% AMS-TeX \documentstyle{amsppt} % \nologo % sorry -- delete this if you wish \magnification=\magstep1 \leftheadtext{Kenneth A. Ribet} \rightheadtext{Progress in Mathematics} \newcount\sectCount % We make a non-outer version of \head: \edef\nonouterhead{\noexpand\head} \def\section#1\endhead{\advance\sectCount by 1% \nonouterhead\the\sectCount. #1\endhead\noindent} % As usual, we automatically number the references \newcount\refCount \def\newref#1 {\advance\refCount by 1 \expandafter\edef\csname#1\endcsname{\the\refCount}} \newref AL %Atkin-Lehner \newref BOSMA %Bosma-Lenstra \newref CAREARLY \newref COX \newref DIAMONDREF \newref FALT \newref FLACH \newref FREYONE \newref FREYTWO \newref GELBART \newref LABESSE %actually by Gerardin and Labesse \newref GOUVEA \newref HAYESRIBET \newref HEARSTRIBET \newref HINDRY \newref AXJ % Allyn Jackson \newref LANGBOOK \newref LANG \newref LANGLANDS \newref HWL \newref MATSUMURA \newref EIS \newref MAZUR %deformation theory \newref GAD \newref MAZURNOTES \newref MAZURTILOUINE \newref MIYAKE \newref OESTERLE \newref PRASAD \newref RIBETSURVEY \newref FERMAT \newref TOULOUSE \newref KOREA \newref NOTICES \newref MOTIVES \newref RUBINSILVER \newref RSNOTE \newref SERRETAU \newref SERREOLD \newref SERRECOURS \newref SERREIM \newref ARCATA \newref DUKE \newref ST \newref SHCRELLE \newref SHIMURABOOK \newref SHIMURAPINK \newref TUNNELL \newref WILES % some of my math macros, most not used \let\phi\varphi \def\U#1{\Z/#1\Z} % integers mod #1 without ( ) \def\UU#1{(\Z/#1\Z)^\ast} %units mod #1 \def\Q{{\bold Q}} \def\Z{{\bold Z}} \def\C{{\bold C}} \def\Qbar{\overline{\Q}} \def\F{{\bold F}} \def\Fbar{\overline{\F}} \def\Fq{\F_q} \def\T{{\Bbb T}} \def\m{{\goth m}} \def\jo#1{J_0(#1)} \def\go#1{\Gamma_0(#1)} \def\ssigma{{}^\sigma\mkern-\thinmuskip} \def\hlinefill{\leaders\hrule height 3pt depth -2.5pt\hfill} \def\emrule{\thinspace\hbox to 0.75em{\hlinefill}\thinspace} \def\GalQ{\Gal(\Qbar/\Q)} \def\seteq{\mathrel{:=}} \def\SL#1{{\bold{SL}}(2,#1)} \def\GL#1{{\bold{GL}}(2,#1)} \def\GLTWO{{\bold{GL}}(2)} % provisional definition -- can one do better? \def\notdivide{\setbox1=\hbox{$|$\llap{%\raise2pt -- nah! \hbox{/}\kern-0.75pt}}\mathrel{\box1}} %% Guillemets \`a la fran\c{c}aise %% by jsv@cs.brown.edu and brouard@frined51 %% modified by bnb@math.ams.com on friday, 10 November 1989 \def\<<{\leavevmode \raise0.28ex\hbox{$\scriptscriptstyle\langle\!\langle$}\nobreak \hskip -.6pt plus.3pt minus.2pt$\,$} \def\>>{$\,$\nobreak\hskip -.6pt plus.3pt minus.2pt \raise0.28ex\hbox{$\scriptscriptstyle\rangle\!\rangle$}} % make the \circ a little smaller \let\tempcirc=\circ \def\circ{\mathord{\raise0.25ex\hbox{$\scriptscriptstyle\tempcirc$}}} % make some mathops \def\newop#1 {\expandafter\def\csname #1\endcsname{\mathop{\roman{#1}}\nolimits}} \newop Aut \newop Gal \newop End \newop Frob \newop tr \topmatter \title Galois representations and modular forms\endtitle \author Kenneth A. Ribet\endauthor \affil University of California, Berkeley\endaffil \address{UC Mathematics Department, Berkeley, CA 94720-3840 USA}\endaddress \email ribet\@math.berkeley.edu \endemail \thanks This work was to some extent supported by NSF Grant \#DMS 93-06898. I am grateful to B.~Mazur, C.~O'Neil, R.~Taylor and A.~Wilkinson for comments and suggestions on drafts of these notes. A special thanks to S.~Ribet for moral support. \endthanks \endtopmatter \document Last fall, the organizers of this Mathfest asked me to discuss ``the mathematics behind Andrew Wiles's solution of the Fermat conjecture.'' At the time it was anticipated that Fermat's Last Theorem would be proved by the methods which Wiles outlined in his lectures last summer. The situation changed dramatically in December when Wiles posted a message to the USENET news group {\tt sci.math} stating that ``the final calculation of a precise upper bound for the Selmer group\dots\ is not yet complete as it stands'' (see \cite\AXJ). At this writing, the required upper bound has not been obtained\emrule Fermat's conjecture is not a theorem. Nevertheless, it is my opinion that the methods introduced by Wiles are significant and continue to merit our attention. The most striking consequence of Wiles's work is that one now has tools for exhibiting infinite families of elliptic curves over the rational field which are attached to modular forms~\cite\RSNOTE. No such families were known before last summer. Because of the intense publicity surrounding Fermat's Last Theorem, most of the background material in these notes has been discussed in news and expository articles which were written on the heels of Wiles's announcement. Among these are the author's news item in the Notices~\cite\NOTICES\ and his article with Brian Hayes in American Scientist~\cite\HAYESRIBET, two pieces in the American Mathematical Monthly \cite{\COX, \GOUVEA}, the report by K.~Rubin and A.~Silverberg in the AMS Bulletin~\cite\RUBINSILVER, and unpublished notes by various authors which can be obtained by gopher from {\tt e-math.ams.org}. In view of the large literature in this subject, I imagined these notes more as an annotated bibliography than as a survey. In the end, it seems that I have written an abbreviated survey which has a relatively large list of~references. I have tried to cite some papers which are still in preparation, or have not yet appeared. \medskip I hope that these notes will be of interest to Mathfest participants. In all likelihood, a corrected version of this text will appear in the AMS Bulletin. So if you find misprints, ambiguities or errors, or if you have suggestions for improvements, please let me know about them either in Minneapolis or by e-mail. My oral presentation on August 17 will provide a general overview, slanted toward developments which have emerged in the last year. \section Modular forms\endhead We begin with some background concerning modular forms. Among reference books in the subject, one might cite \cite\LANGBOOK, \cite\MIYAKE, and~\cite\SHIMURABOOK. For a first reading in the subject, a good starting point is~\cite{\SERRECOURS, Ch.~VII}. The modular forms we will consider are holomorphic functions of one variable, defined on the upper half-plane of complex numbers with positive imaginary part. These functions are usually presented as converging Fourier series $\sum_{n=1}^\infty a_n q^n$, where $q=e^{2\pi i z}$. For the forms which most interest us, the coefficients $a_n$, which are a priori complex numbers, are in fact algebraic integers and often even ordinary integers. Suppose that $N$ is a positive integer. Let $\go N$ be the group of integer matrices with determinant~1 which are upper-triangular mod~$N$. The weight-two cusp forms on~$\go N$ are holomorphic functions $f(z)$ on the complex upper half-plane which obey the functional equation $$f\left({az+b\over cz+d}\right)= (cz+d)^2 f(z)$$ for all $\pmatrix a & b\cr c& d\cr\endpmatrix\in\go N$. This equation implies, of course, that $f(z)$ is invariant under integer translation. In addition to the holomorphy and the functional equation, one imposes a suitable condition at infinity \cite{\SHIMURABOOK, Ch.~2}. The cusp forms of weight two on~$\go N$ form a complex vector space $S(N)$. The dimension of~$S(N)$ is finite and may be calculated easily using the Riemann-Roch theorem. The dimension is~zero for $N\le10$ and is~1 for~$N=11$. A generator for~$S(11)$ may be described in terms of the formal power series with integral coefficients $\sum a_n X^n$ defined by the identity $$ X \prod_{m=1}^\infty (1-X^m)^2(1-X^{11m})^2 = \sum_{n=1}^\infty a_nX^n.$$ The holomorphic function $\sum a_n q^n $ (with $q=e^{2\pi i z}$) is a cusp form $h$ of weight two on~$\go{11}$, i.e., a non-zero element of~$S(11)$. Anticipating somewhat, we refer the reader to Shimura's article~\cite\SHCRELLE, which discusses the relationship between $h$ and a certain elliptic curve, and introduces the Galois groups $G_n$ which appear below. For each integer $n\ge1$, the $n$th {\it Hecke operator\/} on~$S(N)$ is an endomorphism $T_n$ of~$S(N)$, whose action is generally written $f\mapsto f|T_n$. The various $T_n$ commute with each other and are interrelated by identities which express a given $T_n$ in terms of the Hecke operators indexed by the prime factors of~$n$. If $p$ is a prime, the operator $T_p$ may be described easily from the perspective of Fourier coefficients. Namely, if if $p$ is not a divisor of~$N$ and $f=\sum a_n q^n$, then $f|T_p$ has the Fourier expansion $$ \sum_{n=1}^\infty a_{np} q^n + p \sum_{n=1}^\infty a_nq^{pn}.$$ If $p|N$, then $f|T_p$ is given by the simpler formula $\sum_{n=1}^\infty a_{np} q^n$. The elements of~$S(N)$ having special arithmetic interest are the {\it normalized eigenforms\/} in~$S(N)$; these are the non-zero cusp forms $f=\sum a_nq^n$ which are eigenvectors for all the $T_n$ and which satisfy the normalizing condition $a_1=1$. If $f$ is such an eigenform, its Fourier coefficients and its eigenvalues coincide: One has $f|T_n = a_n f$ for all $n\ge1$. The normalized eigenforms are ``arithmetic'' because the~$a_n$ belong to the realm of algebraic number theory: If $f$ is a normalized eigenform, then the subfield of~$\C$ generated by the $a_n$ is a finite (algebraic) extension $E$ of~$\Q$ in~$\C$, and the elements $a_n$ of~$E$ are algebraic integers. It is a mildly complicating fact that the normalized eigenforms in~$S(N)$ do not always form a basis of~$S(N)$. In other words, the operators $T_n$ are not necessarily diagonalizable. This problem arises from those $n$ which have a common factor with~$N$, and can be repaired by the introduction of {\it newforms\/}~\cite\AL. Briefly, a newform is a normalized eigenform $f=\sum a_n q^n$ for which the space $\{\, g\in S(N) \, | \, g|T_n = a_n g \hbox{ for all $n$ prime to $N$}\,\}$ contains only $f$ and its multiples. Atkin and Lehner have shown that $S(N)$ has a basis built out of suitable transforms of the newforms in the spaces $S(M)$, where $M$ runs over the positive divisors of~$N$. Thus $S(N)$ may be described purely in terms of newforms provided that one includes newforms whose ``levels'' $M$ run over the divisors of~$N$. \section The Taniyama-Shimura conjecture\endhead The Taniyama-Shimura conjecture relates elliptic curves over~$\Q$ and certain modular forms. Here is a compact description of the conjecture; another is provided in the recent article by K.~Rubin and A.~Silverberg~\cite{\RUBINSILVER, \S1}. To describe the T-S conjecture, we need to allude to some fundamental concepts pertaining to elliptic curves. For a serious treatment of these concepts, the reader may consult a large number of textbooks and monographs which focus on elliptic curves. (Several of these books discuss modular forms as well.) Rather than list these books here, we refer the reader to the bibliography of a book review by W.~Hearst~III and the author~\cite\HEARSTRIBET. To describe the conjecture, we begin with a construction due to Shimura. Suppose that $f\in S(N)$ is a normalized eigenform and let $E$ be the field generated by the coefficients of~$f$. Shimura has associated to~$f$ an abelian variety $A_f$ over~$\Q$ whose dimension is the degree $[E\:\Q]$ and whose arithmetic incorporates the eigenvalues $a_p$ for $p$ prime to~$N$ \cite{\SHIMURABOOK, Th.~7.14}. %check this The construction $f\mapsto A_f$ is sufficiently flabby that it associates to $f$ a ``clump'' of {\it isogenous\/} abelian varieties, rather than a specific abelian variety which is singled out up to isomorphism. (By definition, two abelian varieties are isogenous if and only if they are obtained from each other by dividing by a finite group. A theorem of Faltings~\cite{\FALT, \S5} justifies the vague statement that two abelian varieties are isogenous if and only if they are equivalent arithmetically.) If $f$ has integral Fourier coefficients, then $E=\Q$ and $A_f$ has dimension~1. This means that $A_f$ is an {\it elliptic curve\/} over~$\Q$, which we can regard as being defined up to isogeny. The eigenvalues $a_p$ can be extracted from the elliptic curve $A=A_f$ via the process of ``reduction mod~$p$.'' More precisely, it is possible to find cubic polynomials with integral coefficients which define~$A$. To each such polynomial, one associates a non-zero discriminant. There is an essentially unique defining equation whose discriminant has smallest absolute value; the discriminant of this equation is said to be the (minimal) discriminant $\Delta$ of~$A$. The prime numbers which divide $\Delta$ are the primes at which $A$ has {\it bad reduction}; the others are the primes at which $A$ has {\it good reduction}. For good primes~$p$, the minimal equation, when viewed mod~$p$, yields an elliptic curve $\tilde{A}_p$ over the finite field $\Z/p\Z$. The reduced curve $\tilde A_p$ over~$\Z/p\Z$. is a projective plane curve over~$\Z/p\Z$; its {\it rational points}, i.e., points with values in~$\Z/p\Z$, form a finite abelian group which we can call~$A(\Z/p\Z)$. A theorem of H.~Hasse states that the integer $$b_p\seteq p{+}1-\#(A(\Z/p\Z))$$ is bounded in absolute value by~$2\sqrt p$. The relation between $f$ and $A=A_f$ is as follows. Firstly, if $p$ does not divide~$N$, then $A$ has good reduction at~$p$. Further, for such $p$, the $p$th Fourier coefficient of~$f$ coincides with the quantity $p{+}1-\#(A(\Z/p\Z))$. In other words, we have $b_p=a_p$ for all $p\notdivide N$. (If $A_f$ has dimension greater than~1, the relation between $f$ and $A_f$ is a bit more complicated to describe, but involves no essentially new ideas.) Incidentally, each elliptic curve $A$ over~$\Q$ has a {\it conductor}, which is a positive integer divisible precisely by the primes at which $A$ has bad reduction. (For the definition, see for example~\cite{\SERREOLD, \S2}.) The conductor and the minimal discriminant of~$A$ are divisible by the same set of prime numbers. Nevertheless, these integers have a completely different feel. For one thing, the conductor is always positive, while the minimal discriminant may be positive or negative. Further, while the latter number may be divisible by a large power of a given prime, the conductor of an elliptic curve tends to be divisible only by low powers of its prime divisors. (In fact, a prime $p\ge5$ can divide the conductor at most to the second power.) Finally, the conductor of an elliptic curve depends only on its $\Q$-isogeny class; the discriminant changes under isogeny. It happens quite frequently that the conductor is square free, i.e., divisible only by the first powers of its prime divisors. The elliptic curve $A$ is then said to be {\it semistable}. Suppose again that $f$ is an eigenform in~$S(N)$ with integral Fourier coefficients and that $A$ is the elliptic curve~$A_f$. It is natural to ask how the conductor of~$A$ is related to~$N$. The answer begins with the fact that $f$ is necessarily built from a newform in~$S(M)$, for some divisor $M$ of~$N$. It is then true that the conductor of~$A$ is precisely this divisor $M$. This theorem was proved by H.~Carayol in~\cite\CAREARLY, following work of Shimura, Igusa, Deligne and Langlands. The Taniyama-Shimura conjecture has several equivalent formulations. In one form, it states that every elliptic curve over~$\Q$ is isogenous to a some curve $A_f$. A more precise statement runs as follows: {\it Let $A$ be given over~$\Q$ and suppose that $N$ is the conductor of~$A$. Define integers $b_p$ for $p\notdivide N$ as above. Then there is a newform $f\in S(N)$ whose Fourier coefficients $a_n$ satisfy the identity $a_p=b_p$ for $p\notdivide N$.} The theory of newforms shows that $f$ is unique, if it exists. Also, if $f$ exists, we may apply ``Tate's conjecture"\emrule now a theorem of Faltings~\cite{\FALT, \S5}\emrule to show that $A$ and~$A_f$ are isogenous over~$\Q$. If the Taniyama-Shimura conjecture is true for~$A$, we say that $A$ is {\it modular}. Mazur's article~\cite\GAD\ formulates the Taniyama-Shimura conjecture purely as a statement about the Riemann surface associated with an elliptic curve over~$\Q$. If $A$ is such an elliptic curve, write $A(\C)$ for the space of points of~$A$ with complex coordinates, so that $A(\C)$ is a complex torus. For each integer~$N$, consider the group $\Gamma(N)$ consisting of integer matrices with determinant~1 which are congruent to the identity matrix modulo~$N$. This group acts naturally on the complex upper half-plane $\Cal H$ by fractional linear transformations, with $\pmatrix a & b\cr c& d\cr\endpmatrix$ giving rise to $$z\mapsto {az+b\over cz+d}.$$ The quotient $\Gamma(N)\backslash\Cal H$ has a natural compactification called~$X(N)$. Mazur shows that an elliptic curve $A$ over~$\Q$ is modular if and only if one can find a non-constant holomorphic map from $X(N)$ to~$A(\C)$, for some positive integer~$N$. In~\cite\KOREA, I consider the set of elliptic curves $A$ over~$\C$ for which there is a non-constant holomorphic map $X(N)\to A(\C)$ for some positive integer~$N$. Answering a question of Serre, I show that the Taniyama-Shimura conjecture may be generalized to a natural conjectural characterization of such elliptic curves. \section Galois representations attached to elliptic curves\endhead Let $A$ be an elliptic curve over~$\Q$. We view $A$ as a commutative {\it algebraic group\/} over~$\Q$. Concretely, one models $A$ as the projective plane curve defined by an affine cubic equation of the form $y^2=x^3+ax+b$ with $a,b\in\Q$. If $K$ is a field containing~$\Q$, the set of points of~$A$ with values in~$K$ is then a subset $A(K)$ of the projective plane over~$K$ with a classical group law, the so-called ``chord and tangent operation.'' There is a unique point $O$ of~$A$ outside the affine plane; this is taken to be the identity element in the group. Three distinct points on $A$ then sum to~$O$ if and only if they are colinear. The composition law on~$A(K)$ can be described in terms of coordinates by a family of polynomial equations with coefficients in~$\Q$. (For a recent discussion concerning families of such equations, the reader may consult~\cite\BOSMA.) A model for $A(K)$ when $K=\C$ is given by Weierstra\ss\ theory: the group $A(\C)$ is the complex torus~$\C/L$, where $L$ is the lattice of periods associated to the given cubic equation. (Explicitly, $L$ is obtained by integrating the differential ${\displaystyle dx\over\displaystyle y}$ on~$A$ over the free abelian group $H_1(A(\C),\Z)$ of rank~two.) Let $n$ be a positive integer, and let $A[n]$ be the group of elements of~$A(\C)$ whose order divides~$n$. This group of {\it $n$-division points\/} on~$A$ may be modeled as ${1\over n}L/L$; it is therefore a free module of rank two over $\U n$, since $L$ is free of rank two over~$\Z$. Further analysis shows that $A[n]$ in fact lies in~$A(\Qbar)$, where $\Qbar$ is the subfield of~$\C$ consisting of all algebraic numbers. Indeed, that $A[n]$ is finite and stable under all automorphisms of~$\C$ implies that none of its elements has a transcendental coordinate. Let $\GalQ$ be the group of automorphisms of~$\Qbar$. Then $A[n]$ is stable under the natural action of~$\GalQ$ on~$A(\Qbar)$, since a point $P\in A(\Qbar)$ has order dividing~$n$ if and only if its coordinates satisfy a certain set of polynomial equations with rational coefficients. This means that $A[n]$ is equipped with a canonical action of the Galois group $\GalQ$. Notice that for each $\sigma\in\GalQ$ the automorphism $P\mapsto\ssigma P$ is a {\it group\/} automorphism of~$A[n]$: We have $\ssigma(P+Q)=\ssigma P+\ssigma Q$ for $P,Q\in A[n]$. This means that when we view the action of~$\GalQ$ on~$A[n]$ as a (continuous) homomorphism $$ \rho_{A,n}\:\GalQ\to\Aut(A[n]),$$ we are entitled to let $\Aut(A[n])$ stand for the group of automorphisms of~$A[n]$ as an {\it abelian group}. We do this. Then since $A[n]$ is isomorphic to the group $(\U n)^2$, $\Aut(A[n])$ is isomorphic to the group $\GL{\U n}$ of two-by-two invertible matrices with coefficients in~$\U n$. While there is no canonical isomorphism $\Aut(A[n])\approx \GL{\U n}$, each choice of basis $A[n]\approx (\U n)^2$ determines such an isomorphism, and the various isomorphisms obtained in this way differ by inner automorphisms of~$\GL{\U n}$. Therefore, each element of $\Aut(A[n])$ has a well-defined trace and determinant in~$\U n$. It is often fruitful to fix a choice of basis $A[n]\approx (\U n)^2$ and to view $\rho_{A,n}$ as taking values in the matrix group $\GL{\U n}$. The kernel of $\rho_{A,n}$ corresponds, via Galois theory, to a finite Galois extension $K$ of~$\Q$ in~$\Qbar$. Concretely, $K$ is the extension of~$\Q$ obtained by adjoining to the rational field the coordinates of the various points in~$A[n]$. The Galois group $\Gal(K/\Q)$ is the image of $\rho_{A,n}$; it is therefore embedded in the target group $\GL{\U n}$. The elliptic curve $A$ and the positive integer $n$ have given rise to a finite Galois extension $K/\Q$ whose Galois group is a subgroup $G_n$ of the group of two-by-two invertible matrices with coefficients in~$\U n$. It is natural to ask for a description of this~$G_n$ as a subgroup of~$\GL{\U n}$. There is a (relatively rare) special case to consider: that where $A$ has {\it complex multiplication\/} (over~$\C$). Viewing $A$ as~$\C/L$, we may define the complex multiplication case to be that for which there is a non-integral complex number $\alpha$ such that $\alpha L\subseteq L$. The group $G_n$ then has an abelian subgroup of index~$\le2$, so it is quite far from the ambient group $\GL{\U n}$. In the much more common case where $A$ has no complex multiplication, Serre has shown that the index of~$G_n$ in~$\GL{\U n}$ is bounded as a function of~$n$. An essentially equivalent piece of information is that one has the equality $G_p = \GL{\U p}$ for all but finitely many primes~$p$ (see \cite\SERREIM). To orient the reader, we might point out that the Taniyama-Shimura conjecture was proved for complex multiplication elliptic curves over~$\Q$ by Shimura in 1971 \cite\SHIMURAPINK. On the other hand, it is relatively easy to show that semistable elliptic curves over~$\Q$ {\it never\/} have complex multiplication. As we shall see below, the elliptic curves which appear in connection with Fermat's Last Theorem are semistable elliptic curves. Accordingly, the result of~\cite\SHIMURAPINK\ sheds no light on the question of whether or not they are modular. A key piece of information about the extension $K/\Q$ (which depends on~$n$ and~$A$) is that its discriminant is divisible only by those prime numbers which divide the product of~$n$ and the conductor of~$A$. In other words, if $p\notdivide n$ is a prime number at which $A$ has good reduction, then $K/\Q$ is unramified at~$p$. In this case, one can introduce a Frobenius element $\sigma_p$ in~$G_n$ which is well defined up to conjugation. Since $G_n$ is a group of matrices, we may view $\sigma_p$ as a matrix whose trace and determinant are well-defined elements of~$\U n$. It is easy to prove that $\det \sigma_p$ is the number $p$ mod~$n$. On the other hand, one has the striking congruence $$ \tr(\sigma_p) \equiv b_p \hbox{ mod }n,$$ where $b_p$ is the number $p{+}1-\#(A(\Z/p\Z))$ introduced above. This means that the representation $\rho_{A,n}$ encapsulates information about the numbers $b_p$ (for $p$ prime to~$n$); more precisely, it determines the numbers $b_p$ mod~$n$. \section Galois representations attached to modular forms\endhead Suppose that $f\in S(N)$ is a normalized eigenform. If the coefficients $a_n$ of~$f$ happen to be integers, the abelian variety $A=A_f$ is an elliptic curve. By considering the family $\rho_{A,n}$, we obtain a series of representations of the Galois group~$\GalQ$. These representations are related to~$f$ by the congruence $ \tr(\sigma_p) \equiv a_p \hbox{ mod }n$, valid for the $n$th representation and all primes $p\notdivide nN$. We are especially interested in the case where $n$ is a prime number~$\ell$; the ring $\U n$ is then the {\it finite field}~$\F_\ell$. The representations $\rho$ are associated to~$A$, which in turn arises from~$f$. Hence we are tempted to write $\rho_{f,\ell}$ for the representations $\rho_{A,\ell}$. The obstacle to doing this arises from the circumstance that $A$ is determined up to isogeny, but not always up to isomorphism. If we replace $A$ by an isogenous elliptic curve, the representations $\rho$ may change! To circumvent this difficulty, we introduce the process of ``semisimplification.'' If $\rho$ is a two-dimensional representation of a group over a field, $\rho$ is either irreducible, or else ``upper-triangular,'' i.e., an extension of a one-dimensional representation $\alpha$ by another, $\beta$. In the case where $\rho$ is irreducible, we declare its semisimplification to be $\rho$ itself. In the reducible case, the semisimplification of~$\rho$ is the {\it direct sum\/} of the two one-dimensional representations $\alpha$ and~$\beta$. Clearly, the trace and determinant are the same for~$\rho$ and for its semisimplification. One shows easily that the semisimplification of~$\rho_{A,\ell}$ depends only on~$f$ and on~$\ell$ (but not on the choice of~$A$). Introducing $$ \rho_{f,\ell} \seteq \hbox{semisimplification of }\rho_{A,\ell},$$ we obtain a sequence of semisimple representations of~$\GalQ$ which are well defined up to isomorphism. The characteristic property of~$\rho_{f,\ell}$ may be summarized in terms of Frobenius elements $\Frob_p$ in the Galois group $\GalQ$: If $p$ is a prime number not dividing $\ell N$, then $\rho_{f,\ell}(\Frob_p)$ has trace $a_p$ mod~$\ell$ and determinant $p$ mod~$\ell$. Our aim will be to reverse this process: Given a two-dimensional representation of~$\GalQ$ over~$\Z/\ell\Z$, we would like to show in some circumstances that this representation arises from some eigenform~$f$. Examples show that it is unreasonable to expect $f$ to have integral coefficients; we must confront the situation where the coefficients of~$f$ are algebraic integers but not ordinary integers. Suppose then that $f=\sum a_n q^n$ is a normalized eigenform, and let $E$ be the field generated by the~$a_n$. Let $\Cal O$ be the ring of integers of~$E$, so that $a_n\in\Cal O$ for all $n\ge1$. Using the abelian variety $A_f$, one constructs representations indexed not by the prime {\it numbers}, but rather by the non-zero prime {\it ideals\/} of~$\Cal O$. If $\lambda$ is such a prime, its residue field $\F_\lambda$ is a finite field, say of characteristic~$\ell$. The prime field $\F_\ell=\U\ell$ is then canonically embedded in~$\F_\lambda$. For each $\lambda$, one finds a semisimple representation $\rho_{f,\lambda}\:\GalQ\to\GL{\F_\lambda}$ which is characterized up to isomorphism by the following property. If $p$ is a prime number not dividing $\ell N$, then $\rho_{f,\lambda}(\Frob_p)$ has trace $a_p$ mod~$\lambda$ and determinant $p$ mod~$\lambda$. The assertion concerning the determinant of the matrices~$\rho_{f,\lambda}(\Frob_p)$ may be rephrased in terms of the determinant of the representation~$\rho_{f,\lambda}$. The latter determinant is a continuous homomorphism $\GalQ\to\F^\ast_\lambda$. This homomorphism turns out to be the composite of the ``cyclotomic character'' $\chi_\ell\:\GalQ\to\F_\ell^\ast$ and the natural inclusion $\F_\ell^\ast\subseteq\F_\lambda^\ast$. The cyclotomic character may be defined by considering the group $\mu_\ell$ of $\ell$th roots of unity in~$\Qbar$. The Galois group $\GalQ$ acts on~$\mu_\ell$ by conjugation, giving rise to a continuous homomorphism $$ \GalQ\to\Aut(\mu_\ell).$$ Since $\mu_\ell$ is a cyclic group of order~$\ell$, its group of automorphisms is the group $\UU\ell=\F_\ell^\ast$. We emerge with a map $\GalQ\to\F_\ell^\ast$, which is the character in question. Suppose now that $c\in\GalQ$ is the automorphism ``complex conjugation.'' Then the determinant of~$\rho_{f,\lambda}(c)$ is~$\chi_\ell(c)$. Now $c$ operates on roots of unity by the map $\zeta\mapsto\zeta^{-1}$, since roots of unity have absolute value~1. Thus we have $\rho_{f,\lambda}(c)=-1$; one says that $\rho_{f,\lambda}$ is {\it odd}. This parity observation generalizes to modular forms of weights other than~two. Here is a quick synopsis of the situation; for details and further references, one may consult~\cite\RIBETSURVEY. For integers $k\ge2$, $N\ge1$ and characters $\epsilon\:\UU N\to\C^\ast$, one considers the space $S_k(N,\epsilon)$ of weight-$k$ cusp forms with character $\epsilon$ on~$\go N$; we have $S(N)=S_2(N,1)$. This space is automatically zero unless $\epsilon(-1)=(-1)^k$, so we will assume that this parity condition is satisfied. The space $S_k(N,\epsilon)$ admits an operation of Hecke operators $T_n$, and we again have the concept of a normalized eigenform in~$S_k(N,\epsilon)$. If $f=\sum a_nq^n$ is such a form, the numbers $a_n$ ($n\ge1$) and the values of~$\epsilon$ all lie in a single integer ring~$\Cal O$. For each non-zero prime ideal $\lambda$ of~$\Cal O$, one constructs a semisimple representation $$\rho_{f,\lambda}\:\GalQ\to\GL{\F_\lambda}.$$ Let $\ell$ again denote the characteristic of~$\F_\lambda$. Then for all $p\notdivide \ell N$, the trace of~$\rho_{f,\lambda}(\Frob_p)$ is again $a_p$ mod~$\lambda$. The determinant of this matrix is $p^{k-1}\epsilon(p)$ mod~$\lambda$. Once the proper definition is made, the determinant of the {\it map\/} $\rho_{f,\lambda}$ becomes the product $\chi_\ell^{k-1}\epsilon$, where $\chi_\ell$ again denotes the mod~$\ell$ cyclotomic character. For this, we regard $\epsilon$ as a map $\GalQ\to\F_\lambda^\ast$. To do this, we compose the natural map $\GalQ\to\UU N$ which gives the action of~$\GalQ$ on the $N$th roots of unity with the character $\epsilon$, thus obtaining a map $\GalQ\to{\Cal O}^\ast$. On reducing this homomorphism mod~$\lambda$, we obtain the desired variant of~$\epsilon$. Evaluating the determinant formula on the ``complex conjugation'' element of~$\GalQ$, we find $$\det(\rho_{f,\lambda}(c)) = (-1)^{k-1}\epsilon(c) = -1.$$ In these equalities, we exploit the fact that $\epsilon(c)$ is another name for~$\epsilon(-1)$ and remember the parity condition $\epsilon(-1)=(-1)^k$. The upshot of this is that the representations $\rho_{f,\lambda}$ are always odd, even in the generalized set-up. Serre has conjectured that all two-dimensional odd irreducible representations of~$\GalQ$ over a finite field are essentially of the form~$\rho_{f,\lambda}$ \cite\DUKE. To give meaning to this statement, we choose a prime number $\ell$ and let $\F$ be an algebraic closure of the prime field $\F_\ell$. Suppose that $\rho\:\GalQ\to\GL\F$ is an odd continuous irreducible representation. We shall say that $\rho$ is {\it modular\/} if one can find: (i) an eigenform $f$ in some space $S_k(N,\epsilon)$; (ii) a prime $\lambda$ dividing $\ell$ in the ring of integers $\Cal O$ associated to~$f$; and (iii) an embedding $\F_\lambda\hookrightarrow\F$ such that $\rho$ is isomorphic to the representation obtained by composing $\rho_{f,\lambda}$ with the inclusion $\GL{\F_\lambda}\hookrightarrow\GL\F$ associated with~(iii). Serre has conjectured that every two-dimensional odd irreducible representation of~$\GalQ$ is modular. It has emerged that this seemingly innocuous conjecture is equivalent to a much more precise statement (also given in~\cite\DUKE), which immediately implies the Taniyama-Shimura conjecture, Fermat's Last Theorem, and a host of other assertions! \section Frey's construction\endhead Almost all recent work on Fermat's Last Theorem begins with the connection between Fermat's equation and elliptic curves. Although Y.~Hellegouarch and others had noted such connections, a decisive step was taken by G.~Frey in an unpublished 1985 manuscript entitled ``Modular elliptic curves and Fermat's conjecture.'' Frey's idea runs as follows. Suppose that we have a non-trivial solution to Fermat's equation $X^\ell+Y^\ell=Z^\ell$. We can assume that the exponent is a prime number different from 2 and~3 and that the solution is given by a triple of relatively prime integers. Changing the sign in the value of~$Z$, we obtain a triple $(a,b,c)$ which satisfies the symmetric equation $a^\ell+b^\ell+c^\ell=0$. The equation $y^2=x(x-a^\ell)(x+b^\ell)$ defines an elliptic curve $A$ with unexpected properties. These properties are catalogued in~\S4.1 of Serre's article~\cite\DUKE. After a possible permutation of the three integers $a$, $b$ and~$c$ defining~$A$, one finds that the conductor of~$A$ is the product of the prime numbers which divide $abc$ (each occurring to the first power), so that $A$ is semistable. On the other hand, the minimal discriminant $\Delta$ of~$A$ is the quotient of~$(abc)^{2\ell}$ by the trivial factor $2^8$. From our point of view, the main ``unexpected'' property of~$A$ is that $\Delta$ is the product of a power of~2 and a perfect $\ell$th power, where $\ell$ is a prime~$\ge5$. (From the point of view of Szpiro's conjecture and the $abc$ conjecture, the surprising feature of~$A$ is that its discriminant is larger than a high power of its conductor.) This property persuaded Frey that $A$ was unlikely to be modular. In other words, Frey concluded heuristically that the existence of~$A$ was incompatible with the Taniyama-Shimura conjecture, which asserts that all elliptic curves over~$\Q$ are modular. Frey's construction spawned several lines of inquiry, in which mathematicians sought either to prove Fermat's Last Theorem outright or to link it to established conjectures. Much of what emerged is off the main topic of these notes\emrule we will not discuss the $abc$ conjecture or Szpiro's conjecture, for instance. Note, however, that these conjectures are treated by such articles as \cite\FREYONE, \cite\FREYTWO, \cite\LANG, \cite\HINDRY, and~\cite\OESTERLE. A first step toward justifying Frey's heuristic conclusion was taken in August, 1985 by Serre in a letter to J-F.~Mestre~\cite\ARCATA. In this letter, Serre formulated two related conjectures about modular forms, which he called $C_1$ and~$C_2$. He pointed out that Fermat's Last Theorem is a consequence of the Taniyama-Shimura conjecture {\it together with\/} the two new conjectures. In other words, Serre exhibited the implication $$ C_1 + C_2 + \hbox{Taniyama-Shimura} \Longrightarrow \hbox{Fermat}.$$ In my article~\cite\FERMAT, I proved Serre's conjectures $C_1$ and~$C_2$. My theorem established the implication $$\hbox{Conjecture of Taniyama-Shimura } \Longrightarrow\hbox{ Fermat}$$ which was the goal of Frey's construction. More generally, let $\ell\ge5$ be a prime number and suppose that $A$ is a semistable elliptic curve over~$\Q$ whose discriminant is the product of a power of~2 and a perfect $\ell$th power. Then the result of~\cite\FERMAT\ shows that $A$ cannot be a modular elliptic curve. (See \cite\PRASAD\ and \cite\TOULOUSE\ for expository accounts of this work.) After writing \cite\ARCATA, Serre presented in~\cite\DUKE\ a broad group of conjectures linking mod~$p$ Galois representations and mod~$p$ modular forms. These conjectures constitute the ``precise statement'' that was mentioned briefly at the end of the last section. As was alluded to above, Serre's conjectures imply Fermat's Last Theorem by a simple direct argument, and they imply a host of other statements about Diophantine equations and elliptic curves. There seems no hope of proving Serre's conjectures in the near term. Nevertheless, certain consequences of the conjectures for the mod~$p$ Galois representations associated to modular forms have seemed quite accessible. A group of these consequences have become known as the ``weak Serre conjecture''; this conjecture is an intricate generalization of the two statements $C_1$ and~$C_2$ formulated in~\cite\ARCATA. Thanks to the work of a large group of mathematicians, the weak Serre conjecture is now a theorem, at least for $p\ge3$; see \cite\MOTIVES\ and~\cite\DIAMONDREF\ for details. The relationship between the weak Serre conjecture and the full group of conjectures of~\cite\DUKE\ is that the conjectures of~\cite\DUKE\ amount to the conjunction of the weak conjecture and the single supplementary statement which was introduced at the end of the previous section. This statement, to the effect that two-dimensional continuous irreducible odd representations of~$\GalQ$ are modular, is the one which appears to be intractable. \section Wiles's strategy\endhead Suppose that $A$ is an elliptic curve over~$\Q$. To verify the Taniyama-Shimura conjecture for~$A$ is to link $A$ to modular forms. For example, we might consider the representations $\rho_{A,\ell}$ obtained from the action of~$\GalQ$ on~$A[\ell]$, when $\ell$ is a prime number. If one shows that an infinite number of these representations are modular (in the broadest possible sense), one can go on to prove that $A$ is modular. The difficulty with this approach, from a philosophical point of view, is that there is no visible program for manufacturing modular forms which can be compared with the~$\rho_{A,\ell}$. Alternatively, one may fix a prime~$\ell$ and consider the family of spaces $A[\ell^\nu]$ for $\nu=1,2,\ldots$. The resulting sequence of representations $$ \rho_{A,\ell^\nu}\:\GalQ\to\GL{\U{\ell^\nu}}$$ may be re-packaged as a single representation $$ \rho_{A,\ell^\infty}\:\GalQ\to\GL{\Z_\ell},$$ where $\Z_\ell$ is the ring of $\ell$-adic integers, i.e., the projective limit of the rings $\U{\ell^\nu}$. The natural map $\Z\to\Z_\ell$ is injective, so that $\Z_\ell$ contains $\Z$ as a subring. To prove that $A$ is a modular elliptic curve, it suffices to show that $\rho_{A,\ell^\infty}$ is modular in an appropriate sense. Indeed, the trace of $\rho_{A,\ell^\infty}(\Frob_p)$ coincides with the rational integer $b_p$ for all $p\notdivide \ell N$, where $N$ is the conductor of~$A$. Once one finds an eigenform $f$ in~$S(N)$ whose Fourier coefficients are related to the traces of~$\rho_{A,\ell^\infty}(\Frob_p)$, one has essentially proved that $A$ is modular. Of course, if $\rho_{A,\ell^\infty}$ is modular, then so is $\rho_{A,\ell}$. Relating $\rho_{A,\ell^\infty}$ to modular forms is therefore at least as hard as the formidable task of relating $\rho_{A,\ell}$ to modular forms! On the other hand, to prove that $A$ is modular by the $\ell$-adic method, we need only work with a {\it single\/} prime~$\ell$. Wiles's approach is based on the lucky circumstance that the representations $\rho_{A,\ell}$ are in fact modular for $\ell\le3$, cf.~\cite{\RUBINSILVER, \S2.3}. This circumstance arises from the theory of base change \`a la Saito-Shintani, as developed by Langlands in~\cite\LANGLANDS\ and then applied by~Tunnell in~\cite\TUNNELL. (For expositions of the results of Langlands, see \cite\GELBART\ and \cite\LABESSE.) Wiles's basic idea is to prove that that if $\ell$ is a prime for which $\rho_{A,\ell}$ is modular, then $\rho_{A,\ell^\infty}$ is automatically modular (and hence $A$ is a modular elliptic curve). In thinking about the jump from~$\rho_{A,\ell}$ to~$\rho_{A,\ell^\infty}$, one seeks to ignore $A$ as much as possible\emrule the aim is to prove a theorem about $\ell$-adic representations with properties generalizing those of~$\rho_{A,\ell^\infty}$. Two theorems in this direction are stated in~\cite\WILES. In each theorem, the prime $\ell$ is taken to be~odd, and the representation $\rho_{A,\ell}$ is required to be irreducible. One theorem pertains to the case where $A$ is semistable; this is the result which would apply to Frey curves arising in connection with Fermat's Last Theorem, and also the theorem which is in difficulty. That there is difficulty means that the theorem is not known to be true for all semistable elliptic curves over~$\Q$. However, it is apparently possible to use the technique of the proof to exhibit a genuinely infinite class of semistable elliptic curves over~$\Q$ which are modular. As I mentioned at the beginning of these notes, no such class was known before Wiles's methods were introduced. The second theorem treats the case where the representation~$\rho_{A,\ell}$ arises from a complex multiplication elliptic curve $A'$ over~$\Q$. Here, there is no ``gap'' in the proof, and one can again use the theorem to exhibit an explicit infinite collection of elliptic curves over~$\Q$ which are modular~\cite\RSNOTE. Wiles seeks to use the first theorem to prove that all semistable elliptic curves are modular. His ingenious argument for doing this is sketched by Rubin and Silverberg in~\cite{\RUBINSILVER, Prop.~2.4}. Suppose that $A$ is a semistable elliptic curve over~$\Q$. If $\rho_{A,3}$ happens to be irreducible, then the results of Langlands and Tunnell, plus the assertion of Wiles's first theorem, give the desired result that $A$ is modular. The hard case is that where $\rho_{A,3}$ is {\it reducible}. If $\rho_{A,5}$ is reducible as well, then it is not hard to show directly that $A$ is modular. So we might as well assume that $\rho_{A,5}$ is irreducible. Wiles shows then that one can find a second elliptic curve $A'$ whose mod~5 representation is isomorphic to that of~$A$ and whose mod~3 representation is irreducible~\cite{\RUBINSILVER, Appendix~B}. Two applications of the ``first theorem'' then suffice to show that $A$ is modular. \section The language of deformations\endhead Suppose that $A$ is a semistable elliptic curve over~$\Q$ and that $\ell\ge3$ is a prime number for which $\rho_{A,\ell}$ is both modular and irreducible. Choosing a basis of~$A[\ell]$, we regard $\rho_{A,\ell}$ as taking values in~$\GL{\F_\ell}$. We seek to show that $\rho_{A,\ell^\infty}$ is modular by showing that {\it all\/} suitable lifts of $\rho_{A,\ell}$ are modular. The idea of looking at all lifts at once stems from a fundamental article of Mazur~\cite\MAZUR. For historical reasons arising from algebraic geometry, isomorphism classes of~lifts are called ``deformations.'' The meaning of ``suitable'' depends on an auxiliary choice: a finite set of prime numbers $\Sigma$ which contains $\Sigma_0$, where $\Sigma_0$ is the set of primes at which $\rho_{A,\ell}$ is ramified. (Thus, by definition, $\Sigma_0$ is the set of primes dividing the discriminant of the number field obtaining by adjoining to~$\Q$ the coordinates of all points in~$A[\ell]$.) A priori, $\Sigma_0$ contains~$\ell$ and is contained in the union of~$\{\ell\}$ and the set of primes at which $A$ has bad reduction. Indeed, this union is the set of primes $\Sigma^\ast$ at which~$\rho_{A,\ell^\infty}$ is ramified, according to the well-known criterion of N\'eron-Ogg-Shafarevich~\cite\ST. That $\Sigma_0$ might be strictly smaller than the union should be regarded as a complicating factor. The reason for this is that the so-called ``minimal'' case $\Sigma=\Sigma_0$ is expected to be the easiest to deal with, while the choice $\Sigma=\Sigma^\ast$ is the one which is relevant to the representation~$\rho_{A,\ell^\infty}$. The following description is derived from~\S3.4 of~\cite\RUBINSILVER. Our discussion differs from that of~\cite\RUBINSILVER\ mainly in our writing $\ell$ for the prime called~$p$ in~\cite\RUBINSILVER\ and in our focus on the auxiliary choice~$\Sigma$. Having fixed this choice for the moment, one considers continuous homomorphisms $\tilde\rho\:\GalQ\to\GL A$, where $A$ is a complete Noetherian $\Z_\ell$-algebra with residue field~$\F_\ell$. (For technical reasons, it might be useful to replace $\Z_\ell$ by the integer ring of a finite extension of the $\ell$-adic field~$\Q_\ell$, but we will ignore this complication.) One demands a number of properties of~$\tilde\rho$, which can be summarized as follows: First, note that the residue map $A\to\F_\ell$ induces a homomorphism $\GL A\to\GL{\F_\ell}$; we require that $\rho_{A,\ell}$ coincide with the composite of~$\tilde\rho$ and this residue homomorphism. Second, we demand that $\tilde\rho$ be unramified outside~$\Sigma$. Third, we ask that $\tilde\rho$ have the same qualitative behavior at the primes in~$\Sigma_0$ as the representation~$\rho_{A,\ell}$. (In the language of~\cite\RUBINSILVER, we ask that $\tilde\rho$ be ordinary at~$\ell$ if $A$ has ordinary or multiplicative reduction at~$\ell$ and that $\tilde\rho$ be flat at~$\ell$ if $A$ has supersingular reduction at~$\ell$.) Finally, we insist that the determinant of~$\tilde\rho$ be the composite of the $\ell$-adic cyclotomic character $\tilde\chi_\ell\:\GalQ\to\Z_\ell^\ast$ and the structural map $\Z_\ell^\ast\to A^\ast$. This latter condition can be removed or modified if one is willing to consider spaces of modular forms which are more general than the spaces~$S(N)$. Representations $\tilde\rho$ with these properties define {\it deformations\/} of~$\rho_{A,\ell}$ of type~$\Sigma$. The deformations are equivalence classes of representations with respect to a natural identification: Two representations give the same deformation if they are conjugate by an element of~$\GL A$ which maps to the identity matrix in~$\GL{\F_\ell}$. According to a theorem of Mazur and Ramakrishna, there is a {\it universal\/} deformation. This is a deformation $$\rho_\Sigma\:\GalQ\to\GL{R_\Sigma}$$ of type~$\Sigma$ which is distinguished by the property that an arbitrary deformation $\tilde\rho\:\GalQ\to\GL A$ of type~$\Sigma$ may be obtained from $\rho_\Sigma$ by a unique homomorphism $R_\Sigma\to A$. Here $R_\Sigma$ should be regarded as a somewhat mysterious ring, which is not given explicitly, at least at first. Its construction depends on an abstract representability theorem, which one applies to the given deformation property. Somewhat analogously, one can define a universal {\it modular\/} deformation of type~$\Sigma$: $$\rho'_\Sigma\:\GalQ\to\GL{\T_\Sigma}.$$ The problem is then to prove that $\rho_\Sigma$ and~$\rho'_\Sigma$ coincide. In contrast to~$R_\Sigma$, the ring $\T_\Sigma$ may be constructed quite directly as an essentially classical Hecke algebra. It is free of finite rank over~$\Z_\ell$. Following a line of inquiry initiated by~Mazur \cite{\EIS, Ch.~II, \S15} (and taken up by a fair number of other mathematicians, including the author), Wiles proves that $\T_\Sigma$ is a Gorenstein ring. The Gorenstein property leads to further speculation as to the precise nature of~$\T_\Sigma$ as an object of commutative ring theory. Wiles realized several years ago that the key to proving that $\rho_\Sigma$ and~$\rho'_\Sigma$ coincide is to show that $\T_\Sigma$ is a complete intersection ring in the sense of~\cite\MATSUMURA. In fact, Wiles shows that if $\T_\Sigma$ is a complete intersection ring, then the universal deformation of type~$\Sigma'$ is modular for all $\Sigma'\supseteq\Sigma$. (For another perspective on questions of this type, see~\cite\MAZURNOTES.) In comparing $\rho_\Sigma$ and~$\rho'_\Sigma$, one begins with the canonical homomorphism $$\phi_\Sigma\: R_\Sigma\to\T_\Sigma$$ which results from the universality of~$\rho_\Sigma$. It is known in most cases, and should be possible to prove in general, that this homomorphism is surjective. To show that all deformations of type~$\Sigma$ are modular is to prove that $\phi_\Sigma$ is an {\it isomorphism}. (Analogous conjectures had been made previously by ~Mazur, cf.~\cite{\MAZURTILOUINE, p.~85}. For a bit more detail see~\cite{\RUBINSILVER, \S4.2}) Using techniques from commutative algebra, Wiles proves a proposition to the effect that $\phi_\Sigma$ is an isomorphism whenever a specific numerical inequality is satisfied~\cite{\RUBINSILVER, Th.~5.2}. Although the statement of the proposition seems to require that $\T_\Sigma$ be Gorenstein, Lenstra has shown how to re-state the proposition so that no Gorenstein hypothesis appears~\cite\HWL. Furthermore, Wiles interprets the numerical condition as an upper bound for the order of a Selmer-like group $S_\Sigma$, whose finiteness is not known a priori~\cite{\RUBINSILVER, \S5.2}. The theory of congruences between modular forms of different levels shows that the desired bound on~$S_\Sigma$, if valid for a given set~$\Sigma$, is automatically verified for all supersets $\Sigma'\supseteq \Sigma$. This means that it will be sufficient to verify the sought-after inequality in the minimal case $\Sigma=\Sigma_0$. Incidentally, I do not know whether $\phi_\Sigma$ is an isomorphism for a given set $\Sigma$ if and only if the inequality is true for~$\Sigma$. Given simply that $\phi_\Sigma$ is an isomorphism, can one conclude that $\phi_{\Sigma'}$ is an isomorphism for all $\Sigma'\supseteq \Sigma$? As the authors of~\cite\RUBINSILVER\ explain, Wiles has sought to verify his inequality for $\Sigma=\Sigma_0$ by constructing a ``geometric Euler system,'' thereby generalizing work of~M.~Flach~\cite\FLACH. At this juncture, no satisfactory construction has been exhibited. 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