%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % padictwist_v15.tex % % % % Robert F. Coleman and William A. Stein % % % % April 6, 2003 % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[11pt]{article} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsthm} \textwidth=1.1\textwidth \hoffset=-0.05\textwidth \textheight=1.1\textheight \voffset=-0.05\textheight \baselineskip=1.1\baselineskip \newcommand{\marginalfootnote}[1]{% \footnote{#1} \marginpar{\hfill {\sf\thefootnote}}% } \newcommand{\edit}[1]{\marginalfootnote{#1}} \newcommand{\an}{{\rm an}} \newcommand{\qe}{\stackrel{\mbox{\tiny ?}}{=}} \newcommand{\e}{\mathbf{e}} \newcommand{\Gam}{\Gamma} \newcommand{\N}{\mathcal{N}} \newcommand{\q}{\mathbf{q}} \newcommand{\union}{\cup} \newcommand{\ra}{\rightarrow} \newcommand{\la}{\leftarrow} \newcommand{\con}{\equiv} \newcommand{\iv}{^{-1}} \newcommand{\alp}{\alpha} \newcommand{\bq}{\mathbf{q}} \newcommand{\vphi}{\varphi} \newcommand{\eps}{\varepsilon} \newcommand{\comment}[1]{} \newcommand{\Q}{\mathbf{Q}} \newcommand{\D}{{\mathbb D}} \newcommand{\C}{\mathbf{C}} \newcommand{\Qbar}{\overline{\Q}} \newcommand{\Z}{\mathbf{Z}} \newcommand{\Qp}{\Q_p} \newcommand{\Qpbar}{\Qbar_p} \newcommand{\Zpbar}{\overline{\Z}_p} \newcommand{\h}{\mathfrak{h}} \newcommand{\p}{\mathfrak{p}} \newcommand{\m}{\mathfrak{m}} \renewcommand{\l}{\ell} \renewcommand{\t}{\tau} \renewcommand{\a}{\mathfrak{a}} \DeclareMathOperator{\new}{new} \DeclareMathOperator{\Gal}{Gal} \DeclareMathOperator{\Norm}{Norm} \DeclareMathOperator{\ord}{ord} \newcommand{\Zbar}{\overline{\Z}} \newcommand{\dia}[1]{\langle #1 \rangle} \newcommand{\pad}{$p$-adic} \newcommand{\zms}[1]{(\Z/#1\Z)^*} \newcommand{\sig}{\sigma} \newcommand{\note}[1]{% \footnote{#1} \marginpar{\hfill {\sf\thefootnote}}% } \renewcommand{\qe}{$q$-expansion} \newcommand{\bM}{\mathbf{M}} \newcommand{\Zm}[1]{\Z/#1\Z} \newcommand{\kap}{\kappa} \newcommand{\hra}{\hookrightarrow} %%%% Theoremstyles \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{claim}[theorem]{Claim} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{question}[theorem]{Question} \newtheorem{guess}[theorem]{Guess} \theoremstyle{remark} \newtheorem{goal}[theorem]{Goal} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \title{\sc Approximation of Eigenforms of Infinite Slope by Eigenforms of Finite Slope} \author{Robert F.~Coleman \and{} William A.~Stein} \begin{document} \maketitle \tableofcontents \section{Introduction} Fix a prime~$p$. Consider a classical newform $$ F = \sum_{n\geq 1} a_n q^n \in S_k\left(\Gamma_1(Np^t),\Qpbar\right) $$ where~$k$ and~$N$ are positive integers and $p\nmid N$ is a prime (by a {\em newform} we mean a Hecke eigenform that lies in the new subspace and is normalized so that $a_1=1$). The {\em slope} of~$F$ is $\ord_p(a_p)$, where $\ord_p(p)=1$. By \cite[Prop.~3.64]{shimura:intro}, the twist $$ F^{\chi} = \sum \chi(n)a_n q^n $$ of~$F$ by any Dirichlet character~$\chi$ of conductor dividing~$p$ is an eigenform on $\Gamma_1({Np^{\max{\{t+1,2\}}} })$. This twist has infinite slope. In Section~\ref{sec:canapprox}, we prove that if~$F$ has finite slope then it is possible to approximate $F^{\chi}$ arbitrarily closely by (classical) finite slope {\em eigenforms}. Assuming refinements of standard conjectures, the best estimate we obtain for the smallest weight of an approximating eigenforms is exponential in the approximating modulus $p^A$. Section~\ref{sec:numerical} contains computations that suggest that the best estimates should have weight that is linear in $p^A$. One motivation for the question of approximation of infinite slope eigenforms by finite slope eigenforms is the desire to understand the versal deformation space of a residual modular representation \cite{mazur:deforming} (the deformation space of an irreducible representation is universal~\cite{mazur:deforming} as is the deformation space of a residual pseudo-representation \cite{eigencurve}). In \cite{gouvea-mazur:density} (see also \cite{mazur:infinite_fern}, and \cite{bockle:density} for a generalization), it was shown that the Zariski closure of the locus of finite slope modular deformations of an absolutely irreducible ``totally unobstructed'' residual modular representation is Zariski dense in the associated representation space but very little is known about the topological closure of this locus. For example, it is not known if it contains any nonempty open sets. Our result implies that it contains tamely ramified twists of modular deformations. We also show in Section~\ref{sec:hatada} that a result of Hatada implies that in at least one (albeit not irreducible) case it does not contain all modular deformations. Our investigation began with with our answer in Section~\ref{sec:canapprox} to a question of Jochnowitz. The idea of studying the \pad\ variation of modular forms began with Serre \cite{serre:antwerp72} and was since developed by Katz \cite{katz:annals75} and Hida \cite{hida:iwasawa} (see also \cite{gouvea:slnm} for a sketch of the theory). It follows, in particular, from their work, that one can approximate all forms on $X_0(p^n)$ with forms on the $j$-line $X_0(1)$, but {\em not} necessarily with {\em eigenforms}. We prove the above result about twists in Section~\ref{sec:canapprox}, then state some questions about approximation by finite slope forms in Section~\ref{sec:questions}. We explain how to reinterpret Hatada's result in Section~\ref{sec:hatada}, then present the results of our computations in Section~\ref{sec:numerical}. Based on the results and computations discussed in this article, Mazur has suggested that it may be the case that an infinite slope eigenform can be approximated by finite slope eigenforms only if the corresponding representation is what he calls {\em tamely semistable} (i.e., semistable, in the sense of \cite{colmez_fontaine}, after a tame extension). {\bf Acknowledgments.} The authors thank Naomi Jochnowitz for provoking this line of thought and for interesting conversations, Barry Mazur for helpful comments and questions, Frank Calegari for conversations, Lo\"\i{}c Merel for his comments on an early draft of this paper, and the referee for a brilliant report. \section{Approximating Teichm\"uller Twists of Finite Slope Eigenforms}% \label{sec:canapprox}% This section is the theoretical heart of the paper. We prove that the infinite slope eigenforms obtained as twists of finite slope eigenforms by powers of the Teichm\"uller character can always be approximated by finite slope eigenforms. We first show that certain overconvergent eigenforms of sufficiently close weight are congruent and have the same slope. Then we use the~$\theta$ operator on overconvergent forms to deduce the main result (Theorem~\ref{thm:twistapprox}) below. Let~$p$ be a prime. All eigenforms in this section will be cusp forms with coefficients in $\Qpbar$ normalized so that $a_1=1$. Suppose $ F = \sum_{n\ge 1}a_nq^n $ is an eigenform and~$\chi:(\Z/M\Z)^* \to \C_p^*$ is a Dirichlet character with modulus~$M$, which we extend to $\Z/M\Z$ by setting $\chi(n)=0$ if $(n,M)\neq 1$. Then the twist of~$F$ by~$\chi$ is the eigenform $$ F^\chi = \sum_{n\ge 1}\chi(n)a_nq^n. $$ Let $\omega\colon \zms p\ra\Z_p^*$ be the Teichm\"uller character (so $\omega(n) \con n \pmod{p}$). The following theorem concerns finite slope approximations of twists of~$F$ by powers of~$\omega$. For example, it concerns the twist $$ F^{\omega^0} = \sum_{(n,p) = 1} a_n(F) q^n $$ of $F$ by the trivial character mod~$p$, which we call the ``$p$-deprivation'' of~$F$ and which has infinite slope. \begin{theorem}\label{thm:twistapprox} Suppose~$F$ is a classical eigenform on $X_1(Np^{t})$, $t\ge 1$, over $\Qbar_p$ of weight~$k$, character~$\psi$, and finite slope at~$p$. Let $A\in \Z_{>0}$ and $r,s\in\Z_{\geq 0}$ with $r,s1$ are not yet defined in the literature). The ring $A^0(X)$ is finite over $A^0(B)$ by Corollary 6.4.1/5 of \cite{bgr}. Let $f_1,\dots,f_n$ be generators. Let $f_0$ be a uniformizing parameter on~$B$ so that $A(B)=\C_p\dia{f_0}$, where $\C_p\dia{f_0}$ is the ring of power series in $f_0$ whose coefficients tend to~$0$ with their degree. Let $Z_L(y)$ be the following Weierstr\"ass subdomain of~$X$: $${\{x\in X\colon |f_i(x)-f_i(y)|\le p^{-L}, 0\le i\le n\}}.$$ Since the functions $x\to \eta_x(T(n))$ lie in $A^{0}(X)$, it follows that if $x\in Z_L(y)$, then $$F_x(q)\con H(q)\pmod{p^L}.$$ Finally, since $Z_L(y)$ is a subdomain of~$X$ and~$X$ is finite over~$B$, the map from $Z_L(y)$ to~$B$ is quasi-finite. It follows from Proposition A5.5 of \cite{coleman:banach} that its image in~$B$ is a subdomain. Since~$\kappa$ is the image of~$y$, its image contains a disk around~$y$. \end{proof} %\begin{remark} %Proposition~\ref{prop:approx} is true even if $H$ is not a cusp form. %\end{remark} \begin{proof}[Proof of Theorem~\ref{thm:twistapprox}] Let $\alpha$ be the slope of $F$. It follows from Proposition~\ref{prop:approx} that if $m\in\Z$ is sufficiently small \pad ally there exists an overconvergent eigenform~$K$ of tame level~$N$, weight-character $ \chi\cdot\dia{\dia{\ }}^{k-m}$ and slope~$\alpha$ such that $K(q)\con F(q)\pmod{p^A}$. Suppose $m\ge k$. Then, by Proposition 4.3 of \cite{coleman:classical} (see also \cite{coleman:classical_higher}) if $F_1=\theta^{m-k+1}K$, then $F_1$ is an overconvergent eigenform of weight-character $$ \kappa_1:=\omega^{2(m-k+1)}\cdot\chi\cdot\dia{\dia{\ }}^{k_1}, $$ where $k_1=m-k+2$, and $F_1$ has finite slope $\alp_1=\alp+m-k+1$. Applying this same process to $F_1$, for $\ell\in\Z$ sufficiently small \pad ally such that $\ell\ge k_1$, we obtain an overconvergent finite slope eigenform $F_2$ of weight-character $\kappa_2$, where $\kappa_2=\omega^{2\ell}\cdot\chi\cdot\dia{\dia{\ }}^{k_2}$ and where $k_2=\ell-k_1+2=k+\ell-m$, such that if $F_2(q)=\sum_{n\ge 1}b_nq^n$, then \begin{align*} b_n&\con n^{\ell-k_1+1}n^{m-k+1}a_n\\ &\con n^\ell a_n\pmod{p^A}. \end{align*} The latter is congruent to $\omega^r(n)a_n\pmod{p^A}$ if $\ell\con r\pmod{\vphi(p^A)}$ and $\ell+v(a_p)\ge A$. It follows from \cite[\S8]{coleman:classical}, \cite{coleman:classical_higher}, and \cite{coleman:banach} that if~$c$ is an integer sufficiently small \pad ally, such that $c+k_2>v(b_p)+1$ (note that $v(b_p)$ is the slope of $F_2$ so is finite) there exists a classical eigenform~$G$ on $X_1(Np^t)$ of weight $k_2+c=k+\ell-m+c$, slope $v(b_p)$ and character $\omega^{m+r-c}\cdot\psi$ such that $G(q)\con{} F_2(q)\con{} F^{\omega^r}(q)\pmod{p^A}$. We can choose~$c$ so that $m+r-c\con s\pmod{p-1}$ and then $k_2+c\con k+2r-s\mod (p-1)$. \end{proof} The following corollary addresses a question of Jochnowitz, which motivated this entire investigation: \begin{corollary} Suppose~$R$ is a classical eigenform of weight~$k$ on $X_1(N)$, let $A\in \Z_{>0}$, and let $r \in \Z_{\geq 0}$ with $r2v(b_p)+1$. Then~$G$ is old of weight congruent to $k\mod (p-1)$ and~$G$ is congruent to an eigenform~$S$ of the same weight on $X_1(N)$ modulo $p^{v(b_p)}$. Since $b_p\con 0\pmod{p^A}$, we obtain the corollary. \end{proof} \begin{remark}\label{remark:estimate} Assuming a natural refinement of the Gouv\^ea-Mazur conjectures, the best estimate we obtain for the weight of~$H$ in the above proof is exponential in~$p^A$. Computational evidence suggests that the best estimates should have weights that are linear in $p^A$ (see Section~\ref{sec:numerical}). \end{remark} \begin{remark}\label{remark:jochnowitz_mazur} Jochnowitz and Mazur have independently observed that the above argument can be used to prove the following result: {\em Suppose~$F$ is an overconvergent eigenform of arithmetic weight-character~$\kappa$, which is a limit of overconvergent eigenforms of finite slope. If $\iota\colon \Z_p^*\ra\Z_p^*$ is the identity character, then the twist $F^{\iota/\kappa}(q)$ of $F$ by $\iota/\kappa$, which is the $q$-expansion of a convergent eigenform of weight-character ${\iota}^2/\kappa$, is the limit of overconvergent eigenforms of finite slope.} \end{remark} \begin{remark}\label{remark:twinninig} One can also approach the $p$-deprivation (the twist by the $0$th power of Teichm\"uller) of a finite slope eigenform~$F$ by using the evil twins of eigenforms approaching~$F$. \end{remark} \subsection{Questions}\label{sec:questions} Some natural questions arise: \begin{enumerate} \item Is every $p$-adic convergent eigenform which is the limit of finite slope overconvergent eigenforms an overconvergent eigenform? (We can show the twist of an overconvergent eigenform by a Dirichlet character is overconvergent.) \item Which infinite slope eigenforms are limits of finite slope eigenforms? \item If $F(q)$ is the $q$-expansion of an overconvergent eigenform of weight-character~$\kappa$, is $F^{\iota/\kappa}(q)$ the $q$-expansion of an overconvergent eigenform of weight-character $\iota^2/\kappa$ (recall that $\iota$ is the identity character $\Z_p^*\xrightarrow{\sim}\Z_p^*$)? Another closely related question is as follows: Suppose~$\rho$ is the representation of the absolute Galois group of~$\Q$ attached to an overconvergent eigenform and let~$\chi$ denote the cyclotomic character. Then is the representation $\rho\otimes{\chi\cdot\det(\rho)\iv}$ attached to an overconvergent eigenform? \end{enumerate} \section{An Infinite Slope Eigenform that is Not Approximable}\label{sec:cantapprox} In Section~\ref{sec:hatada}, we prove an extension to higher level of a theorem of Hatada about the possibilities for systems of Hecke eigenvalues modulo~$8$. We use this result to deduce that the normalized weight~$2$ cusp form on $X_0(32)$ is not $2$-adically approximable by normalized eigenforms of tame level~$1$ and finite slope. In Section~\ref{sec:nonapproxmaybe} we give an example of an infinite slope eigenform of level~$27$ that computer computations suggest cannot be approximated by finite slope forms. For related investigations, see \cite{calegari_emerton}. \subsection{An Extension of a Theorem of Hatada}\label{sec:hatada} \begin{theorem}\label{thm:genhatada} If~$F=\sum a_n q^n$ is a normalized cuspidal newform over $\C_2$ of finite slope on $X_0(2^n)$, then $a_2 \con 0\pmod{8}$ and $a_p \con p+1\pmod{8}$ for all odd primes~$p$. \end{theorem} \begin{proof} Suppose~$F$ has weight~$k$ and finite slope~$\alp$. The assumption that~$F$ has finite slope implies $n\le 1$. If $n=0$ the assertion of Theorem~\ref{thm:genhatada} was proved by Hatada in \cite{hatada:eigenvalues79}, so we may assume that $n=1$ and $\alp=(k-2)/2$ (in general, the slope of a newform on $\Gamma_0(p)$ of weight~$k$ is $(k-2)/2$). Note that $\alp\geq 3$ since there are no newforms on $X_0(2)$ of weight $<8$. It follows from Theorems A of \cite{coleman:banach} (see \S B2 of \cite{coleman:banach} for the extension to $p=2$) and Theorem B5.7 of \cite{coleman:banach} that if~$j$ is an integer sufficiently close $2$-adically to~$k$, then there exists a classical normalized cuspidal eigenform~$G$ on $X_0(2)$ of weight~$j$ and slope~$\alp$ such that $$G(q)\con F(q)\pmod{8}.$$ If in addition we assume that $j>2(\alp+1)$, then~$G$ must be old (since the slope of a newform of weight~$j$ is $(j-2)/2\neq \alpha$). Thus there is a cuspidal eigenform $H=\sum b_n q^n$ of level~$1$ such that $G$ is a linear combination of $H(q)$ and $H(q^2)$. More precisely, $$ G(q) = H(q) - \rho H(q^2) $$ where $\rho$ is a root of $P(X) = X^2 - b_2 X +2^{j-1}$. By Hatada's theorem $\ord_2(b_2)\geq 3$, and $j\geq 12$, so the slopes of the Newton polygon of $P(X)$ at~$2$ are both at least~$3$. Thus $G(q)\con H(q)\pmod{8}$, which proves the theorem because~$H$ has level~$1$. \end{proof} \begin{corollary} Let $G$ be the normalized weight $2$ cusp form on $X_0(32)$. Then $G$ is not 2-adically approximable by normalized eigenforms of tame level~$1$ and finite slope. \end{corollary} \begin{proof} If $F_{32}$ were approximable there would have to be a normalized eigenform $F$ on $X_0(2)$ such that $F_{32}(q)\con F(q)\pmod{8}$. However, $F_{32}(q)=\sum_{n=1}^\infty a_nq^n$ where, $$a_p= \begin{cases} 2x&\text{if }p=x^2+y^2,\quad\text{written so } x+y\con x^2\!\!\pmod{4}\\ 0&\text{otherwise}. \end{cases} $$ As $a_3=0\not\con 4\pmod{8}$, we see from Theorem~\ref{thm:genhatada} that~$F$ does not exist. \end{proof} \begin{remark} If $p\con 1 \pmod{4}$ then the coefficient of $a_p$ in $F_{32}$ agrees modulo~$8$ with $p+1$. If~$p$ is $3$ mod $4$ it does not because for $F_{32}$ the coefficient vanishes. What is happening is that there is a reducible mod 8 pseudo-representation (namely the trivial one-dimensional representation plus the cyclotomic character) such that any finite slope level $2^n$ form gives this pseudo-representation mod 8. Conversely the mod 8 representation associated to $F_{32}$ is the direct sum of the quadratic character associated to $\Q(i)$ and the cyclotomic character. Hence the congruence works when $p=1 \mod 4$ but not otherwise. \end{remark} \subsection{Another Eigenform that Conjecturally Cannot be Approximated}% \label{sec:nonapproxmaybe} In this section we consider an infinite slope eigenform that is not a Teichm\"uller twist of a finite slope eigenform. We conjecture that this eigenform cannot be approximated arbitrarily closely by finite slope eigenforms. %The infinite slope weight-two normalized eigenform attached to $X_0(27)$ is % $$F = q - 2q^4 - q^7 + 5q^{13} + 4q^{16} - 7q^{19} - \cdots$$ \comment{ Let~$G$ be a newform in either $S_k(\Gamma_0(1),\Z_3)$, for $k\leq 110$, or $S_k(\Gamma_0(3),\Z_3)$, for $k\leq 50$. Then $$\min\{\ord_p(a_p(G) - a_p(F)) \,\,:\,\, p \in \{2,5,7,11,13, 17\} \} = 1.$$ Thus none of these newforms~$G$ approximate~$F$ modulo $3^2$. Note that we have omitted forms in $S_k(\Gamma_0(1),\Zbar_3)$ that are not defined over $\Z_3$, but there are very few such forms. This computation suggests that~$F$ cannot be approximated by finite slope eigenforms. % and supports Mazur's suggestion as the %representation attached to~$F$ is not tamely semi-stable. } \begin{conjecture}\label{conj:systems3} There are exactly five residue classes in $(\Z/9\Z)[[q]]$ of normalized eigenforms in $S_k(\Gamma_0(N))$ where $k\geq 1$ and $N=1,3,9$. They are given in the following table, where the indicated weight is the smallest weight where that system of eigenvalues occurs (the level is $1$ in each case): {\rm\tt \begin{center}\begin{tabular}{|c|l|}\hline {\rm Weight} & {\rm [ $a_2, a_3, \ldots, a_{43} \mod 9$ ]}\\\hline % 2 3 5 7 11 13 17 19 23 29 31 37 41 43 % 2 2 1 2 1 2 1 2 2 1 1 2 1 12 & [ 3, 0, 6, 5, 3, 8, 0, 2, 6, 3, 8, 2, 6, 5 ]\\ 16 & [ 0, 0, 0, 2, 0, 2, 0, 2, 0, 0, 2, 2, 0, 2 ]\\ 20 & [ 6, 0, 3, 8, 6, 5, 0, 2, 3, 6, 5, 2, 3, 8 ]\\ 24 & [ 6, 0, 3, 5, 6, 8, 0, 2, 3, 6, 8, 2, 3, 5 ]\\ 32 & [ 3, 0, 6, 8, 3, 5, 0, 2, 6, 3, 5, 2, 6, 8 ]\\ % [ 0, 0, 0, 8, 0, 5, 0, 2, 0, 0, 5, 2, 0, 8 ] \hline\end{tabular}\end{center}} \noindent{}The system of eigenvalues mod~$9$ associated to the weight~$2$ form $F$ on $X_0(27)$ is \begin{center} {\tt[ 0, 0, 0, 8, 0, 5, 0, 2, 0, 0, 5, 2, 0, 8 ],} \end{center} so we conjecture that there is no eigenform~$f$ on $\Gamma_0(N)$ with $N\mid 9$ such that $f\con F\pmod{9}$. \end{conjecture} As evidence, we verified that each of the mod~$9$ reductions of each newform of level~$1$ and weight~$k\leq 74$ has one of the five systems of Hecke eigenvalues listed in the table. We also verified that all newforms of levels~$3$ and $9$ and weight $k\leq 40$ have corresponding system of eigenvalues mod $9$ in the above table. We checked using the method described in Section~\ref{sec:numerical} that there is no newform of level~$1$ with weight $k\leq 300$ that approximates the weight~$2$ form on $X_0(27)$ modulo~$9$. We now make some remarks about pseudo-representations when $p=3$. Let $$\chi : \Z/27\Z \to \Z/9\Z$$ be the mod~$9$ cyclotomic character, so $\chi$ has order~$6$ and if $\gcd(n,3)=1$ then $\chi(n) = n \in \Z/9\Z$. The pseudo-representation corresponding to a form of weight~$k$ giving the system of eigenvalues in the table in Conjecture~\ref{conj:systems3} are \begin{center}\begin{tabular}{|c|c|}\hline Weight & Pseudo-representation \\\hline 12 & $\chi^2\oplus \chi^3$ \\ 16 & $1 \oplus \chi^3$\\ 20 & $\chi^3\oplus\chi^4$\\ 24 & $1\oplus \chi^5$ \\ 32 & $1\oplus \chi$\\ $S_2(\Gamma_0(27))$& $\chi^2\oplus \chi^5$ \\ \hline\end{tabular}\end{center} Note that the square of any pseudo-representation of level~$1$ in the above table has~$1$ as an eigenvalue, but the square of the pseudo-representation attached to $S_2(\Gamma_0(27))$ does not have~$1$ as an eigenvalue. Also, $$F \con f_{16} \otimes \chi^2\pmod{9},$$ where $f_{16}$ is of weight~$16$. The order of~$\chi^2$ is~$3$, so $\chi^2$ is not a power of the Teichm\"uller character (which has order~$2$) and Theorem~\ref{thm:twistapprox} does not apply. Further computations {\em suggest} that the pseudo-representations attached to forms of level~$1$ with coefficients in $\Z_9$ are \begin{center}\begin{tabular}{|c|c|}\hline Weight & Pseudo-representations \\\hline $k\con 0\pmod{6}$ & $1\oplus \chi^5$, $\quad\chi^2\oplus \chi^3$ \\ $k\con 2\pmod{6}$ & $1\oplus \chi$, $\quad\chi^3\oplus \chi^4$\\ $k\con 4\pmod{6}$ & $1 \oplus \chi^3$\\ \hline\end{tabular}\end{center} The pseudo-representations attached to forms of level~$27$ with coefficients in $\Z_9$ seem to be \begin{center}\begin{tabular}{|c|c|}\hline Weight & Pseudo-representations \\\hline $k\con 0\pmod{6}$ & $\chi\oplus \chi^4$\\ $k\con 2\pmod{6}$ & $\chi^2\oplus \chi^5$\\ $k\con 4\pmod{6}$ & $\chi\oplus\chi^2$, $\quad\chi^4\oplus\chi^5$\\ \hline\end{tabular}\end{center} Also note that if $\chi^i\oplus\chi^j$ is one of the pseudo-representations of level $27$ in the table, then the sum of the orders of $\chi^i$ and $\chi^j$ is~$9$, whereas at level $1$ the sum of the orders is at most~$7$. \section{Computations About Approximating Infinite Slope Eigenforms}\label{sec:numerical} In this section, we investigate computationally how well certain infinite slope form can be approximated by finite slope eigenforms. \subsection{A Question About Families} The following question is an analogue of \cite[\S8]{gouvea-mazur:families} but for eigenforms of infinite slope. Fix a prime~$p$ and an integer~$N$ with $(N,p)=1$. \begin{question}\label{ques:family} Suppose $f\in S_{k_0}(\Gamma_0(Np^r))$ is an eigenform having infinite slope (note that $f$ need not be a newform). Is there a ``family'' of eigenforms $\{f_k\}$, with $f_k\in S_k(\Gamma_0(Np))$, where the weights~$k$ run through an arithmetic progression $$ k \in \mathcal{K} = \{k_0 + m p^{\nu}(p-1) \text{ for } m=1,2,\ldots\} $$ for some integer~$\nu$, such that $$ f_k \con f\pmod{p^{n}}, $$ where $n=\ord_p(k-k_0) + 1$? (When $p=2$ set $n=\ord_2(k-k_0)+2$.) \end{question} Our question differs from the one in \cite[\S8]{gouvea-mazur:families} because there the form being approximated has finite slope, whereas our form $f$ does not. We know, as discussed in the previous section, that our question sometimes has a negative answer since it might not be possible to approximate~$f$ at all. \subsection{An Approximation Bound} Let $$ f = \sum_{n\geq 1} a_n q^n\in K[[q]] $$ be a $q$-expansion with coefficients that generate a number field~$K$. Fix a prime~$p$ and an even integer~$k\geq 2$. In order to gather some data about Question~\ref{ques:family}, we now define a reasonably easy to compute upper bound on how well~$f$ can be approximated by an eigenform in $S_k(\Gamma_0(p))$. Suppose $\ell\geq 1$, let $F$ be the characteristic polynomial of $T_\ell$ acting on the space $S_k(\Gamma_0(p))$ of classical cusp forms of weight~$k$ and tame level~$1$, and let~$H$ be the characteristic polynomial of $a_\ell\in K$. Let~$G$ be the resultant of $F(Y)$ and $H(X+Y)$ with respect to the variable $Y$, normalized so that $G$ is monic. Thus the roots of $G$ are the differences $\alpha - \beta$ where $\alpha$ runs through the roots of $F$ and $\beta$ runs through the $\Gal(\Qbar/\Q)$-conjugates of $a_\ell$. We can easily compute the $p$-valuations of the roots of~$G$ without finding the roots, because the $p$-valuations of the roots are the slopes of the newton polygon of~$G$. Let $m_\ell\in \Q\union\{\infty\}$ be the {\em maximum} of the slopes of the Newton polygon of~$G$. Let $$ c_k(f,r) = \min \{m_\ell : \text{ $\ell\leq r$ is prime} \}. $$ We note that computing $c_k(f,r)$ requires knowing only the characteristic polynomials of Hecke operators $T_\ell$ on $S_k(\Gamma_0(p))$ and of $a_\ell$ for primes $\ell\leq r$. \begin{proposition} If there is a normalized eigenform $g\in S_k(\Gamma_0(p))$ such that $f\con g\pmod{p^A}$, then $A\leq c_k(f,r)$ for any~$r$. \end{proposition} \begin{proof} To see this observe that $c_k(f,r)$ is the minimum of the $$ \ord_p(a_n(f) - a_n(g)) $$ where $1\leq n\leq r$ and~$g$ runs through all normalized eigenforms in $S_k(\Gamma_0(p))$, and we run through all possible embeddings of~$f$ and~$g$ into $\Zpbar[[q]]$. \end{proof} The motivation for our definition of $c_k(f,r)$ is that it is straightforward to compute from characteristic polynomials of Hecke operators, even when the coefficients of~$f$ lie in a complicated number field. The number $c_k(f,r)$ could overestimate the true extent to which~$f$ is approximated by an eigenform in $S_k(\Gamma_0(p))$ in at least two ways: \begin{enumerate} \item There is an $r'>r$ such that $c_k(f,r')1$ and all $k$ in an arithmetic progression $\mathcal{K} = \{k_0 + m p^{\nu}(p-1) \text{ for } m=0,1,2,\ldots\}$. (When $p=2$ we should have $\ord_2(k-k_0) + 2\leq c_k(f,r)$.) The following or the results of some computations of $c_k(f,r)$. %We order newforms in dictionary order by the trace of %their $q$-expansion in $\Z[[q]]$. \hspace{-2ex}$\mathbf{p=2}$\vspace{-1ex}: \begin{enumerate} \item For $k_0=6,10,12,14,16,20$ let $f\in S_{k_0}(\Gamma_0(4))$ be the unique newform. Then for all~$k$ with $k_0 < k\leq 100$ we have $c_k(f,47) = \ord_2(k-k_0) + 2$. \item For $k_0=18,22$ let $f\in S_{k_0}(\Gamma_0(4))$ be the unique, up to Galois conjugacy, newform. Then for all $k$ with $k_0 < k\leq 100$ we have $c_k(f,7) = \ord_2(k-k_0) + 2$. \item Let $f\in S_4(\Gamma_0(8))$ be the unique newform. For most $4k_0$ with $k\con k_0\pmod{p-1}$, there is an eigenform $f_k$ on $X_0(Np)$ of weight~$k$ such that $f\con f_k\pmod{p^n}$ where $n=\ord_p(k-k_0)+1$ (or $+2$ if $p=2$). (In general one might have to restrict to~$n$ sufficiently large.) \end{guess} \providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} \begin{thebibliography}{BGR84} \bibitem[B{\"o}c01]{bockle:density} G.~B{\"o}ckle, \emph{On the density of modular points in universal deformation spaces}, Amer. J. Math. \textbf{123} (2001), no.~5, 985--1007. \bibitem[BCP97]{magma} W.~Bosma, J.~Cannon, and C.~Playoust, \emph{The {M}agma algebra system. {I}. {T}he user language}, J. 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