\documentstyle[12pt]{article} \newcommand{\note}[1]{\marginpar{\small \sf#1}} %\def\baselinestretch{1.49} %\textwidth 15cm %\textheight 22cm %\topmargin 0pt %\headsep 52pt \makeatletter \def\@oddfoot{}\def\@oddhead{\rm\hfil\thepage\hfil} \def\@evenfoot{}\let\@evenhead\@oddhead \def\thebibliography#1{\section*{% \hbox to \textwidth{\hss Bibliography\hss}\@mkboth {REFERENCES}{REFERENCES}}\list {[\arabic{enumi}]}{\settowidth\labelwidth{[#1]}\leftmargin\labelwidth \advance\leftmargin\labelsep \usecounter{enumi}} \def\newblock{\hskip .11em plus .33em minus .07em} \sloppy\clubpenalty4000\widowpenalty4000 \sfcode`\.=1000\relax} \makeatother \let\GGGamma\Gamma \def\GGamma{{\mit\GGGamma}} \let\Gamma\GGamma \let\DDDelta\Delta \def\DDelta{{\mit\DDDelta}} \let\Delta\DDelta \let\OOOmega\Omega \def\OOmega{{\mit \OOOmega}} \let\Omega\OOmega \let\SSSigma\Sigma \def\SSigma{{\mit\SSSigma}} \let\Sigma\SSigma \let\LLLambda\Lambda \def\LLambda{{\mit\LLLambda}} \let\Lambda\LLambda %\pagestyle{empty} \vfuzz 10pt \font\datefont=cmdunh10 \setcounter{secnumdepth}{1} \def\doublemap{\mathrel{\null _{\raise.2ex\hbox{$\textstyle\rightarrow$}} ^{\lower.2ex\hbox{$\textstyle\rightarrow$}}}} \let\hold=\circ \def\jirk{{\scriptstyle\hold}} \let\circ=\jirk %\textheight=38.75\baselineskip \newcommand{\dual}{{\displaystyle\check{\hbox to 1 ex{}}}} \newcommand{\F}{{\bf F}} \newcommand{\End}{\mathop{\rm End}\nolimits} \newcommand{\et}{{\sf et}} \newcommand{\R}{{\cal R}} \newcommand{\V}{{\cal V}} \newcommand{\W}{{\cal W}} \newcommand{\J}{{\cal J}} \newcommand{\Hom}{\mathop{\rm Hom}\nolimits} \newcommand{\Aut}{\mathop{\rm Aut}\nolimits} \newcommand{\card}{\mathop{\rm card}\nolimits} \newcommand{\tr}{\mathop{\rm tr}\nolimits} \newcommand{\Pic}{\mbox{\rm Pic}} \newcommand{\Pico}{{\rm Pic}^{0}} \newcommand{\Alb}{\mbox{\rm Alb}} \newcommand{\Zp}{{\bf Z}_p} \newcommand{\Rf}{{R_{\sf f}}} \newcommand{\Hf}{{H_{\sf f}}} \newcommand{\Sh}{{\sf Sh}} \newcommand{\boldE}{{\bf E}} \newcommand{\crazyE}{{\underline{\bf E}}} \newcommand{\Zq}{{{\bf Z}_q}} \newcommand{\Qq}{{{\bf Q}_q}} \newcommand{\Mtwoql}{{\rm M}(2,{\bf Q}_\ell)} \newcommand{\Mtwozl}{{\rm M}(2,{\bf Z}_\ell)} \newcommand{\GLtwoql}{{\bf GL}(2,{\bf Q}_\ell)} \newcommand{\Fq}{{{\bf F}_q}} \newcommand{\Fqbar}{{{\overline{\bf F}}_q}} \newcommand{\Fpbar}{{{\overline{\bf F}}_p}} \newcommand{\Qpbar}{{{\overline{\bf Q}}_p}} \newcommand{\Tbar}{{{\overline{\bf T}}}} \newcommand{\Tunderline}{{\underline{T}}} \newcommand{\Z}{{\bf Z}} \newcommand{\C}{{\bf C}} \newcommand{\Q}{{\bf Q}} \newcommand{\Qbar}{\overline{{\bf Q}}} \newcommand{\GL}{{\bf GL}} \newcommand{\Qp}{{{\bf Q}_p}} \newcommand{\Fp}{{\bf F}_p} \newcommand{\Gm}{{\bf G}_{\rm m}} \newcommand{\JtJ}{J_{0}(N) \times J_{0}(N) } \newcommand{\T}{{\bf T}} \newcommand{\m}{{\sf m}} \newcommand{\Frob}{{\sf Frob}} \newcommand{\GalQ}{{\rm Gal}(\Qbar /\Q)} \newtheorem{theorem}{{\sc Theorem}} \newtheorem{lemma}{{\sc Lemma}} \newtheorem{prop}{{\sc Proposition}} \newtheorem{cor}{{\sc Corollary}} \let\thecor=\relax % We don't want the corollaries to have numbers \newenvironment{proof}{ \medskip \noindent {\em Proof\/}. }{\nolinebreak[2]\rule{.45em}{.9em}\medskip} \begin{document} {\large \begin{center} {{\sc Raising the Levels of Modular Representations}}\\ %\thanks{Paris Number Theory Seminar, 1987--88}} {Kenneth A. {\sc Ribet}} \note{Personal version of William Stein.} \par \end{center} } \section{Introduction} Let $\ell$ be a prime number, and let $\F$ be an algebraic closure of the prime field $\F_\ell$. Suppose that \[ \rho :\GalQ \to {\bf GL}(2,\F)\] is an irreducible (continuous) representation. We say that $\rho$ is {\em modular of level $N$}, for an integer $N\ge 1$, if $\rho$ arises from cusp forms of weight 2 and trivial character on $\Gamma_0(N)$. The term ``arises from'' may be interpreted in several equivalent ways. For our present purposes, it is simplest to work with maximal ideals of the Hecke algebra for weight-2 cusp forms on $\Gamma_0(N)$. Namely, let $S(N)$ be the $\C$-vector space consisting of such forms, and for each $n\ge 1$ let $T_n \in \End S(N)$ be the $n$th Hecke operator. Let $\T = \T_N$ be the subring of $\End S(N)$ generated by these operators. As is well known (\cite{D-S}, Th.~6.7 and \cite{Fermat}, \S5), for each maximal ideal $\m$ of $\T$, there is a semisimple representation \[ \rho _\m: \GalQ \to {\bf GL}(2,\T/\m),\] unique up to isomorphism, satisfying \[ \tr \rho_\m (\Frob _r) = T_r \pmod{\m}, \qquad \det \rho _\m (\Frob _r) = r \pmod{\m} \] for almost all primes $r$. (Here $\Frob _r$ is a Frobenius element in $\GalQ$ for the prime $r$.) This representation is in fact unramified at every prime $r$ prime to $\ell N$, and the indicated relations hold for all such primes. We understand that $\rho$ is modular of level $N$ if there is a maximal ideal $\m$ of $\T$, together with an inclusion $\omega\colon\T/\m \hookrightarrow \F$, so that the representations $\rho$ and $\rho _\m\otimes_\omega\F$ are isomorphic. (Cf.~\cite{Fermat}, \S5.) The representations $\rho_\m$ are nothing other than the Galois representations attached to mod $\ell$ eigenforms of weight 2 on $\Gamma_0(N)$. Indeed, let $\cal L$ be the space of forms in $S(N)$ which have rational integral $q$-expansions. As is well known, $\cal L$ is a lattice in $S(N)$, cf.~\cite{D-S}, Proposition~2.7. The space $\overline{\cal L} = {\cal L}/\ell {\cal L}$ is the space of mod $\ell$ cusp forms on $\Gamma_0(N)$. The ${\bf F}_\ell$-algebra $\cal A$ generated by the Hecke operators $T_n$ in $\End \overline{\cal L}$ may be identified with $\T/\ell\T$ (see, for example, \cite{Fermat}, \S5). To give a pair $(\m,\omega)$ as above is to give a character (i.e., homomorphism) \[ \epsilon : {\cal A} \to \F.\] If $f$ is a non-zero element of $\overline{\cal L}\otimes _{{\bf F}_\ell}\F$ which is an eigenvector for all $T_n$, the action of $\cal A$ on the line generated by $f$ defines such a character $\epsilon$. It is an elementary fact that all characters $\epsilon$ arise in this manner. Assume now that $\rho$ is modular of level $Np$, where $p$ is a prime number not dividing $N$. We say that $\rho$ is $p$-new (of level $pN$) if $\rho$ arises in a similar manner from the $p$-new subspace $S(pN)_{p\rm{-new}}$ of $S(pN)$. Recall that there are two natural inclusions (or degeneracy maps) $S(N)\doublemap S(pN)$ and dually two trace maps $S(pN)\doublemap S(N)$. (See \cite{A-L} for the former maps.) The two maps $S(N)\doublemap S(pN)$ combine to give an inclusion $S(N)\oplus S(N)\hookrightarrow S(pN)$, whose image is known as the $p$-old subspace $S(pN)_{p\rm{-old}}$ of $S(pN)$. The space $S(pN)_{p\rm{-new}}$ is defined as the orthogonal complement to $S(pN)_{p\rm{-old}}$ in $S(pN)$, under the Petersson inner product on $S(pN)$. It may also be characterized algebraically as the intersection of the kernels of the two trace maps; this definition is due to Serre. The space $S(pN)_{p\rm{-new}}$ is $\T_{pN}$-stable. The image of $\T_{pN}$ in $\End S(pN)_{p\rm{-new}}$ is the {\em $p$-new quotient\/} \[\Tbar _{pN} = \T_{pN/p\rm{-new}}\] of $\T_{pN}$. We say that $\rho$ is $p$-new if $\m\subset\T_{pN}$ and $\omega$ may be found, as above, in such a way that the maximal ideal $\m$ of $\T_{pN}$ is the inverse image of a maximal ideal of $\Tbar _{pN}$, under the canonical quotient map $\T_{pN}\to\Tbar _{pN}$. On a concrete level, this means that the character \[ \epsilon : \T _{pN} \to \F \] coming from $(\m,\omega)$ is defined by an eigenform in the mod $\ell$ reduction of the space $S(pN)_{p\rm{-new}}$, i.e., in the ${\bf F}$-% vector space $\Lambda\otimes _\Z\F$, where $\Lambda$ is the lattice in $S(pN)_{p\rm{-new}}$ consisting of forms with rational integral coefficients. \begin{theorem}\label{theorem1} Let $\rho$ be modular of level $N$. Let $p\not| \ell N$ be a prime satisfying one or both of the identities \begin{equation} \label{equation1} \tr \rho(\Frob_p) = \pm (p+1) \pmod{\ell}. \end{equation} Then $\rho$ is $p$-new of level $pN$. \end{theorem} \medskip \noindent \hbox to \textwidth{{\em Remarks.}\hss} \par\smallskip\noindent {\bf 1.} In the Theorem, and in the discussion below, we assume that $\rho$ is irreducible, as above. \par\smallskip\noindent {\bf 2.} A slightly stronger conclusion may be obtained if one assumes that $\rho$ is $q$-new of level $N$, where $q$ is a prime number which divides $N$, but not $N/q$. Under this hypothesis, plus the hypothesis of Theorem~\ref{theorem1}, one may show that $\rho$ is $pq$-new of level $pN$, in a sense which is easy to make precise as above. (See \cite{Fermat}, \S7, where a theorem to this effect is proved, under the superfluous additional hypothesis $p \equiv -1 \hbox{ (mod $\ell$)}$.) The interest of Theorem~\ref{theorem1} is that no hypothesis is made about the existence of a prime number $q$. \par\smallskip\noindent {\bf 3.} The case $p=\ell$ can be included in the Theorem if its hypothesis (\ref{equation1}) is reformulated. Namely, (\ref{equation1}) tacitly relies on the fact that $\rho$ is unramified outside the primes dividing $\ell N$. Choose a maximal ideal $\m$ for $\rho$ as in the definition of ``modular of level $N$.'' Then (\ref{equation1}) may be re-written as the congruence \[T_p \equiv \pm(p+1) \pmod{\m}.\] Assuming simply that $p$ is prime to $N$, but permitting the case $p=\ell$, one proves that $\rho$ is $p$-new of level $pN$ if this congruence is satisfied (with at least one choice of $\pm$). \begin{cor} Let $\rho$ be modular of level $N$. Then there are infinitely many primes $p$, prime to $\ell N$, such that $\rho$ is $p$-new of level $pN$. \end{cor} \smallskip\noindent Indeed, suppose that $p$ is prime to $\ell N$. Then $p$ is unramified in $\rho$, so that a Frobenius element $\Frob_p$ is well defined, up to conjugation, in the image of $\rho$. By the Cebotarev Density Theorem, there are infinitely many such $p$ such that $\Frob_p$ is conjugate to $\rho(c)$, where $c$ is a complex conjugation in $\GalQ$. Both sides of the congruence (\ref{equation1}) are then $0$, so that (\ref{equation1}) is satisfied. (Cf.~\cite{Fermat}, Lemma~7.1.) Our corollary is stated (in terms of mod $\ell$ eigenforms) as ``Th\'eor\`eme~(A)'' in a recent preprint of Carayol \cite{car}. Carayol describes his Th\'eor\`eme~(A) as having been proved in preliminary versions of \cite{Fermat}, as an application of results in \cite{ICM}. In later versions of \cite{Fermat}, Th\'eor\`eme (A) was replaced by a theorem involving $pq$-new forms (alluded to above), which is proved by methods involving Shimura curves. The aim of this present note is to resurrect Th\'eor\`eme (A). Our derivation of Theorem~\ref{theorem1} is based on the results of \cite{ICM}. Although we couch our results in the language of Jacobians of modular curves, it should be clear to the reader that we use no fine arithmetic properties of these Jacobians: the argument is entirely cohomological. As F.~Diamond has recently shown \cite{Diamond}, an elaboration of these methods leads to results for cusp forms of weight $k\ge 2$. \section{Summary of \protect\cite{ICM}} First let $N$ be a positive integer, and consider the modular curve ${X_0(N)}_\C$, along with its Jacobian $J_0(N) = \Pico(X_0(N))$. The curve $X_0(N)$ comes equipped with standard Hecke correspondences $T_n$, which induce endomorphisms of $J_0(N)$ by Pic functoriality (cf.~\cite{Fermat}, \S3). These endomorphisms, in turn, act on the space of holomorphic differentials on the abelian variety dual to $J_0(N)$, which is the Albanese variety of $X_0(N)$. This space of differentials is canonically identified with $S(N)$, and via this identification the endomorphism $T_n$ of $J_0(N)$ acts on the space of differentials as the usual Hecke operator $T_n$ of $S(N)$. Since the action of $\End(J_0(N))$ on $S(N)$ is faithful, it follows that the subring of $\End(J_0(N))$ generated by the $T_n$ is ``nothing other'' than the ring $\T_N$. We now choose a prime $p$ prime to $N$ and consider $X_0(pN)$ and $J_0(pN)$, to which the same remarks apply. The two curves $X_0(pN)$ and $X_0(N)$ are linked by a pair of natural degeneracy maps $\delta_1,\delta_p\colon X_0(pN) \doublemap X_0(N)$, with the following (naive) modular interpretation. The curve $X_0(pN)$ is associated to the moduli problem of classifying elliptic curves $E$ which are furnished with cyclic subgroups $C_N$ and $C_p$ of order $N$ and $p$, respectively. Similarly, $X_0(N)$ classifies elliptic curves with cyclic subgroups of order $N$. The degeneracy map $\delta_1$ maps $(E,C_N,C_p)$ to $(E,C_N)$, while $\delta_p$ maps $(E,C_N,C_p)$ to $(E/C_p,C^{\prime}_N)$, where $C^{\prime}_N$ is the image of $C_N$ on $E/C_p$. In a similar vein, we recall the modular interpretation of the correspondences $T_p$ on $X_0(N)$ and on $X_0(pN)$. First, for $X_0(N)$ we have \[ T_p : (E,C_N) \mapsto \sum_D (E/D,(C_N\oplus D)/D),\] where the sum is taken over the $(p+1)$ different subgroups $D$ of order $p$ in $E$. For $X_0(pN)$, we have a sum of $p$ terms \[ T_p : (E,C_N,C_p) \mapsto \sum_{D\ne C_p} (E/D,(C_N\oplus D)/D,E[p]/D),\] where $E[p]$ is the group of $p$-division points on $E$. (This latter group is the direct sum $C_p\oplus D$.) These formulas lead immediately to the relations among correspondences \begin{equation}\label{cr} \delta_1\circ T_p = T_p\circ\delta_1 - \delta_p, \qquad \delta_p\circ T_p = p\cdot \delta_1. \end{equation} The maps $\delta_1$ and $\delta_p$ combine to induce a map on Jacobians \[ \alpha : J_0(N) \times J_0(N) \to J_0(pN),\qquad (x,y)\mapsto \delta_1^*(x) + \delta_p^*(y) .\] The image of this map is by definition the $p$-old subvariety $A$ of $J_0(pN)$; the kernel of $\alpha$ is a certain finite group which is calculated in \cite{ICM}. Namely, let $\Sh$ be the Shimura subgroup of $J_0(N)$, i.e., the kernel of the map $J_0(N) \to J_1(N)$ which is induced by the covering of modular curves $X_1(N) \to X_0(N)$. The group $\Sh$ is a finite group which may be calculated in the following way: Consider the maximal unramified subcovering $X \to X_0(N)$ of $X_1(N) \to X_0(N)$, and let $\cal G$ be the covering group of this subcovering. Then $\cal G$ and $\Sh$ are canonically $\Gm$-dual. Let $\Sigma \subset \JtJ$ be the image of $\Sh$ under the antidiagonal embedding \[ J_0(N) \to \JtJ, \qquad x \mapsto (x,-x).\] According to \cite{ICM}, Theorem~4.3, we have \note{This prop. implies that the map $J_0(N)\rightarrow J_0(Np)$ is injective!!?} \begin{prop} The kernel of $\alpha$ is the group $\Sigma$. \end{prop} The map $\alpha$ is equivariant with respect to Hecke operators $T_n$ with $(n,p)=1$. Namely, we have $\alpha\circ T_n = T_n \circ\alpha$ for all $n$ prime to $p$, with the understanding that the endomorphism $T_n$ of $J_0(N)$ acts diagonally on the product $\JtJ$. On the other hand, this formula must be modified when $n$ is replaced by $p$, as one sees from (\ref{cr}). Before recording the correct formula for $T_p$, we introduce the notational device of reserving the symbol $T_p$ for the $p^{\rm th}$ Hecke operator at level $N$, and the symbol $U_p$ for the $p^{\rm th}$ Hecke operator at level $pN$. With this notation, we have (as a consequence of (\r