$ having the same $j$-invariant are isomorphic over $\mathbf{Q}$. Thus, the classification of elliptic curves over $\mathbf{C}$ does not lead directly to the desired enumerative classification of elliptic curves defined over $\mathbf{Q}$, but it does bring to the fore the notion of modular form, which is central in the study of elliptic curves defined over $\mathbf{Q}$. What can be said easily is that, according to the Shimura-Taniyama-Weil conjecture, the isogeny classes of elliptic curves defined over $\mathbf{Q}$ are parameterized by certain modular forms.

4.  Modular forms

The group ${\mathrm{SL}}_2\left(\mathbf{Z}\right)$ is an infinite group that is generated by the two elements $$\left(\begin{array}{cc}\hfill 0& \hfill 1\\ \hfill -1& \hfill 0\end{array}\right),\left(\begin{array}{cc}\hfill 0& \hfill -1\\ \hfill 1& \hfill 1\end{array}\right),$$ which have orders 4 and 6 respectively. The action of ${\mathrm{SL}}_2\left(\mathbf{Z}\right)$ on the upper-half plane $H$ by linear fractional transformations has kernel $$\left\{\pm \left(\begin{array}{cc}\hfill 1& \hfill 0\\ \hfill 0& \hfill 1\end{array}\right)\right\},$$ and the quotient of ${\mathrm{SL}}_2\left(\mathbf{Z}\right)$ by this kernel is the group ${\mathrm{PSL}}_2\left(\mathbf{Z}\right)$. It is not difficult to see that the set $$\{\tau \in H|-1⁄2\le \mathrm{Re}\left(\tau \right)\le 1⁄2,\left|\tau \right|\ge 1\}$$ is a “fundamental domain” for the action of ${\mathrm{PSL}}_2\left(\mathbf{Z}\right)$ on $H$. More precisely, this set meets each orbit, and the only redundancies are the boundary identifications arising from the maps $\tau \mapsto \tau +1$ and $\tau \mapsto -1⁄\tau$. The quotient $H⁄{\mathrm{PSL}}_2\left(\mathbf{Z}\right)$ is not compact since the fundamental domain is “open at the top”. Beyond that the modular invariant $j$ induces a bicontinuous biholomorphic isomorphism of the quotient $H⁄{\mathrm{PSL}}_2\left(\mathbf{Z}\right)$ with the affine line over $\mathbf{C}$. Since $j\left(\tau +1\right)=j\left(\tau \right)$, and since for $q=e^{2\pi i\tau }$ one has $\left|q\right|<1$ for $\tau \in H$, there is a holomorphic function $\stackrel{\sim }{j}$ in the punctured unit disk such that $\stackrel{\sim }{j}\left(q\right)=j\left(\tau \right)$. Likewise $\Delta$ may be regarded as function of $q$, and one may use the calculus of residues to show that $\Delta$ has a simple zero at $q=0$; hence, $\stackrel{\sim }{j}$ has a simple pole at $q=0$, or, equivalently, $j$ has a simple pole at $\infty$ (the “missing top” of the fundamental domain). Thus, $j$ gives rise to a bicontinuous biholomorphic isomorphism $$H⁄{\mathrm{PSL}}_2\left(\mathbf{Z}\right)\cup \left\{\infty \right\}\to {\mathbf{P}}^1\left(\mathbf{C}\right).$$

A non-trivial element of ${\mathrm{PSL}}_2\left(\mathbf{Z}\right)$ has a fixed point in $H$ if and only if it has finite order, and one's explicit knowledge of the fundamental domain makes it possible to see that the only elements of finite order are of order $2$ or $3$⁵. A congruence subgroup of ${\mathrm{SL}}_2\left(\mathbf{Z}\right)$ is a subgroup $\Gamma$ that contains one of the principal congruence subgroups; the principal congruence subgroup $\Gamma \left(N\right)$ of level $N$ is the set of all elements $\gamma$ of ${\mathrm{SL}}_2\left(\mathbf{Z}\right)$ that are congruent $\left(modN\right)$ to the identity matrix. The group ${\Gamma }_0\left(N\right)$ is the congruence subgroup of ${\mathrm{SL}}_2\left(\mathbf{Z}\right)$ consisting of all elements $$\left(\begin{array}{cc}\hfill a& \hfill b\\ \hfill c& \hfill d\end{array}\right)$$ for which $c\equiv 0\left(modN\right)$. It is obvious that each congruence group $\Gamma$ has finite index in ${\mathrm{SL}}_2\left(\mathbf{Z}\right)$, and, consequently the quotient $H⁄\Gamma$ is a non-compact Riemann surface. Observe that for each level $N$ the group ${\Gamma }_0\left(N\right)$ contains the parabolic element $$T=\left(\begin{array}{cc}\hfill 1& \hfill 1\\ \hfill 0& \hfill 1\end{array}\right),$$ which gives rise to the holomorphic map $\tau \mapsto \tau +1$ that fixes the point $\infty$.

A modular form⁶ of weight $w$ for $\Gamma$ is a holomorphic function $f$ in $H$ that satisfies the functional equation $$\begin{array}{cc}\text{(7)}& f\left(\gamma ·\tau \right)={\left(c\tau +d\right)}^{w}f\left(\tau \right),\gamma \in \Gamma \end{array}$$ and that is holomorphic at each cusp of $\Gamma$. The role of cusps for $\Gamma$ is to provide a slightly larger set $H^{*}$ than $H$, $$H^{*}=H\cup \left\{\text{cusps}\right\},$$ where $\Gamma$ acts such that $H^{*}⁄\Gamma$ is a compact Riemann surface containing $H⁄\Gamma$ as the open complement of a finite set of points arising from cusps. The cusps of $\Gamma$ are the points of the closure of the boundary of $H$ in ${\mathbf{P}}^1\left(\mathbf{C}\right)$ = $\mathbf{C}\cup \left\{\infty \right\}$ that are fixed by some non-trivial parabolic element of $\Gamma$. When $\Gamma$ = ${\mathrm{SL}}_2\left(\mathbf{Z}\right)$, the set of cusps is $\mathbf{Q}\cup \left\{\infty \right\}$. In view of (7) applied to the case $\gamma =T$ one sees that a modular form $f$ of any weight for the group ${\Gamma }_0\left(N\right)$ satisfies $$\begin{array}{cc}\text{(8)}& f\left(\tau +1\right)=f\left(\tau \right),\end{array}$$ and, therefore, $f$ defines a holomorphic function in the variable $q=e^{2\pi i\tau }$ for $0<q<1$. The condition in the definition of modular form that $f$ should be holomorphic at $\infty$ means that $f$ as a function of $q$ is holomorphic at $q=0$. Consequently, $f$ admits an absolutely convergent Fourier expansion $$\begin{array}{cc}\text{(9)}& f\left(\tau \right)=\sum _{m=0}^{\infty }{c}_m{e}^{2\pi im\tau },\end{array}$$ which is a Taylor series in $q$.

For any cusp $\rho$ of a congruence group $\Gamma$ one may define the notion holomorphic at $\rho$ for a modular form $f$ by an analogous procedure using an arbitrary parabolic element of $\Gamma$ that fixes $\rho$ instead of $T$. For a given congruence group $\Gamma$ two cusps $\rho$ and $\sigma$ are equivalent if there is some element $\gamma$ in $\Gamma$ such that $\sigma =\gamma ·\rho$. A modular form $f$ is holomorphic at any cusp that is equivalent to another where it is holomorphic. The modular form $f$ is a cuspform if, in addition to being holomorphic at each cusp, $f$ vanishes at each cusp. For a given congruence group $\Gamma$ a modular form vanishes at any cusp that is equivalent to another where it vanishes. The set of modular forms of given weight $w$ forms a finite-dimensional vector space over $\mathbf{C}$ in which the set of cuspforms is a linear subspace of codimension bounded by the number of equivalence classes of cusps. In fact, using “Eisenstein series” one may show that the codimension of the space of cuspforms in the space of modular forms is often equal to the number of equivalence classes of cusps. For example, with the group $\Gamma \left(1\right)={\mathrm{SL}}_2\left(\mathbf{Z}\right)$ there are no modular forms of odd weight, there is an Eisenstein series of every even weight greater than $2$ that is not a cuspform, and every cusp is equivalent to $\infty$. Furthermore, since $\infty$ is the only zero of the cusp form $\Delta$ (of the preceding section) in the quotient $H^{*}⁄\Gamma \left(1\right)$ and since $\infty$ is a simple zero of $\Delta$, every cuspform for $\Gamma \left(1\right)$ is divisible by $\Delta$. Thus, in this case, there are no cuspforms of weight less than $12$.

It is not difficult to see that the cuspforms of weight $2$ for a congruence group $\Gamma$ correspond to holomorphic differential $1$-forms (differentials of the first kind) on the compact Riemann surface $X=H^{*}⁄\Gamma$. Thus, the dimension of the space of cuspforms of weight $2$ is the genus of $X$. The fact that there are no cuspforms of weight $2$ for the group $\Gamma \left(1\right)$ matches the previously mentioned fact that $X$ is ${\mathbf{P}}^1$. It is certain of the cuspforms of weight two for the groups ${\Gamma }_0\left(N\right)$ that, according to the Shimura-Taniyama-Weil conjecture, parameterize the isogeny classes of elliptic curves defined over $\mathbf{Q}$.

5.  Euler products

It will be recalled that the infinite series $$\sum _{n=1}^{\infty }\frac{1}{n^s}$$ converges for $\mathrm{Re}\left(s\right)>1$ and gives rise by analytic continuation to a meromorphic function $\zeta \left(s\right)$ in $\mathbf{C}$. For $\mathrm{Re}\left(s\right)>1$ $\zeta \left(s\right)$ admits the absolutely convergent infinite product expansion $$\prod _p\frac{1}{1-p^{-s}},$$ taken over the set of primes. This “Euler product” may be regarded as an analtyic formulation of the principle of unique factorization in the ring $\mathbf{Z}$ of integers. It is, as well, the product taken over all the non-archimedean completions of the rational field $\mathbf{Q}$ (which completions ${\mathbf{Q}}_p$ are indexed by the set of primes) of the “Mellin transform”⁷ in ${\mathbf{Q}}_p$ $${\xi }_p\left(s\right)=\frac{1}{1-p^{-s}},$$ of the canonical “Gaussian density” $${\Phi }_p\left(x\right)=\left\{\begin{array}{cc}1\hfill & \text{if }x\in \text{ closure of}\mathbf{Z}\text{in}{\mathbf{Q}}_p\text{.}\hfill \\ 0\hfill & \text{otherwise.}\hfill \end{array}\right.,$$ which Gaussian density is equal to its own Fourier transform. For the archimedean completion ${\mathbf{Q}}_{\infty }=\mathbf{R}$ of the rational field $\mathbf{Q}$ one forms the classical Mellin transform $${\xi }_{\infty }\left(s\right)={\pi }^{-\left(s⁄2\right)}\Gamma \left(s⁄2\right)$$ of the classical Gaussian density $${\Phi }_{\infty }\left(x\right)=e^{-\pi x^2},$$ (which also is equal to its own Fourier transform). Then the function $$\xi \left(s\right)={\xi }_{\infty }\left(s\right)\zeta \left(s\right)=\prod _{p\le \infty }{\xi }_p\left(s\right)$$ is meromorphic in $\mathbf{C}$, and satisfies the functional equation $$\begin{array}{cc}\text{(10)}& \xi \left(1-s\right)=\xi \left(s\right)\text{.}\end{array}$$

The connection of Riemann's $\zeta$-function with the subject of modular forms begins with the observation that $\zeta \left(2s\right)$ is essentially the Mellin transform of ${\theta }_I\left(x\right)=\theta \left(ix\right)-1$, where $\theta$, which is a modular form of weight $1⁄2$ and level $8$, is defined in the upper-half plane $H$ by the formula $$\theta \left(\tau \right)=\sum _{m\in \mathbf{Z}}\exp \left(\pi i\tau m^2\right).$$ In fact, one of the classical proofs of the functional equation (10) is given by applying the Poisson summation formula⁸ to the function $x\mapsto \exp \left(\pi i\tau x^2\right)$, while observing that the substitution $s\mapsto \left(1⁄2\right)-s$ for $\zeta \left(2s\right)$ corresponds in the upper-half plane to the substitution $\tau \mapsto -1⁄\tau$ for the theta series.

If $f$ is a cuspform for a congruence group $\Gamma$ containing $$T=\left(\begin{array}{cc}\hfill 1& \hfill 1\\ \hfill 0& \hfill 1\end{array}\right),$$ and so, consequently, $f\left(\tau +1\right)=f\left(\tau \right)$, then, as previously explained, one has the Fourier expansion (9) $$f\left(\tau \right)=\sum _{m=1}^{\infty }{c}_m{e}^{2\pi im\tau }.$$ The Mellin transform $\phi \left(s\right)$ of $f_I$ leads to the Dirichlet series $$\begin{array}{cc}\text{(11)}& \phi \left(s\right)=\sum _{m=1}^{\infty }{c}_m{m}^{-s},\end{array}$$ which may be seen to have a positive abscissa of convergence. One is led to the questions:

1. For which cuspforms $f$ does the associated Dirichlet series $\phi \left(s\right)$ admit an analytic continuation with functional equation?

2. For which cuspforms $f$ does the associated Dirichlet series $\phi \left(s\right)$ have an Euler product expansion?

For the “modular group” $\Gamma \left(1\right)$ the Dirichlet series associated to every cuspform of weight $w$ admits an analtyic continution with functional equation under the substitution $s\mapsto w-s$. Since $\Gamma \left(1\right)$ is generated by the two matrices $T$ and $$W=\left(\begin{array}{cc}\hfill 0& \hfill 1\\ \hfill -1& \hfill 0\end{array}\right)$$ and since the functional equation of a modular form $f$ relative to $T$ is reflected in the formation of the Fourier series (9), the condition that an absolutely convergent series (9) is a modular form for $\Gamma \left(1\right)$ is the functional equation for a modular form relative solely to $W$. This is equivalent to the (properly formulated) functional equation for the associated Dirichlet series $\phi$ together with a “growth condition”. For the group ${\Gamma }_0\left(N\right)$, with $N>1$, the question of a functional equation is more complicated since, although $T$ is available, there is no reason for a cuspform to satisfy a law of transformation relative to $W$. But note that for any $\Gamma$ the set of cuspforms of given weight for which the associated Dirichlet series have analytic continuations satisfying a given finite set of functional equations is a vector space. On the other hand, there is no reason to believe, even for level $1$, that the cuspforms admitting an Euler product expansion form a vector space.

In a nutshell the cuspforms admitting Euler products are those which arise as eigenforms for an arithmetically defined commutative algebra of semi-simple operators on the space of cuspforms of a given weight introduced by E. Hecke. The theory of Hecke operators is reasonably simple for level $1$ but somewhat more complicated in general (see, e.g., Shimura's book [24]).

Observing that the formula $$d{s}^2=\frac{d{x}^2+d{y}^2}{y^2},\text{for}\tau =x+iy\in H,$$ gives a (the hyperbolic) ${\mathrm{SL}}_2\left(\mathbf{R}\right)$-invariant metric in $H$ with associated invariant measure $$d\mu =\frac{dxdy}{y^2},$$ one introduces the Petersson (Hermitian) inner product in the space of cuspforms of weight $w$ for $\Gamma$ with the definition: $$\begin{array}{cc}\text{(12)}& \langle f,g\rangle ={\int }_{H⁄\Gamma }f\left(\tau \right)\overline{g}\left(\tau \right){\mathrm{Im}\left(\tau \right)}^{w}d\mu \left(\tau \right)\text{.}\end{array}$$ (Integration over the quotient $H⁄\Gamma$ makes sense since the integrand $$f\left(\tau \right)\overline{g}\left(\tau \right)y^w$$ is $\Gamma$-invariant.)

For the modular group $\Gamma \left(1\right)$ the $n^{th}$ Hecke operator $T\left(n\right)=T_w\left(n\right)$ is the linear endomorphism of the space of cuspforms of weight $w$ arising from the following considerations. Let $S_n$ be the set of $2\times 2$ matrices in $\mathbf{Z}$ with determinant $n$. For $$M=\left(\begin{array}{cc}\hfill a& \hfill b\\ \hfill c& \hfill d\end{array}\right)\in S_n$$ and for a function $f$ in $H$ one defines $$\begin{array}{cc}\text{(13)}& \left(M{·}_{w}f\right)\left(\tau \right)=\det {\left(M\right)}^{w-1}{\left(c\tau +d\right)}^{-w}f\left(\tau \right),\end{array}$$ and then, observing that $\Gamma \left(1\right)$ under ${·}_w$ acts trivially on the modular forms of weight $w$, one may define the Hecke operator $T_w\left(n\right)$ by $$\begin{array}{cc}\text{(14)}& T_w\left(n\right)\left(f\right)=\sum _{M\in S_n⁄\Gamma \left(1\right)}\left(M{·}_{w}f\right)\left(\tau \right),\end{array}$$ where the quotient $S_n⁄\Gamma \left(1\right)$ refers to the action of $\Gamma \left(1\right)$ by left multiplication on the set $S_n$. One finds for $m$, $n$ coprime that $$T\left(mn\right)=T\left(m\right)T\left(n\right),$$ and furthermore one has $$T\left(p^{e+1}\right)=T\left(p^e\right)T\left(p\right)-p^{w-1}T\left(p^{e-1}\right).$$ Consequently, the operators $T\left(n\right)$ commute with each other, and, therefore, generate a commutative algebra of endomorphisms of the space of cusp forms of weight $w$ for $\Gamma \left(1\right)$. It is not difficult to see that the Hecke operators are self-adjoint for the Petersson inner product on the space of cuspforms. Consequently, the space of cuspforms of weight $w$ admits a basis of simultaneous eigenforms for the Hecke algebra. A “Hecke eigencuspform” is said to be normalized if its Fourier coefficient $c_1=1$. If $f$ is a normalized Hecke eigencuspform, then

• The Fourier coefficient $c_m$ of $f$ is the eigenvalue of $f$ for $T\left(m\right)$.

• The Fourier coefficients $c\left(m\right)=c_m$ of $f$ satisfy

  $c\left(mn\right)=c\left(m\right)c\left(n\right)$ for $m,n$ coprime, and

  $c\left(p^{e+1}\right)=c\left(p^e\right)c\left(p\right)-p^{w-1}c\left(p^{e-1}\right)$ for $p$ prime.

Consequently, the Dirichlet series associated with a simultaneous Hecke eigencuspform of level $1$ and weight $w$ admits an Euler product $$\begin{array}{cc}\text{(15)}& \phi \left(s\right)=\prod _p\frac{1}{1-c_p{p}^{-s}+p^{w-1-2s}}\text{.}\end{array}$$ For example, when $f$ is the unique normalized cuspform $\Delta$ of level $1$ and weight $12$, one has $$\phi \left(s\right)=\prod _p\frac{1}{1-\tau \left(p\right)p^{-s}+p^{11-2s}},$$ where $c_p=\tau \left(p\right)$ is the function $\tau$ of Ramanujan.

For the congruence group ${\Gamma }_0\left(N\right)$ a Hecke eigencuspform of weight $w$ gives rise to a Dirichlet series $\phi \left(s\right)$ that admits an Euler product expansion whose factors at primes $p$ coprime to $N$ resemble those given by (15). In order for $\phi \left(s\right)$ to satisfy a functional equation under the substitution $s\mapsto w-s$, one needs to require that the eigencuspform $f$ admits a functional equation not only with respect to each element of the group ${\Gamma }_0\left(N\right)$ but also with respect to the substitution in the upper-half plane $H$ given by the matrix $$W_N=\left(\begin{array}{cc}\hfill 0& \hfill -1\\ \hfill N& \hfill 0\end{array}\right).$$ A. Weil ([31]) showed that the cuspforms of weight $2$ for the group ${\Gamma }_0\left(N\right)$ satisfying the appropriate functional equation under the mapping of $H$ given by $W_N$ correspond precisely to Dirichlet series with certain growth conditions that admit analytic continuations as meromorphic functions in $\mathbf{C}$ satisfying a finite number of “twisted” functional equations.

The reader will have noticed that it is not extremely easy to characterize the cuspforms of weight $2$ that conjecturally (Shimura-Taniyama-Weil) parameterize the isogeny classes of elliptic curves defined over the rational field $\mathbf{Q}$. The Euler product is an extremely important part of the characterization since the Dirichlet series given by such an elliptic curve, as will be made explicit in the next section, is, by its very nature, an Euler product. Weil conjectures explicitly that the Dirichlet series with Euler product given by each elliptic curve defined over $\mathbf{Q}$ satisfies these conditions, i.e., is the Dirichlet series associated to some $W_N$-compatible Hecke eigencuspform for the group ${\Gamma }_0\left(N\right)$, where $N$ is the “conductor” of $E$. This has led to efforts, related to the “Langlands program” to understand the $W_N$-compatible Hecke eigencuspforms in a more intrinsic way as objects of representation theory over $\mathbf{Q}$ (see, e.g., the survey of Gelbart [4]).

6.  Elliptic curves over the rational field $\mathbf{Q}$

Let $E$ be an elliptic curve defined over $\mathbf{Q}$. One may clear denominators from its cubic equation, if necessary, in order to arrive at an equation with integer coefficients having no common factor. While the Weierstrass normal form (3) is available to represent the isomorphism class of any elliptic curve over a field of characteristic different from $2$ and $3$, one needs the generalized Weierstrass form $$\begin{array}{cc}\text{(16)}& y^2+a_{1}xy+a_{3}y=x^3+a_2{x}^2+a_{4}x+a_6\end{array}$$ over an arbitrary field, and, moreover, for each elliptic curve $E$ defined over $\mathbf{Q}$ there is a “best possible” equation (e.g., see Silverman [27]) of the form (16) with integer coefficients called the Neron model of $E$. With an abuse of notation $E$ will denote the Neron model, which may be regarded as a “curve over $\mathbf{Z}$”. (One would want to call it an “elliptic curve over $\mathbf{Z}$” if it were “smooth over $\mathbf{Z}$”, i.e., if it had good reduction at each prime $p$; the fact that every Neron model has bad reduction at least once corresponds under the “dictionary” to the fact that there are no cuspforms of weight two and level 1.) It then may be observed that for each prime $p$ the Neron model gives rise to a cubic equation over the finite field ${\mathbf{F}}_p$. For all but a finite number of $p$ the equation over ${\mathbf{F}}_p$ is non-singular over ${\overline{\mathbf{F}}}_p$, i.e., determines an elliptic curve $E_p$ defined over ${\mathbf{F}}_p$. One says in this case that $E$ has “good reduction” mod $p$. Following Tate ([30]) one introduces $$b_2={a_1}^2+4a_2,$$ $$b_4=a_1{a}_3+2a_4,$$ $$b_6={a_3}^2+4a_6\text{, and}$$ $$b_8=b_2{a}_6-a_1{a}_3{a}_4+a_2{a_3}^2-{a_4}^2.$$ Then one has $$\Delta =-{b_2}^2{b}_8-8{b_4}^3-27{b_6}^2+9b_2{b}_4{b}_6.$$ The non-vanishing of $\Delta$ mod $p$ is necessary and sufficient for $E$ to have good reduction mod $p$. It follows that a prime $p$ divides $\Delta$ if and only if $E$ does not have good reduction mod $p$. If $p$ is a prime for which $E$ has “bad reduction”, then there is a single singular point of the reduced curve $E_p$, and either (a) $E_p$ has distinct tangent lines at the singular point (semi-stable reduction) or (b) $E_p$ has a single tangent line occurring with multiplicity $2$. $E$ is called semi-stable if it has either good or semi-stable reduction at each prime. The conductor of $E$ is the integer $N$ defined by $$N=\prod _p{p}^{{\nu }_p},$$ where $${\nu }_p=\left\{\begin{array}{cc}0\hfill & \text{if }E\text{ has good reduction at }p\text{.}\hfill \\ 1\hfill & \text{if }E\text{ has semi-stable reduction at }p\text{.}\hfill \\ 2+{\lambda }_p\ge 2\hfill & \text{otherwise.}\hfill \end{array}\right.$$ The non-negative integer ${\lambda }_p$ cannot be positive unless $p$ is $2$ or $3$. Tautologically, $E$ is semi-stable if and only if its conductor $N$ is square-free.

One defines the “L-series” of $E$ by $$\begin{array}{cc}\text{(17)}& L\left(E,s\right)=\prod _{p|N}\frac{1}{1-c_p{p}^{-s}}\prod _{p\nmid N}\frac{1}{1-c_p{p}^{-s}+p^{1-2s}},\end{array}$$ where $c_p$ is defined when $E$ has good reduction mod $p$ by the formula $$c_p=p+1-\left|E\left({\mathbf{F}}_p\right)\right|,$$ and $c_p$ is defined when $E$ has bad reduction mod $p$ by $$c_p=\left\{\begin{array}{cc}1\hfill & \text{if }{\nu }_p=1\text{ and the tangents are defined over }{\mathbf{F}}_p\text{.}\hfill \\ -1\hfill & \text{if }{\nu }_p=1\text{ with “irrational” tangents.}\hfill \\ 0\hfill & \text{if }{\nu }_p>1\text{.}\hfill \end{array}\right.$$ One observes readily that the L-function of $E$ codifies information about the number of points of $E$ in the finite field ${\mathbf{F}}_p$. Quite generally for an algebraic variety defined over $\mathbf{Q}$ the analogous codification of information obtained by counting points in the various reductions mod $p$ of the variety yields the “Hasse-Weil zeta function”, which reflects “cohomological” information about $E$. The L-series of $E$ is the essential part, corresponding to cohomology in dimension $1$, of the Hasse-Weil zeta function of $E$. The Hasse-Weil zeta function is a special case of the general notion (Serre [19]) of “zeta function” for a scheme of finite type over $\mathbf{Z}$.

One observes that $L\left(E,s\right)$ resembles, at least insofar as one considers its Euler factors for primes $p$ corresponding to good reductions of $E$, the Dirichlet series associated to a cuspform of weight $2$ that admits an Euler product expansion. The observation of this resemblance is the beginning of an appreciation of the Shimura-Taniyama-Weil conjecture. One is led to ask to what extent the two classes of Dirichlet series with Euler products coincide. The conjecture states that the L-function of an elliptic curve defined over $\mathbf{Q}$ with conductor $N$ arises from a cuspform for the group ${\Gamma }_0\left(N\right)$ that is compatible with the substitution in the upper-half plane