\documentstyle[12pt, verbatim]{article} \font\teneuf=eufm10 \font\fiveeuf=eufm5 \font\seveneuf=eufm7 \newfam\euffam \def\euf{\fam\euffam\teneuf} \textfont\euffam=\teneuf \scriptfont\euffam=\seveneuf \scriptscriptfont\euffam=\fiveeuf \renewcommand{\baselinestretch}{1.3} \newcommand{\itemspace}{\hspace{8pt}} \newcommand{\hfhf}{\left(\frac{1}{2},\frac{1}{2}\right)} \newcommand{\overp}{\overline{P}} \def\thefigure{\arabic{section}.\arabic{subsection}.\arabic{figure}[p]} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% %%% Welcome to the macros of Ilan Vardi! %%% %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%% %%%%%% 1 in margins produced by this %%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \setlength{\textwidth}{6.5in} \setlength{\textheight}{8.9in} \setlength{\topmargin}{0pt} \setlength{\oddsidemargin}{0pt} \setlength{\evensidemargin}{0pt} \setlength{\headheight}{0pt} \setlength{\headsep}{0pt} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%% Bold 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\def\bbbt{{\bf T}} \def\diag{{\rm diag}\,} \def\per{{\rm per}\,} \def\pr{{\rm Pr}\,} \def\calj{{\cal J}} \def\caln{{\cal N}} \def\calg{{\cal G}} \def\calx{{\cal X}} \def\calf{{\cal F}} \def\calm{{\cal M}} \def\cali{{\cal I}} \def\calb{{\cal B}} \def\calh{{\cal H}} \def\calo{{\cal O}} \def\calk{{\cal K}} \def\call{{\cal L}} \def\cala{{\cal A}} \def\calp{{\cal P}} \def\calq{{\cal Q}} \def\cals{{\cal S}} \def\cald{{\cal D}} \def\lll{\prec} \def\ggg{\succ} \def\da{\downarrow} \def\ua{\uparrow} \def\llangle{\langle\!\langle} \def\rrangle{\rangle\! \rangle} \def\doty{\dot{y}} \def\ddoty{\ddot{y}} \def\bfm{{\bf m}} \def\bfw{{\bf w}} \def\bfone{{\bf 1}} \def\bfA{{\bf A}} \def\var{{\rm Var}\,} \def\bfk{{\bf k}} \def\hatX{\hat{X}} \def\hatS{\hat{S}} \def\bfh{{\bf h}} \def\bfY{{\bf Y}} \def\calu{{\cal U}} \def\tr{{\rm tr}\,} \def\re{{\rm Re}\,} \def\im{{\rm Im}\,} \def\deg{{\rm deg}\,} \def\cov{{\rm cov}\,} \def\s{\sigma} %\def\S{\Sigma} \def \bfZ{{\bf Z}} \begin{document} %\title{\Large\bf The Dimension of the Space of Cusp Forms \\ % of Weight One} \title{\large\bf THE DIMENSION OF THE SPACE OF CUSP FORMS \\ OF WEIGHT ONE} \author{\large \bf W. Duke% \thanks{Research supported in part by NSF Grants DMS--9202022 and DMS--9022140} } \maketitle \subsection*{1. Introduction} It is a basic problem to determine the dimension of the space of cusp forms of a given type. For classical holomorphic forms of integral weight larger than one, the dimension is well understood by means of either the Riemann--Roch theorem or the Selberg trace formula. The case of weight one, however, remains mysterious. From the point of view of spectral theory this is because these forms belong to an eigenvalue of the Laplacian which is not isolated; the difficulty of estimating nontrivially the multiplicity of such an eigenvalue is well known. \medskip Suppose, for example, that $q$ is a prime and that $S_1(q)$ is the space of holomorphic cusp forms for $\Gamma_0(q)$ of weight 1 with character $\left({\cdot\over q}\right )$, the Legendre symbol. No nonzero cusp forms may exist unless $\left({-1\over q}\right ) = -1$, so assume $q\equiv 3\; (\mod \, 4)$. Hecke discovered that the existence of such cusp forms is tied up with the class number $h$ of ${\bf Q}(\sqrt{-q})$; if $\chi$ is any nontrivial (hence non--real) class character then $$ \sum_{{\euf a}} \chi({\euf a}) e(N({\euf a})\, z) \in S_1(q)\,, \leqno(1) $$ where ${\euf a}$ runs over all nonzero integral ideals of ${\bf Q}(\sqrt{-q})$. There are $(h-1)/2$ independent forms of this type so by Siegel's theorem we have the (ineffective) lower bound $$ {\rm dim}\, S_1(q) \mathop{\gg}_{\varepsilon} q^{1/2-\e} $$ for all $\e > 0$. In general, $S_1(q)$ is not spanned by forms of Hecke's type. The construction of specific examples demonstrating this is an active area of research (see \cite{F} and its references). Such exotic forms seem quite rare, however, and it appears reasonable to expect that in fact $$ {\rm dim}\, S_1(q) =\frac{1}{2}\, (h-1)+O(q^\e)\,. $$ In particular, this would imply that ${\rm dim}\, S_1(q) \ll q^{1/2} \log q$. Rather less, however, is actually known. Serre has shown \cite{S} that if $q = 24m-1$ or $24m + 7$ then ${\rm dim}\, S_1(q) \le m - (h-1)/2$, while if $q = 24m + 11 $ or $24m + 19$ then ${\rm dim}\, S_1(q) \le m - h+1$. Sarnak has informed me that the Selberg trace formula for weight one \cite[Chapter 9]{Hej} with a suitably chosen test function yields the bound $$ {\rm dim}\, S_1(q)\ll \frac{q}{\log q} $$ which, for large values of $q$, is currently the best known.\footnote{In an unpublished note J.-M. Deshouillers and H. Iwaniec obtained in this way the bound $O(q \log^{-3} q)$ for the multiplicity of the eigenvalue $1/4$ in the case of weight zero Maass cusp forms for $\Gamma_0(q)$ with trivial character.} The main object here is to improve this estimate. \medskip\noindent {\bf Theorem 1.} {\sl For $q$ prime $$ {\rm dim}\, S_1(q)\ll q^{11/12}\, \log^4 q\,, $$ with an absolute implied constant. } \bigskip Roughly speaking, the idea of the proof is to exploit two conflicting properties of the Fourier coefficients of newforms in $S_1(q)$ not of Hecke's type (1): their approximate orthogonality and the finiteness of the number of their possible values at primes. The first property is a consequence of their belonging to automorphic forms while the second is a consequence of Deligne--Serre theorem. Taken together, these properties limit the number of possible newforms which may exist. Following Serre \cite{S}, Theorem~1 has an application to the quotient $X_0^*(q)$ of the modular curve $X_0(q)$ by the Fricke involution $z\mapsto -1/qz$ when $q = 24 m - 1$ is prime. In this case the genus of $X_0^*(q)$ is $m-(h-1)/2$. \bigskip\noindent {\bf Corollary.} {\sl For $q = 24m - 1$ prime, the space of differential forms of the first kind on $X_0^*(q)$ with a zero of order at least $m$ at the cusp has dimension which is $O(q^{11/12} \log^4 q)$. } \bigskip A different kind of application of these ideas is to bound the number $m _4(-q)$ of quartic number fields of discriminant $-q$. \bigskip\noindent {\bf Theorem 2.} {\sl For $q$ prime $$ m _4(-q) \ll q^{7/8} \log ^4 q\,, $$ with an absolute implied constant. } \bigskip\noindent To put this result in context it may be of interest to show what follows from algebraic number theory and trivial bounds for class numbers. By means of class field theory Heilbronn \cite{Hei} showed that $$ m_4(-q) = \frac{4}{3}\, \sum_k h_2(k)\,, $$ where $k$ runs over all cubic number fields of discriminant $-q$. Here, for any $\ell$ and any number field $k$, $h_\ell(k)$ denotes the number of ideal classes of $k$ of (exact) order $\ell$. Furthermore, by \cite{Ha}, the number of cubic fields in the sum is $\frac{3}{2}\, h_3({\bf Q}(\sqrt{-q}))$ . For the class number $h(k)$ of any number field $k$ of degree $n>1$ and discriminant $D$ we have the bound $$ h(k) \ll |D|^{1/2}\log^{n-1}|D| $$ where the implied constant depends only on $n$ (see \cite[p.153.]{N}). Since $h_\ell(k) \leq h(k)$ we deduce that $$ m _4(-q) \ll q\, \log^3 q $$ with an absolute implied constant. The improvement of this given in Theorem~2 requires both the classification of quartic fields of discriminant $-q$ by odd octahedral Galois representations of conductor $q$ given in \cite{S} and the proof in this case of the Artin conjecture given in \cite{T}. If we assume the Artin conjecture for icosahedral representations then similarly we can deduce that the number of non-real quintic fields of discriminant $q^2$ whose normal closure has Galois group $A_5$ is $O(q^{11/12}\log^4 q)$. \subsection*{2. Approximate orthogonality of Fourier Coefficients} For $q\in {\bf Z}^+$ and $\e$ an odd Dirichlet character $\mod q$, let $S_1(q)=S_1(q,\e)$ be the set of all holomorphic cusp forms for $\Gamma_0(q)$ of weight 1 with character $\e$. Thus $f\in S_1(q)$ satisfies for $\gamma\in {a\; b\choose c\; d}\in \Gamma_0(q)$ $$ f(\gamma z) = \e(d) \, (cz+d)\, f(z) $$ and $({\rm Im} \, z)^{1/2} \, |f(z)|$ is uniformly bounded on the upper half--plane ${\bf H}$. The vector space $S_1(q)$ is finite dimensional and has an inner product $$ \langle f, g\rangle = \int\limits_{\Gamma_0(q)\backslash {\bf H}} f(z) \bar g(z)\, y^{-1}dxdy\,. $$ Each $f\in S_1(q)$ has the Fourier expansion at $\infty$ $$ f(z) =\sum_{n\ge 1} a_f(n)\, e(nz)\,. $$ The object of this section is to establish the following mean value result which expresses the approximate orthogonality of the $a_f(n)$ over any fixed orthonormal basis ${\cal B}$ for $S_1(q)$. \medskip\noindent {\bf Proposition 1.} {\sl For arbitrary $c_n\in {\bf C}$ with $1\le n\le N$ we have $$ \sum_{f\in {\cal B}} \, \left|\sum_{n\le N} c_n\, a_f(n)\right|^2 \ll \left(1+\frac{N}{q}\right)\, \sum_{n\le N}\, |c_n|^2 $$ with an absolute implied constant. } \medskip\noindent The proof we give of this uses the following duality principle. \medskip\noindent {\bf Lemma 1.} {\sl Suppose that $V$ is a finite dimensional inner product space over ${\bf C}$ with an orthonormal basis ${\cal B}$. Let $\{v_1,\ldots,v_N\}\subset V$ be a fixed set of vectors and $\Delta$ be a positive number. Then the inequality $$ \sum_{n\le N}\, |\langle u,v_n \rangle |^2 \le \Delta\, \langle u,u\rangle \leqno(2) $$ holds for all $u\in V$ if and only if $$ \sum_{f\in {\cal B}}\, \left| \sum_{n\le N} c_n\, \langle f, v_n\rangle \right|^2\le \Delta\, \sum_{n\le N}\, |c_n|^2 \leqno(3) $$ holds for all $c_n\in {\bf C}$. } \medskip\noindent {\bf Proof:} By Parseval's equality $$ \eqalign{ \sum_{f\in {\cal B}} \left|\sum_{n \le N} c_n \, \langle f, v_n\rangle \right|^2 &= \mathop{\sum\sum}_{n,m\le N}\, c_n\bar c_m \sum_{f\in {\cal B}} \langle f, v_n\rangle\, \overline{\langle f, v_m\rangle} \cr\noalign{\vskip 5pt} &= \mathop{\sum\sum}_{n,m\le N} c_n \bar c_m \langle v_n,v_m\rangle\,. \cr } \leqno(4) $$ Now the Lemma follows from Boas' generalization of Bessel's inequality as given in \cite[p.~151]{D}. For the sake of completeness we will include the rest of the proof that $(2)\Rightarrow (3)$ here since this is the part which we need and refer to \cite{D} for the converse. Take $u=\sum_{m\le N} c_m v_m$ in (2); then $$ \eqalign{ \langle u, u\rangle &= \mathop{\sum\sum}_{n,m\le N} c_n \bar c_m\langle v_n, v_m\rangle =\sum_{n\le N} c_n \langle v_n, u\rangle \cr\noalign{\vskip 5pt} &\le \left(\sum_{n\le N} |\langle u, v_n\rangle|^2\right)^{1/2}\, \left(\sum_{n\le N}|c_n|^2\right)^{1/2} \cr\noalign{\vskip 5pt} &\le \Delta^{1/2} \langle u, u\rangle ^{1/2}\, \left(\sum |c_n|^2\right)^{1/2} \cr } \leqno(5) $$ by (2). Hence $$ \langle u, u\rangle \le \Delta\, \sum_{n\le N} |c_n |^2 $$ and (3) follows by (4) and (5). $\Box$ \medskip Taking $v_n = \sum_{f\in {\cal B}} \bar a_f(n)\, f$, we see that Proposition~1 is reduced to the following Lemma. \medskip\noindent {\samepage {\bf Lemma 2.} {\sl For any $f\in S_1(q)$ $$ \sum_{n \le N} \, |a_f(n)|^2 \ll \left(1 +\frac{N}{q}\right)\, \langle f, f\rangle $$ with an absolute implied constant. }} \medskip\noindent {\bf Proof:} We employ a technique of Iwaniec (unpublished) to bound such sums uniformly. For any $y> 0$ we have $$ \sum_{n\ge 1} e^{-4\pi n y} \, |a_f(n)|^2 =\int_0^1 |f(x + iy)|^2 \, dx. $$ Thus, for any $Y>0$, $$ \sum_{n\le N} F_Y(n)\, |a_f(n)|^2 \leq \int _Y^\infty \int _0^1 |f(z)|^2 y^{-1} \, dxdy\,, \leqno(6) $$ where $$ F_Y(n)=\int_1^\infty e^{-4\pi n Y y}y^{-1} \, dy\, \ge \int_1^\infty e^{-4\pi N Yy}\, y^{-1}\, dy \leqno(7) $$ for $n\le N$. Setting $$P(Y) = \{z\in {\bf H}\, ;\, 0 < {\rm Re}\, z\le 1,\; {\rm Im}\, z> Y\} $$ we have $$ \int_Y^\infty \int_0^1 |f(z)|^2\, y^{-1}\, dxdy \le \mathop{\max}_{z\in P(Y)\,}\# \{\gamma \in \Gamma_0(q)/\{\pm 1\}\, ;\, \gamma z\in P(Y)\} \, \langle f, f\rangle \,. \leqno(8) $$ Now for fixed $z=x+iy \in {\bf H}$ the condition that $\gamma z = (az+b)/(cqz+d) \in P(Y)$ is imposed by requiring that $$ {\rm Im}\,((az+b)/(cqz+d)) = y |cqz+d|^{-2}>Y \leqno(9) $$ and that $$ 0<{\rm Re}\,((az+b)/(cqz+d))\leq 1. \leqno(10) $$ For a fixed $c$ and $d$ satisfying (9) $a$ and $b$ are determined by (10). It follows that for $z \in {\bf H}$ $$ \eqalign{ \#\{\gamma \in \Gamma_0(q)/\{\pm 1\}\, ; \gamma z\in P(Y)\} &\le 1 + \# \{c > 0, d\,;\, |cqz+d|^2 < yY^{-1}\} \cr\noalign{\vskip 5pt} &\le 1 +\# \{c, d\,;\, 00$. Thus $$ \eqalign{ \sum_{n\ge 1} b(n) F(n/q^2) &= \frac{1}{2\pi i } \int\limits_{{\rm Re}\, s=2} \hat F(s)\varphi(s) q^{2s} ds \cr\noalign{\vskip 5pt} &= 2\pi^2 q\, \langle f, f\rangle + \frac{1}{2\pi i} \int\limits_{{\rm Re}\, s = -1} \hat F(s) \varphi(s) q^{2s} ds. } \leqno(17) $$ It follows from (16), (15) and (12) that $$ |\varphi(-1+it)| \leq (4\pi^2)^{-3} q^3 \frac{|\Gamma(2-it)|^2}{|\Gamma(-1+it)|^2} (1+q^{-2})|\zeta(4-2it)| \sum_{n \geq 1} d(n)^2 n^{-2} $$ and by Stirling's formula this is $ \ll q^3 (|t|+ 1)^6$ with an absolute constant. Hence $$ \frac{1}{2\pi i}\int_{{\rm Re}\,(s)= -1}\hat{F}(s)\varphi(s)q^{2s}ds \ll q \int_{-\infty}^{\infty}|\hat{F}(-1+it)|(|t|+1)^6dt \ll q \leqno(18) $$ since $\hat{F}(-1+it)\ll (1+|t|)^{-8}$. By (17) and (18) we get $$ \langle f, f\rangle = \frac{1}{2\pi^2 q}\, \sum_{n\ge 1} b(n) \, F(n/q^2) + O(1)\, $$ where the implied constant depends only on $F$. Now, by (15) and (12) $$ 0\le b(n) \le 2\, \sum_{n=m\ell^2} d^2(m) = 2d_4(n) $$ where $d_4(n)$ is the number of factorizations of $n$ into $4$ factors. Thus $$ \langle f,f\rangle \ll \frac{1}{q}\, \sum_{n\ge 1} d_4(n)\, F(n/q^2)\ll q\, \log^3 q. $$ Fixing $F$ gives the result with an absolute implied constant. $\Box$ \bigskip\noindent (b) The fact that $f$ cannot be tetrahedral is shown in \cite[Theorem 7(c)]{S}. It is also observed in \cite[p.~362]{S} that if $p\ne q$ and $f\in N_{{\rm oct}}$ then $$ \left(\frac{p}{q}\right)\, a(p^2)\in \{-1, 0, 1, 3\}\,, \leqno(19) $$ while if $f\in N _{{\rm ico}}$ then $$ \left(\frac{p}{q}\right)\, a(p^2) \in \left\{-1, 0, 3, \frac{1+\sqrt{5}}{2},\frac{1-\sqrt{5}}{2}\right\}\,. \leqno(20) $$ For general $f\in N_1(q, \e)$ it follows from (11) that if we set $x=\bar \e(p) \, a_f(p^2)$ then for $p\ne q$ and $n\ge 0$ $$ \bar \e ^n (p)\, a_f(p^{2n})=P_n(x)\,, \leqno(21) $$ where $P_0(x)=1$, $P_1(x)=x$, and $P_{n+1}(x) = (x-1)\, P_n(x) - P_{n-1}(x)$ for $n\ge 1$. Thus $P_2(x) = x^2 - x - 1$, $P_3(x) = x^3 - 2 x^2 - x + 1$, etc. One may check that $$ x(x+1)(x-1) (x-3) + 1 = P_4(x) - P_2(x) - P_1(x)\,, $$ while $$ x(x+1)(x-2) (x-3) (x^2-x-1) + 1 = P_6(x) - P_4(x) -P_1(x)\,. $$ Thus by (19)--(21) we finish the proof of (b).~$\Box$ \medskip\noindent {\bf Remark:} It has been checked that no linear form $\sum_{\ell=1}^7 c_\ell\, a_f(p^{\ell})$ takes positive values for all possible values of $a_f(p)$, $p\ne q$, when $f\in N_{{\rm oct}}$. A similar remark applies to $f\in N_{{\rm ico}}$. \subsection*{4. Counting newforms and quartic fields} By combining Propositions 1 and 2 we may now estimate the number of newforms of octahedral or icosahedral type. \medskip\noindent {\bf Proposition 3.} {\sl For $q$ prime $$ \leqalignno{ &\# N_{{\rm oct}} \ll q^{7/8} \log^4 q &{\rm (a)} \cr\noalign{\vskip 5pt} &\# N_{{\rm ico}} \ll q^{11/12} \log^4 q &{\rm (b)} \cr } $$ with absolute implied constants. } \medskip\noindent {\bf Proof:} By Propositions 1 and 2(a) we have for any $c_n \in {\bf C}$ $$ \sum_{f\in N_1(q)}\, |\sum_{n\le N}c_n\, a_f(n)|^2 \ll (q+N)\, \log^3 q \sum_{n\le N}|c_n|^2\,. $$ Thus we deduce by positivity the inequality $$ \sum_{f\in N_{{\rm oct}}}\, |\sum_{n\le N}c_n\, a_f(n)|^2 \ll (q+N)\, \log^3 q \sum_{n\le N}|c_n|^2. \leqno(22) $$ Choose $N=q$ and $$ c_n =\cases{ 1, & $n=p^8\le q$,\cr -1, & $n=p^4\le q^{1/2}$,\cr -\left(\frac{p}{q}\right), & $n=p^2 \le q^{1/4}$, \cr 0, & {\rm otherwise}, \cr } $$ for $p$ prime. By the prime number theorem and (13) $$ \sum_{n\le N} c_n a_f(n) \sim \frac{8q^{1/8}}{\log q},\qquad {\rm for }\; f\in N_{{\rm oct}}\,, $$ while $$ \sum_{n\le N}|c_n|^2 \sim \frac{24 q^{1/8}}{\log q}\,. $$ Hence, by (22) $$ \# N_{{\rm oct}}\ll q^{7/8}\, \log^4 q\,. $$ A similar argument works for $N_{{\rm ico}}$.~$\Box$ \medskip Theorem 1 now follows from Propositions 2(b) and 3 since $$ \# N_{{\rm dih}} = \frac{h-1}{2}\ll q^{1/2}\log q\,. $$ \medskip For its Corollary we use the fact, shown in \cite{S}, that the required space of differentials has dimension $$ \frac{1}{2}\, {\rm dim}\, S_1(q) - \frac{1}{4}\, (h-1)\,. $$ \medskip Theorem 2 follows from Proposition 3(a), Theorem 8 of \cite{S} and Tunnell's proof \cite{T} of the Artin conjecture for octahedral Galois representations. Here we also use the fact that the Galois closure of any quartic extension with discriminant $-q$ has Galois group $S_4$, since otherwise the discriminant would not be square free (e.g., see \cite{B}). \medskip\noindent {\it Acknowledgement.} I would like to thank J-P. Serre for his many helpful comments. This paper was written while I was participating in the 1994-95 program on Automorphic Forms at MSRI. I thank the organizers and the Institute for providing financial support and a congenial atmosphere in which to work. %\newpage \begin{thebibliography}{xxxxxxxx} \bibitem[B]{B} A.M. Baily, {\sl On the density of discriminants of quartic fields,} J. Reine Angew. Math. {\bf 315} (1980) 190--210. \bibitem[D]{D} H. Davenport, {\sl Multiplicative Number Theory,} Springer 1980. \bibitem[DS]{DS} P. Deligne and J--P. Serre, {\sl Formes modulaires de poids 1,} Ann. Sci. Ec. Norm. Sup. {\bf 7} (1974) 507--530, in Serre's Collected Papers, III, 193--216. \bibitem[F]{F} G. Frey, {\sl Construction and arithmetical applications of modular forms of low weight,} in C.R.M. Proc. and Lecture Notes, V.~4, Elliptic Curves and Related Topics, A.M.S. (1994), 1--21. \bibitem[Ha]{Ha} H. Hasse, {\sl Arithmetische Theorie der Kubischen Zahlk\"orper auf Klassenk\"orper--theoretischer Grundlage,} Math. Z. {\bf 31} (1930) 565--582. \bibitem[Hei]{Hei} H. Heilbronn, {\sl On the $2$--classgroup of cubic fields,} in Studies in Pure Math. (L. Mirsky, ed.), Academic Press 1971, 117--119. \bibitem[Hej]{Hej} D. A. Hejhal, {\sl The Selberg Trace Formula for $\mbox{PSL}(2, {\bf R})$, Volume 2,} Springer LNM \# 1001, 1983. \bibitem[N]{N} W. Narkiewicz, {\sl Elementary and Analytic Theory of Numbers,} Springer 1990. \bibitem[L]{L} W.--C. Li, {\sl $L$--series of Rankin type and their functional equations,} Math. Ann. {\bf 244} (1979), 135--166. \bibitem[S]{S} J--P. Serre, {\sl Modular forms of weight one and Galois representations,} Algebraic Number Fields, ed. by A. Fr\"ohlich, Academic Press 1977, 193--268, in Serre's Collected Papers, III, 292--367. \bibitem[T]{T} J. Tunnell, {\sl Artin's conjecture for representations of octahedral type,} Bull. A.M.S. {\bf 5} (1981), 173--175. \end{thebibliography} \bigskip\bigskip \renewcommand{\baselinestretch}{0.5} \noindent {\sc William Duke} Department of Mathematics, Rutgers University New Brunswick, NJ 08903 {\it email\/}: duke\verb+@+math.rutgers.edu \bigskip \noindent{\it Current Address\/}: Mathematical Sciences Research Institute 1000 Centennial Drive Berkeley, CA 94720--5070 \end{document} \address{William Duke\endgraf Department of Mathematics, Rutgers University \endgraf New Brunswick, NJ 08903 \endgraf {\it Current Address\/}\endgraf Mathematical Sciences Research Institute\endgraf 1000 Centennial Drive \endgraf Berkeley, CA 94720--5070} \email{duke\@math.rutgers.edu}