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Author: William A. Stein
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% by Kevin Buzzard and William Stein
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\title{A mod five approach to modularity of icosahedral
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Galois representations}
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\author{Kevin Buzzard and William A. Stein}
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\begin{document}
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\maketitle
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\begin{abstract}
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We give eight new examples of icosahedral Galois representations that
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satisfy Artin's conjecture on holomorphicity of their $L$-function.
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We give in detail one example of an icosahedral representation of
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conductor ${\bf 1376}=2^5\cdot 43$ that satisfies Artin's conjecture.
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We also briefly explain the computations behind seven additional
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examples of conductors ${\bf 2416}=2^4\cdot 151$, ${\bf 3184}=2^4\cdot
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199$, ${\bf 3556}=2^2\cdot 7\cdot 127$, ${\bf 3756}=2^2\cdot 3\cdot
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313$, ${\bf 4108}=2^2\cdot 13\cdot 79$, ${\bf 4288}=2^6\cdot 67$, and
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${\bf 5373}=3^3\cdot 199$.
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\end{abstract}
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\section*{Introduction}
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Consider a continuous irreducible Galois representation
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$$\rho:\galq\ra\GL_n(\C)$$
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with $n > 1$.
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Inspired by his reciprocity law,
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Artin conjectured in~\cite{artin:conjecture} that
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$L(\rho,s)$ has an analytic continuation to the whole complex plane.
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Many of the known cases of this conjecture were obtained by
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proving the apparently stronger assertion that~$\rho$ is \defn{automorphic},
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in the sense that the $L$-function of~$\rho$ is equal to the $L$-function
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of a certain automorphic representation (whose $L$-function is known to have
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analytic continuation). In the special case where $n=2$ and $\rho$ is in
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addition assumed to be odd, the automorphic representation in question
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should be the one associated to a classical weight~$1$
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modular eigenform, and in fact there is conjectured to be a
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bijection between such~$\rho$ and the set of all weight~$1$
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cuspidal newforms, which should
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preserve $L$-functions. It is this bijection
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that we are concerned with in this paper, so assume for the rest
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of the paper that $n=2$ and~$\rho$ is odd.
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In this special case, the construction
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of~\cite{deligne-serre} shows how to construct a continuous irreducible
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odd 2-dimensional representation from a weight~$1$ newform, and the problem
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is to go the other way. Say that a representation is \defn{modular}
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if it arises in this way.
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If the image of~$\rho$ is solvable,
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then~$\rho$ is known to be modular
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\cite{langlands:basechange, tunnell:artin};
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if the image is not solvable, then $\im(\rho)$ in $\PGL_2(\C)$
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is isomorphic to the
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alternating group~$A_5$, and the modularity of~$\rho$
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is, in general, unknown. We call such a 2-dimensional representation an
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``icosahedral representation''.
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The published literature contains only eight examples (up to twist)
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of odd icosahedral Galois representations that are known to satisfy Artin's
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conjecture: one of conductor $800=2^5\cdot 5^2$
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(see \cite{buhler:thesis}), and seven of conductors:
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$2083,\, 2^2\cdot 487,\, 2^2\cdot 751,\,
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2^2 \cdot 887,\, 2^2\cdot 919,\,
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2^5\cdot 73,\,\text{ and } 2^5\cdot 193$
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(see \cite{freyetal}).
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After the first draft of this paper was written, the
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preprint~\cite{bdsbt} appeared, which contains a general theorem that
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yields infinitely many (up to twist) modular icosahedral representations.
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However, we feel that our work, although much less powerful, is still
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of some worth, because it gives an effective computational approach to
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proving that certain mod~5 representations are modular, without
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computing any spaces of weight~1 forms or using effective versions of
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the Chebotar\"ev density theorem. We also note that the
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main theorem of~\cite{bdsbt} does not apply to any of the examples
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considered in the present paper. Very recently,
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the preprint~\cite{taylor:artin2}
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appeared, which gives local conditions under which an icosahedral
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representation is modular. In particular, \cite{taylor:artin2} also
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proves that the first three
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examples in the present paper, of conductors 1376, 2416, 3184,
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are modular; these correspond to the first, third, and fourth
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equations at the end of~\cite{taylor:artin2}.
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However, \cite{taylor:artin2} does not apply to
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our remaining five examples.
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In this paper we give eight new examples of modular icosahedral
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representations that were computed
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by applying the main theorem of~\cite{buzzard-taylor} to
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the mod~$5$ reduction of~$\rho$.
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We verify modularity mod~$5$ on a case-by-case basis. Later we shall
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explain our approach more carefully, but let us briefly summarise it here.
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By~\cite{buzzard-taylor},
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the problem is to show that the mod~5 reduction of~$\rho$ is modular.
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We do this by finding a candidate mod~5 modular form at weight~5
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and then, using the table of icosahedral extensions of $\Q$ in~\cite{freyetal}
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and what we know about the 5-adic representation attached to our candidate
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form, we deduce that the mod~5 representation attached to our candidate
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form must be the reduction of~$\rho$. In particular, this paper gives
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a computational methods for checking the modularity of certain mod~5
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representations whose conductors are not too large. We now give
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more details.
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In each of our examples it is easy to compute a few Hecke operators
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and be morally convinced that a mod~$5$ representation should be modular;
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it is far more difficult to prove this.
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Effective variants of the Chebotarev density theorem require
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that we check vastly more traces of Frobenius than is practical.
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Instead we use the Local Langlands theorem for $\GL_2$, the
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theory of companion forms, and Table~2 of~\cite{freyetal},
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to provide proofs of modularity
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in certain cases.
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More precisely, let~$K$ be an icosahedral extension of~$\Q$ that is not
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totally real, and consider a minimal lift $\rho:\GQ\ra \GL_2(\C)$
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of
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$$\GQ\ra \Gal(K/\Q)\ncisom{}A_5\subset \PGL_2(\C);$$
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the lift is minimal in the sense that its conductor is minimal.
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Assume that~$5$ does not ramify in~$K$, and that
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a Frobenius element at~$5$ in $\Gal(K/\Q)$ does not have order~$1$ or~$5$.
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Inspired by the possibility that~$\rho$ is modular,
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we search for a mod~$5$ modular form of weight~$5$ whose existence would
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be forced by modularity of~$\rho$. Indeed, we find
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a candidate mod~$5$ form~$f$, and then prove that the fixed field
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of the kernel of the projective mod~$5$ representation
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associated to a certain twist of~$f$ must be~$K$.
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This proves that the mod~$5$ reduction of a twist
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of~$\rho$ is modular, and the main theorem
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of \cite{buzzard-taylor} then implies
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that~$\rho$ is modular.
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We carried out this program for icosahedral representations
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of the following conductors:
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${\bf 1376} = 2^5\cdot 43$,
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${\bf 2416}=2^4\cdot 151$,
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${\bf 3184}=2^4\cdot 199$,
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${\bf 3556}=2^2\cdot 7\cdot 127$,
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${\bf 3756}=2^2\cdot 3\cdot 313$,
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${\bf 4108}=2^2\cdot 13\cdot 79$,
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${\bf 4288}=2^6\cdot 67$, and
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${\bf 5373}=3^3\cdot 199$.
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We choose an icosahedral field~$K$ and representation~$\rho$,
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then proceed as follows:
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\vspace{.5ex}
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\begin{numlist}
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\item Search for a form~$f \in S_5(N,\eps;\Fbar_5)$ whose
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associated mod~$5$ Galois representation looks like
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it is the mod~$5$ reduction of~$\rho$.
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\item Twist~$f$ to obtain an eigenform~$g$ with coefficients in~$\F_5$.
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\item Prove that~$\rho_g$ is unramified at~$5$ by finding a companion form.
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\item Prove that the image of $\proj\rho_g$ is~$A_5$ by ruling out all
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other possibilities.
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\item Prove that the fixed field~$L$ of $\proj\rho_g$ has
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root field of discriminant at most $2083^2$,
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so~$L$ is in Table~2 of~\cite{freyetal}; deduce that~$L=K$.
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\item Apply the main theorem of~\cite{buzzard-taylor}
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to a lift of $\rhobar=\rho_g$
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to conclude that~$\rho$ is modular.
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\end{numlist}
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\section{Modularity of an icosahedral representation of
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conductor~$1376=2^5\cdot 43$}\label{sec:1376}
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In this section we prove the following theorem.
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\begin{theorem}\label{thm:1376}
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The icosahedral representations whose corresponding
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icosahedral extension
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is the splitting field of $x^5 + 2x^4+6x^3+8x^2+10x+8$
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are modular.
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\end{theorem}
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Let~$K$ be the splitting field of $h=x^5 + 2x^4+6x^3+8x^2+10x+8$.
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The Galois group of~$K$ is~$A_5$, so we obtain a homomorphism
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$G_\Q\ra{}A_5\subset \PGL_2(\C)$;
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let $\rho:G_\Q\ra\GL_2(\C)$ be a minimal lift, minimal
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in the sense that the Artin conductor of~$\rho$ is minimal.
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By Table~$A_5$ of~\cite{buhler:thesis}, the conductor of~$\rho$
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is $N=1376=2^5\cdot 43$. Since
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$h\con (x-1)(x^2-x+1)(x^2-x+2)\pmod{5}$,
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and ${\rm disc}(h)$ is coprime to~$5$,
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any Frobenius element at~$5$ in $\Gal(K/\Q)$ has order~$2$.
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We use the notation of Tables 3.1 and 3.2 of~\cite[pg. 46]{buhler:thesis};
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from Table 3.2 we see that the type of~$\rho$ at~$2$
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is~$17$ and the type at~$43$ is~$2$.
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The mod~$N$ Dirichlet character~$\eps=\det(\rho)$
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factors as~$\eps=\eps_2\cdot \eps_{43}$ where~$\eps_2$ is
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a character mod~$2^5$ and~$\eps_{43}$ is a character mod~$43$.
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Corresponding to each type in Buhler's table, there is a character,
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and fortunately Buhler's level $800$ example also was of type~$17$ at~$2$
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(see the first line of~\cite[Table~3.2]{buhler:thesis}).
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By~\cite[pg.~80]{buhler:thesis} $\eps_2$ is the unique
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character of conductor~$4$ and order~$2$.
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A local computation shows that the image
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of~$\eps_{43}$ has order~$3$.
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If~$\rho$ is modular, then there is a weight~$1$
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newform $f_?\in S_1(N,\eps;\Qbar)$ that gives rise to~$\rho$.
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Suppose for the moment that~$\rho$ is modular, so that~$f_?$ exists.
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Choose a prime of~$\overline{\Z}$ lying
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over~$5$, and denote by~$\fbar_?$ the reduction
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of $f_?$ modulo this prime. The Eisenstein series
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$E_4\in M_4(1;\F_5)$ is congruent to~$1$ modulo~$5$, so
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$E_4\cdot{}\fbar_?\in S_5(N,\eps;\Fbar_5)$ has the same $q$-expansion
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as $\fbar_?$. Using a computer, we can search for a
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form $f\in S_5(N,\eps,\Fbar_5)$ that has the same
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$q$-expansion as the conjectural form $E_4\cdot{}\fbar_?$.
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Instead of multiplying $\fbar_?$ by~$E_4$, we could have multiplied
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it by an Eisenstein series of weight~$1$, level~$5$, and character $\eps'$.
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We used $E_4$ because the dimension of $S_5(N,\eps;\Fbar_5)$
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is~$696$ whereas the dimension of the relevant space
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$S_2(5\cdot 1376, \eps_{43})$ of weight~$2$ cusp forms is~$1040$.
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\subsection{Searching for the newform~$f$}
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Using modular symbols (see Section~\ref{sec:modsym}) we
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compute (at least up to semi-simplification) the space
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$S_5(1376,\eps;\F_{25})$. Note that there is injective map
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from the image of~$\eps$ into $\F_{25}^*$. By computing
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the kernels of various Hecke operators on this space, we find~$f$.
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In the following computations, we represent nonzero elements of~$\F_{25}$
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as powers of a generator~$\alp$ of~$\F_{25}^*$, which satisfies
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$$\alp^2 + 4\alp + 2=0.$$
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Our character $\eps_{43}$ was represented
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as the map sending $(1,3)\in(\Z/2^5\Z)^*\cross(\Z/43\Z)^*$ to
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$2\alp+1$. Note that~3 is a primitive root mod~43, and that $2\alp+1$
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has order~3.
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If the least common multiple of the degrees of the factors of
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the polynomial~$h$ modulo an
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unramified prime~$p$ is~$2$, then $\Frob_p\in\Gal(K/\Q)$
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has order~$2$. The minimal polynomial of $\rho(\Frob_p)\in\GL_2(\C)$
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is then $x^2-1$, so $\rho(\Frob_p)$ has trace~$0$.
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The first three primes $p \nmid 5\cdot 1376$ such
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that $\rho(\Frob_p)$ has order~$2$ are $p= 19,31,97$.
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We computed the mod~$5$ reduction $\sS_5(1376,\eps;\F_{25})^{+}$
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of the $\Z_5[\zeta_3]$-lattice of
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modular symbols of level~$1376$ and
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character~$\tilde{\eps}$ where
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complex conjugation acts as $+1$.
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Here~$\tilde{\eps}$ denotes the Teichm\"uller lift of~$\eps$.
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Let~$V$ be the intersection of the kernels of $T_{19}$, $T_{31}$, and
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$T_{97}$ inside of the space $\sS_5(1376,\eps;\F_{25})^{+}$ of mod~5
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modular symbols.
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The space~$V$ is $8$-dimensional,
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and no doubt all the eigenforms in this space give rise to~$\rho$ or one
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of its twists. One of the eigenvalues of~$T_3$ on this space
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is~$\alp^{16}$, and the kernel $V_1$ of $T_3-\alp^{16}$ is $2$-dimensional
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over $\F_{25}$. The Hecke operator~$T_5$ acted as a diagonalisable matrix on
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$V_1$, with eigenvalues $\alp^{10}$ and $\alp^{22}$, so the corresponding
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two systems of eigenvalues must correspond to mod~$5$ modular eigenforms,
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and furthermore we must have found all mod~$5$ modular eigenforms
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of this level, weight and character,
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such that $a_{19}=a_{31}=0$ and $a_3=\alp^{16}$.
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\begin{remark}
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The careful reader might wonder how we know that the
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systems of mod~$5$ eigenvalues really do correspond to mod~$5$ modular
362
forms, and not to perhaps some strange mod~$5$ torsion in the space of
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modular symbols. However, we eliminated this possibility by
364
computing the dimension of the full space of mod~$5$ modular
365
symbols where complex conjugation acts as~$+1$, and checking that it
366
equals $696$, the dimension of $S_5(1376,\tilde{\eps},\C)$, which we
367
computed using the formula in \cite{cohen-oesterle}.
368
\end{remark}
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Let~$f$ be the eigenform in~$V_1$ that satisfies
371
$a_5=\alp^{22}$; the $q$-expansion of~$f$ begins
372
$$f=q + \alp^{16}q^3 + \alp^{22}q^5 + \alp^{14}q^7
373
+ \alp^{14}q^9 + 4q^{11}+\cdots.$$
374
Further eigenvalues are given in Table~\ref{table:1376}.
375
The primes~$p$ in the table such that~$a_p=0$ are
376
exactly those
377
predicted by considering the splitting behavior of~$h$.
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This is strong evidence that~$\rho$ is modular,
379
and also that our modular symbols algorithm have been correctly
380
implemented.
381
382
\begin{table}
383
\caption{\label{table:1376}Eigenvalues of~$f$}
384
\begin{center}
385
$$\begin{array}{|rl|}\hline
386
2&0\\
387
3&\alpha^{16}\\
388
5&\alpha^{22}\\
389
7&\alpha^{14}\\
390
11&4\\
391
13&\alpha^{14}\\
392
17&\alpha^{14}\\
393
19&0\\
394
23&\alpha^{16}\\
395
29&\alpha^{8}\\
396
31&0\\
397
37&\alpha^{10}\\
398
41&1\\
399
43&\alpha^{10}\\
400
47&1\\
401
53&\alpha^{22}\\
402
\hline\end{array}
403
\begin{array}{|rl|}\hline
404
59&4\\
405
61&\alpha^{14}\\
406
67&\alpha^{4}\\
407
71&\alpha^{20}\\
408
73&\alpha^{2}\\
409
79&\alpha^{20}\\
410
83&\alpha^{4}\\
411
89&\alpha^{10}\\
412
97&0\\
413
101&\alpha^{8}\\
414
103&\alpha^{14}\\
415
107&0\\
416
109&\alpha^{10}\\
417
113&2\\
418
127&0\\
419
131&2\\
420
\hline\end{array}
421
\begin{array}{|rl|}\hline
422
137&0\\
423
139&\alpha^{22}\\
424
149&\alpha^{4}\\
425
151&1\\
426
157&\alpha^{14}\\
427
163&0\\
428
167&\alpha^{22}\\
429
173&4\\
430
179&\alpha^{2}\\
431
181&\alpha^{14}\\
432
191&\alpha^{10}\\
433
193&4\\
434
197&0\\
435
199&3\\
436
211&0\\
437
223&0\\
438
\hline\end{array}
439
\begin{array}{|rl|}\hline
440
227&\alpha^{10}\\
441
229&0\\
442
233&\alpha^{14}\\
443
239&0\\
444
241&\alpha^{2}\\
445
251&\alpha^{2}\\
446
257&3\\
447
263&\alpha^{16}\\
448
269&2\\
449
271&\alpha^{8}\\
450
277&0\\
451
281&\alpha^{16}\\
452
283&0\\
453
293&3\\
454
307&\alpha^{4}\\
455
311&\alpha^{22}\\
456
\hline\end{array}
457
\begin{array}{|rl|}\hline
458
313&0\\
459
317&0\\
460
331&\alpha^{14}\\
461
337&0\\
462
347&\alpha^{16}\\
463
349&\alpha^{4}\\
464
353&0\\
465
359&0\\
466
367&\alpha^{22}\\
467
373&0\\
468
379&3\\
469
383&3\\
470
389&1\\
471
397&\alpha^{16}\\
472
401&0\\
473
409&2\\
474
\hline\end{array}
475
\begin{array}{|rl|}\hline
476
419&3\\
477
421&\alpha^{20}\\
478
431&4\\
479
433&\alpha^{4}\\
480
439&\alpha^{20}\\
481
443&0\\
482
449&0\\
483
457&0\\
484
461&0\\
485
463&\alpha^{10}\\
486
467&0\\
487
479&0\\
488
487&\alpha^{8}\\
489
491&\alpha^{2}\\
490
499&\alpha^{20}\\
491
503&\alpha^{2}\\
492
\hline\end{array}
493
\begin{array}{|rl|}\hline
494
509&\alpha^{8}\\
495
521&\alpha^{10}\\
496
523&\alpha^{14}\\
497
541&\alpha^{20}\\
498
547&\alpha^{22}\\
499
557&3\\
500
563&1\\
501
569&\alpha^{16}\\
502
571&\alpha^{22}\\
503
577&\alpha^{14}\\
504
587&\alpha^{20}\\
505
593&0\\
506
599&\alpha^{22}\\
507
601&0\\
508
607&\alpha^{16}\\
509
613&2\\
510
\hline\end{array}
511
\comment{\begin{array}{|rl|}\hline
512
617&0\\
513
619&\alpha^{20}\\
514
631&\alpha^{20}\\
515
641&4\\
516
643&1\\
517
647&4\\
518
653&1\\
519
659&\alpha^{14}\\
520
661&2\\
521
673&\alpha^{8}\\
522
677&4\\
523
683&0\\
524
691&\alpha^{16}\\
525
701&\alpha^{14}\\
526
709&4\\
527
719&\alpha^{4}\\
528
\hline\end{array}
529
}
530
$$
531
\end{center}
532
\end{table}
533
534
\subsection{Twisting into $\GL(2,\F_5)$}
535
Although there is a representation
536
$\rho_f:\GQ\ra\GL(2,\F_{25})$ attached to $f$,
537
it is difficult to say anything about its image without further
538
work. We use a trick to show that the image of $\rho_f$ is small.
539
Firstly, for a character~$\chi:\GQ\to\Fbar_5$, let~$\tilde\chi$
540
denote its Teichm\"uller lift to~$\Qbar_5$. By a result of Carayol,
541
there is a characteristic 0 eigenform
542
$\tilde{f}\in S_5(N,\tilde{\eps};\Qbar_5)$ lifting $f$.
543
The twist $\tilde{g}=\tilde{f} \tensor \tilde{\eps}_{43}$ is, by
544
\cite[Prop. 3.64]{shimura:intro}, an eigenform in
545
$S_5(43N, \tilde{\eps}_2; \Qbar_5)$, and its reduction is
546
a form $g\in S_5(43N,\eps_2,\F_{25})$.
547
The eigenvalues $a_p(g) = a_p(f) \eps_{43}(p)$, for the
548
first few
549
$p\nmid 5N$, are given in Table~\ref{table:1376twist}.
550
551
\begin{table}
552
\caption{\label{table:1376twist}Eigenvalues of~$g=f\tensor\eps_{43}$}
553
\begin{center}
554
$$
555
\begin{array}{|rl|}\hline
556
2&*\\%0\\
557
3&1\\
558
5&*\\%3\\
559
7&2\\
560
11&4\\
561
13&2\\
562
17&2\\
563
19&0\\
564
23&1\\
565
29&1\\
566
31&0\\
567
37&3\\
568
41&1\\
569
43&*\\%0\\
570
47&1\\
571
53&2\\
572
\hline\end{array}
573
\begin{array}{|rl|}\hline
574
59&4\\
575
61&2\\
576
67&4\\
577
71&4\\
578
73&3\\
579
79&4\\
580
83&4\\
581
89&3\\
582
97&0\\
583
101&1\\
584
103&2\\
585
107&0\\
586
109&3\\
587
113&2\\
588
127&0\\
589
131&2\\
590
\hline\end{array}
591
\begin{array}{|rl|}\hline
592
137&0\\
593
139&2\\
594
149&4\\
595
151&1\\
596
157&2\\
597
163&0\\
598
167&2\\
599
173&4\\
600
179&3\\
601
181&2\\
602
191&3\\
603
193&4\\
604
197&0\\
605
199&3\\
606
211&0\\
607
223&0\\
608
\hline\end{array}
609
\begin{array}{|rl|}\hline
610
227&3\\
611
229&0\\
612
233&2\\
613
239&0\\
614
241&3\\
615
251&3\\
616
257&3\\
617
263&1\\
618
269&2\\
619
271&1\\
620
277&0\\
621
281&1\\
622
283&0\\
623
293&3\\
624
307&4\\
625
311&2\\
626
\hline\end{array}
627
\begin{array}{|rl|}\hline
628
313&0\\
629
317&0\\
630
331&2\\
631
337&0\\
632
347&1\\
633
349&4\\
634
353&0\\
635
359&0\\
636
367&2\\
637
373&0\\
638
379&3\\
639
383&3\\
640
389&1\\
641
397&1\\
642
401&0\\
643
409&2\\
644
\hline\end{array}
645
\begin{array}{|rl|}\hline
646
419&3\\
647
421&4\\
648
431&4\\
649
433&4\\
650
439&4\\
651
443&0\\
652
449&0\\
653
457&0\\
654
461&0\\
655
463&3\\
656
467&0\\
657
479&0\\
658
487&1\\
659
491&3\\
660
499&4\\
661
503&3\\
662
\hline\end{array}
663
\begin{array}{|rl|}\hline
664
509&1\\
665
521&3\\
666
523&2\\
667
541&4\\
668
547&2\\
669
557&3\\
670
563&1\\
671
569&1\\
672
571&2\\
673
577&2\\
674
587&4\\
675
593&0\\
676
599&2\\
677
601&0\\
678
607&1\\
679
613&2\\
680
\hline\end{array}
681
\begin{array}{|rl|}\hline
682
617&0\\
683
619&4\\
684
631&4\\
685
641&4\\
686
643&1\\
687
647&4\\
688
653&1\\
689
659&2\\
690
661&2\\
691
673&1\\
692
677&4\\
693
683&0\\
694
691&1\\
695
701&2\\
696
709&4\\
697
719&4\\
698
\hline\end{array}
699
\comment{
700
\begin{array}{|rl|}\hline
701
727&4\\
702
733&0\\
703
739&2\\
704
743&2\\
705
751&4\\
706
757&3\\
707
761&3\\
708
769&0\\
709
773&0\\
710
787&4\\
711
797&1\\
712
809&3\\
713
811&1\\
714
821&2\\
715
823&3\\
716
827&3\\
717
\hline\end{array}
718
\begin{array}{|rl|}\hline
719
829&2\\
720
839&0\\
721
853&2\\
722
857&0\\
723
859&0\\
724
863&4\\
725
877&1\\
726
881&1\\
727
883&0\\
728
887&2\\
729
907&0\\
730
911&1\\
731
919&1\\
732
929&0\\
733
937&3\\
734
941&4\\
735
\hline\end{array}
736
\begin{array}{|rl|}\hline
737
947&2\\
738
953&1\\
739
967&4\\
740
971&3\\
741
977&0\\
742
983&0\\
743
991&3\\
744
997&3\\
745
&\\
746
&\\
747
&\\
748
&\\
749
&\\
750
&\\
751
&\\
752
&\\
753
\hline\end{array}}
754
$$
755
\end{center}
756
\end{table}
757
758
\begin{proposition}\label{prop:1376-g}
759
Let $g=f\tensor \eps_{43}$. Then $a_p(g)\in \F_5$
760
for all~$p\nmid \ell N$.
761
\end{proposition}
762
\begin{proof}
763
Consider an eigenform $\tilde{f}\in S_5(N,\tilde{\eps};\Qbar_5)$
764
lifting~$f$ as above.
765
Associated to~$\tilde{f}$ there is an automorphic
766
representation~$\pi=\tensor_v'\pi_v$ of $\GL(2,\bA)$, where~$\bA$
767
is the ad\`{e}le ring of~$\Q$.
768
Because $43\mid\mid N$, and~$43$ divides the conductor
769
of $\eps$, we see that the local component $\pi_{43}$ of $\pi$ at
770
$43$ must be ramified principal series. By Carayol's theorem,
771
$\rho_{\tilde{f}}|_{D_{43}} \sim
772
\abcd{\Psi_1}{0}{0}{\Psi_2}$
773
with, without loss of generality,~$\Psi_2$ unramified. We have
774
$(\Psi_1\cdot \Psi_2)|_{I_{43}}=\tilde{\eps}|_{I_{43}}=\tilde{\eps}_{43}$,
775
therefore, $\rho_{\tilde{f}}|_{I_{43}} \sim
776
\abcd{\tilde{\eps}_{43}}{0}{0}{1}$.
777
778
Now twist~$\tilde{f}$ by $\tilde{\eps}_{43}^{-1}$; we find that
779
$\rho_{\tilde{f}\tensor\tilde{\eps}_{43}^{-1}}|_{I_{43}} \sim
780
\abcd{1}{0}{0}{\tilde{\eps}^{-1}_{43}}$.
781
In particular, there is an
782
eigenform~$\tilde{f}'\in S_5(N,\tilde{\eps}_2\tilde{\eps}^{-1}_{43},\Qbar_5)$
783
whose associated Galois representation is the twist by $\tilde{\eps}^{-1}_{43}$
784
of that of $\tilde{f}$ (recall that $N=1376$ and so~$43$ divides~$N$
785
exactly once). Let~$f'$ denote the mod~$5$ reduction of~$\tilde{f}'$. Then
786
one checks easily that $f'\in S_5(N,\eps_2\eps^{-1}_{43},\F_{25})=S_5(N,\eps^5,\F_{25})$.
787
788
For all primes $p\nmid5N$ we have $a_p(f')=\eps_{43}(p)^{-1}a_p(f)$.
789
In particular, we have $a_p(f')=0$ for
790
$p=19,31$.
791
Also, $\eps_{43}(3)=\alp^8$ and $\eps_{43}(5) =\alp^8$, so
792
$$a_3(f')=\alp^{16}/\alp^8 = \alp^8 = (\alp^{16})^5$$
793
$$a_5(f')=\alp^{22}/\alp^8 = \alp^{14} = (\alp^{22})^5.$$
794
Now if $\sigma$ is the non-trivial automorphism of $\F_{25}$,
795
then $\sigma(f')$ and $f$ both lie in
796
$S_5(1376,\eps;\F_{25})$ and have same~$a_p$ for
797
$p=3,5,19,31$, so they are equal because we found~$f$
798
by computing the unique eigenform with given~$a_p$ for $p=3,5,19,31$.
799
So $g = f\tensor\eps_{43} = \sigma(f)\tensor\eps_{43}^2$.
800
Thus for all $p\nmid 5N$, we see that
801
$a_p(g) = a_p(f)^5 \eps_{43}^2$ has fifth power
802
$a_p(g)^5 = a_p(f)^{25} \eps_{43}^{10}
803
= a_p(f) \eps_{43} = a_p(g)$.
804
\end{proof}
805
806
\subsection{Proof that~$\rho_g$ is unramified at~$5$}
807
808
We begin with a generalisation of~\cite{sturm:cong}.
809
Let $M>4$ be an integer, and let $h=\sum_{n\geq1}c_nq^n$ be a
810
normalised cuspidal eigenform
811
of some weight~$k\geq1$, level~$M$ and character~$\chi$, defined over some
812
field of characteristic not dividing~$M$. Even though the base field
813
might not have characteristic zero, we may still define the conductor
814
of $\chi$ to the the largest divisor $f$ of $M$ such that~$\chi$
815
factors through $(\Z/f\Z)^\times$.
816
Let~$I$ be a set of primes, with the property that for all~$p$
817
in~$I$, one of the following conditions hold:
818
819
(i) $p$ divides~$M$ but~$p$ does not divide~$M/\cond(\chi)$, or
820
821
(ii) $p$ divides~$M$ exactly once, and $h$ is $p$-new, in the sense
822
that there is no eigenform $h'$ of level $M/p$ such
823
that the $T_n$-eigenvalues of $h$ and $h'$ agree for all $n$
824
prime to $p$.
825
826
Let~$C$ denote the orbit of the cusp~$\infty$ in $X_1(M)$
827
under the action of the group generated by $w_p$ for $p\in I$, and
828
the Diamond operators $\langle d\rangle_M$. The orbit of~$\infty$
829
under the Diamond operators has size $\phi(M)/2$, and each
830
$w_p$ increases the size of the orbit by a factor of~2. In this
831
situation, we have
832
833
\begin{lemma} The first~$t$ terms of the $q$-expansion
834
of~$h$ at any cusp in~$C$ are determined by~$M$,~$k$, $\chi$, $c_p$
835
for~$p$ in~$I$, and $c_n$ for $1\leq n\leq t$.
836
837
\end{lemma}
838
839
\begin{remark} Our proof is just a translation of Corollary~4.6.18
840
of \cite{miyake} into the language of moduli problems (Miyake's argument
841
technically is only valid over the complex numbers).
842
\end{remark}
843
\begin{proof}
844
If $J\subseteq I$ is any subset, and $w_J$ denotes the product
845
of $w_p$ for $p\in J$, then $h|w_J$ is an eigenform for all the
846
Diamond operators, and this observation reduces the proof
847
of the lemma to showing that for $p\in I$, if $h|w_p=\sum_n d_nq^n$,
848
then $d_j$ for $1\leq j\leq n$ and $d_q$ for all $q\in I$
849
are determined by $M$, $k$, $\chi$, $p$, $c_j$ for $1\leq j\leq n$
850
and $c_q$ for all $q\in I$.
851
852
We first deal with primes $p$ of the form (i).
853
Say $M=p^mR$, where $R$ is prime to $p$.
854
Thinking of~$h$ as a rule for attaching
855
$k$-fold differentials to elliptic curves equipped with points
856
of order $p^m$ and $R$, we have by definition that
857
$$h(\G_m/q^\Z,\zeta,\zeta_R)=\bigg(\sum c_nq^n\bigg)(dt/t)^k,$$
858
where $\zeta=\zeta_{p^m}$ and $\zeta_R$ are fixed $p^m$th and $R$th roots
859
of unity in $\G_m$ which correspond to the cusp~$\infty$,
860
and $dt/t$ is the canonical differential on the Tate
861
curve $\G_m/q^\Z$. We normalise things such that
862
$$h(\G_m/q^{p^m\Z},q,\zeta_R)=\bigg(\sum d_nq^n\bigg)(dt/t)^k,$$
863
and remark that because $h$ is an action for the diamond operators,
864
we do not have to worry too much about whether this corresponds to
865
the standard normalisation of the $w_p$-operator.
866
867
We recall that the operator $pU_p$ in this setting can be thought
868
of as being defined by the rule:
869
$$(pU_ph)(E,P,Q)=\sum_C\pi^*h(E/C,\overline{P},\overline{Q}),$$
870
where $C$ runs through the subgroups of $E$ of order $p$ which have
871
trivial intersection with $\langle P\rangle$, and $\pi$ denotes the canonical
872
projection $E\to E/C$.
873
We see that
874
\begin{align*}
875
(pc_p)^m\big(\sum d_nq^n\big)(dt/t)^k&=(p^mU_{p^m}h)(\G_m/q^{p^m\Z},q,\zeta_R)\\
876
&=\sum_{c=0}^{p^m-1}\pi^c*h(\G_m/\langle q^{p^m},\zeta q^c\rangle,q,\zeta_R),
877
\end{align*}
878
where $\pi$ denotes the canonical projection from $\G_m/\langle q^{p^m}\rangle$
879
to the appropriate quotient. This last sum can be written as a
880
double sum
881
\begin{align*}
882
&\sum_{c\in(\Z/p^m\Z)^\times}\pi^*h(\G_m/\langle q^{p^m},\zeta q^c\rangle,q,\zeta_R)+\sum_{a=0}^{p^{m-1}-1}\pi^*h(\G_m/\langle q^{p^m},\zeta q^{pa}\rangle,q,\zeta_R)\\
883
=&\sum_{b\in(\Z/p^m\Z)^\times}\pi^*h(\G_m/\langle q^{p^m},\zeta^{-b}q\rangle,q,\zeta_R)+p^{m-1}\pi^*U_{p^{m-1}}h(\G_m/\langle q^{p^m},\zeta^{p^{m-1}}\rangle,q,\zeta_R)\\
884
=&\sum_{b\in(\Z/p^m\Z)^\times}\pi^*h(\G_m/\langle\zeta^{-b}q\rangle,\zeta^b,\zeta_R)+(pc_p)^{m-1}\pi^*h(\G_m/\langle q^{p^m},\zeta^{p^{m-1}}\rangle,q,\zeta_R)\\
885
=&\sum_{b}\chi_p(b)\sum_{n\geq1}c_n(\zeta^{-b}q)^n(dt/t)^k+p^k(pc_p)^{m-1}\pi^*h(\G_m/\langle q^{p^{m+1}}\rangle,q^p,\zeta_R^p),
886
\end{align*}
887
where we have written $\chi=\chi_R\chi_p,$ for $\chi_R$ a character of
888
level~$R$ and $\chi_p$ a character of level~$p^m$. We deduce that
889
\begin{align*}
890
&(pc_p)^m\big(\sum d_nq^n\big)(dt/t)^k-p^k(pc_p)^{m-1}\chi_R(p)\pi^*h(\G_m/\langle q^{p^{m+1}}\rangle,q^p,\zeta_R)\\
891
=&\bigg(\sum_n\big(\sum_b\chi_p(b)\zeta^{-bn}\big)c_nq^n\bigg)(dt/t)^k\\
892
=&W(\chi_p)\big(\sum_{p\nmid n}\chi_p(-n)^{-1}c_nq^n\big)(dt/t)^k
893
\end{align*}
894
where $W(\chi_p)=\sum_{b\in(\Z/p^m\Z)^\times}\chi_p(b)\zeta^b$ can be
895
checked to be nonzero because the conductor of $\chi_p$ is $p^m$.
896
Hence
897
$$(pc_p)^m\sum_n d_nq^n-p^k(pc_p)^{m-1}\chi_R(p)\sum_n d_nq^{np}=W(\chi_p)\chi_p(-1)\sum_{p\nmid n}\chi_p(n)^{-1}c_nq^n.$$
898
Equating coefficients of $q$ we deduce that $W(\chi_p)\chi_p(-1)=(pc_p)^md_1$,
899
and because $h|w_p$ is an eigenform for $T_n$ for all $n$ prime to $p$,
900
with eigenvalues determined by $\chi$ and $c_n$, we deduce that we can
901
determine $d_n$ for $n$ prime to $p$ from $c_n$. It remains to establish
902
what $d_p$ is, and equating coefficients of $q^p$ in the above equation
903
gives us that $(pc_p)^md_p=p^k(pc_p)^{m-1}\chi_R(p)d_1$ and hence
904
that $d_p$ is determined by $\chi$ and $c_p$.
905
Note that as a consequence we see that $d_p/d_1=p^{k-1}\chi_R(p)/c_p$,
906
a classical formula if the base field is the complexes.
907
908
Now we deal with primes of the form (ii) (note that we never use
909
this case in the rest of the paper). We think of $h$ as a rule associating
910
$k$-fold differentials to triples $(E,C,Q)$ where $C$ a cyclic subgroup of
911
order~$p$ and $Q$ a point of order~$R=M/p$. Because $h$ is $p$-new, the
912
trace of $h$ down to $X_1(M/p)$ must be zero, and hence we see
913
that for any elliptic curve $E$ equipped with a point $Q$ of order $R$,
914
$$\sum_C\pi^* h(E/C,E[p]/C,\overline{Q})=0.$$ As before, normalise things so
915
that
916
$$h(\G_m/q^\Z,\mu_p,\zeta_R)=\bigg(\sum_n c_nq^n\bigg)(dt/t)^k$$
917
and
918
$$h(\G_m/q^{p\Z},\langle q\rangle,\zeta_R)=\bigg(\sum_n d_nq^n\bigg)(dt/t)^k.$$
919
The fact that the trace of~$h$ is zero implies that
920
$$(pU_p)h(\G_m/q^{p\Z},\langle q\rangle,\zeta_R)+\pi^*h(\G_m/q^\Z,\mu_p,\zeta_R)=0,$$
921
and hence that
922
$$c_p\sum d_nq^n+p^{k-1}\sum c_nq^n=0$$
923
from which we deduce that the $d_n$ can be read off from $c_p$ and the $c_n$.
924
\end{proof}
925
\begin{remark} The size of~$C$ is
926
$\phi(M).2^{|I|-1}$, and the usefulness of this lemma is that
927
if $h_1$ and $h_2$ are two normalised eigenforms of the same level,
928
weight and character as above, both new at all primes in~$I$,
929
and the coefficients of $q^n$ in the
930
$q$-expansions of $h_1$ and $h_2$ agree for $n\in I$ and $n\leq t$,
931
then $h_1-h_2$ has a zero of order at least $t+1$ at all cusps in~$C$,
932
and in particular if
933
$\phi(M).2^{|I|-1}(t+1)>k/12[\SL_2(\Z):\Gamma_1(M)]=\deg(\omega^k)$ on
934
$X_1(M)$ then $h_1=h_2$. Using the fact that
935
$[\Gamma_0(M):\Gamma_1(M)]=\phi(M)/2$,
936
we deduce
937
\end{remark}
938
\begin{corollary}\label{cor:bound}
939
Let $h_1$ and $h_2$ be two normalised eigenforms as above.
940
If the coefficients of $q^n$ in the $q$-expansions of $h_1$ and $h_2$ agree
941
for all primes in $I$ and for all
942
$n\leq\frac{k}{12}[\SL_2(\Z):\Gamma_0(M)]/2^{|I|}$ then $h_1=h_2$.
943
\end{corollary}
944
\begin{remark} One can certainly do better than this corollary
945
in many cases. For example, when $n>1$ and
946
$p^n$ exactly divides both the level
947
of an eigenform and the conductor of its character, then one can compute
948
the $q$-expansion of the eigenform at many ``middle cusps'' too,
949
and hence increase the size of $C$ in the result above.
950
\end{remark}
951
952
We now go back to the explicit situation we are concerned with.
953
Although~$g$ is an eigenform of level $59168=2^5\cdot 43^2$,
954
we can still consider the corresponding representation
955
$\rho_g :\GQ\ra \GL(2,\F_5)$, and then directly analyze
956
its ramification.
957
\begin{proposition}
958
The representation~$\rho_g$ is unramified at~$5$.
959
\end{proposition}
960
\begin{proof}
961
Continuing the modular symbols computations as above,
962
we find that~$V_1$ is spanned by the two eigenforms
963
\begin{align*}
964
f\,\,&=q + \alp^{16}q^3 + \alp^{22}q^5 + \alp^{14}q^7
965
+ \alp^{14}q^9 + 4q^{11}+\cdots\\
966
f_1&=q + \alp^{16}q^3 + \alp^{10}q^5 + \alp^{14}q^7
967
+ \alp^{14}q^9 + 4q^{11}+\cdots.
968
\end{align*}
969
For $p\neq 5$ and $p\leq 997$, we have $a_p(f_1)=a_p(f)$.
970
To check that $a_p(f) = a_p(f_1)$ for all $p\neq 5$,
971
it suffices to show that the difference~$f-f_1$ has
972
$q$-expansion involving only powers of~$q^5$;
973
for this we use the $\theta$-operator
974
$q\frac{d}{dq}:S_5(1376,\eps,\F_{25})\ra S_{11}(1376,\eps;\F_{25})$.
975
Since~$\theta$ sends normalized eigenforms to normalized eigenforms,
976
it suffices to check that the subspace of
977
$S_{11}(1376,\eps;\F_{25})$ generated by~$\theta(f)$
978
and~$\theta(f_1)$ has dimension~$1$.
979
Corollary~\ref{cor:bound} implies that it suffices to verify that the
980
coefficients $a_p(\theta(f))$ and $a_p(\theta(f_1))$ are equal for all
981
$$p \leq \frac{11}{12}\cdot [\SL_2(\Z):\Gamma_0(1376)]\cdot \frac{1}{2}
982
= 968.$$
983
The eigenform~$f$ must be new because we computed it by finding
984
the intersections of the kernels of Hecke operators $T_p$ with
985
$p\nmid 1376$; if~$f$ were an oldform then the intersection of the
986
kernels of these Hecke operators
987
would necessarily have dimension greater than~$1$.
988
Because it takes less than a second
989
to compute each $a_p(\theta(f))$, we were easily able to verify that the
990
space generated by $\theta(f)$ and $\theta(f_1)$ has dimension~$1$.
991
992
\begin{remark}
993
It is possible to avoid appealing to Corollary~\ref{cor:bound} by using
994
one of the following two alternative methods:
995
\begin{enumerate}
996
\item Define~$\theta$ directly on modular symbols and compute it.
997
\item Compute the intersection
998
$$\bigcap_{p\geq 2} \ker(T_p - pa_p(f))
999
\subset S_{11}(1376,\eps;\F_{25}).$$
1000
Since~$\theta(f)$ and~$\theta(f_1)$ both lie
1001
in the intersection, the moment the dimension
1002
of a partial intersection is~$1$, it follows
1003
that $\theta(f-f_1)=0$.
1004
\end{enumerate}
1005
We successfully carried out both alternatives.
1006
For the first, we showed that~$\theta$ on modular symbols is
1007
induced by multiplication by
1008
$X^5Y - Y^5X$.
1009
For the second, we find that after intersecting
1010
kernels for $p\leq 11$, the dimension is already~$1$.
1011
The first of these two methods took much less
1012
time than the second.
1013
\end{remark}
1014
1015
Next we use that $\theta(f-f_1)=0$ to show that $\rho_g$ is unramified,
1016
thus finishing the proof of the proposition.
1017
Since~$f$ is ordinary, Deligne's theorem (see~\cite[\S12]{gross:tameness})
1018
implies that
1019
$$\rho_f|_{D_5}\sim
1020
\mtwo{\alp}{*}{0}{\beta}\qquad\text{over $\Fbar_5$}$$
1021
with both~$\alp, \beta$ unramified,
1022
$\alp(\Frob_5)=\eps(5)/a_5=\alp^8/\alp^{22}=\alp^{10}$, and
1023
$\beta(\Frob_5)=\alp^{22}$.
1024
Since $a_p(f_1)=a_p(f)$, for $p\neq 5$, we have
1025
$${\rho_f}|_{D_5}\sim {\rho_{f_1}}|_{D_5} \sim \mtwo{\alp'}{*}{0}{\beta'}$$
1026
with
1027
$\alp'(\Frob_5)=\alp^8/\alp^{10}=\alp^{22}$ and
1028
$\beta'(\Frob_5)=\alp^{10}$;
1029
in particular, $\alp'=\beta$.
1030
Thus $\rho_f|_{D_5}$ contains $\alp\oplus \beta$, so
1031
$\rho_f|_{D_5}\sim\alp\oplus\beta$ and hence there is a choice
1032
of basis so that $*=0$.
1033
1034
\end{proof}
1035
1036
1037
1038
\subsection{The image of $\proj \rho_g$}
1039
\begin{proposition}\label{prop:image_is_A5}
1040
The image of $\proj \rho_g$ is $A_5$.
1041
\end{proposition}
1042
\begin{proof}
1043
1044
The image~$H$ of $\proj \rho_g$ in $\PGL_2(\F_5)$ is easily checked to
1045
lie in $\PSL_2(\F_5)\cong A_5$ because of what we know about the
1046
determinant of $\rho_g$. Hence $H$ is a subgroup of $A_5$ that
1047
contains an element of order~$2$ (complex conjugation) and an element
1048
of order~$3$ (for example, $\rho_g(\Frob_7)$ has characteristic
1049
polynomial $x^2-2x-1$). This proves that~$H$ is isomorphic to
1050
either~$S_3$,~$A_4$, or~$A_5$. Let $L$ be the number field cut out
1051
by~$H$. If~$L$ were an $S_3$-extension, then there would be a
1052
quadratic extension contained in it which is unramified outside
1053
$2\cdot 5\cdot 43$; it is furthermore unramified at~$5$ by the
1054
previous section and unramified at $43$ because $I_{43}$ has
1055
order~$3$. Thus it is one of the three quadratic fields unramified
1056
outside~$2$.
1057
In particular, the trace of $\Frob_p$
1058
would be zero for all primes in a certain congruence class
1059
modulo~8.
1060
However, there are primes~$p$ congruent to $3$, $5$, and $7$
1061
mod $8$ such that $a_p(g)\neq 0$, e.g., $3$, $7$, and $13$.
1062
1063
1064
If $H$ were isomorphic to $A_4$, then let~$M$ denote the cyclic
1065
extension of degree~3 over~$\Q$ contained in~$L$. Now~$M$ is unramified
1066
at~2 and~5, and hence is the subfield of $\Q(\zeta_{43})$ of degree~3.
1067
Choose $p\nmid 1376\cdot 5$ that is inert in~$M$, i.e., so that
1068
$p$ is not a cube mod $43$. The order of
1069
$\rho_g(\Frob_p)$ in $\GL_2(\F_5)$ must be divisible by~$3$. However,
1070
a quick check using Table~\ref{table:1376twist} shows that this is
1071
usually not the case, even for $p=3$.
1072
\end{proof}
1073
1074
1075
\subsection{Bounding the ramification at~$2$ and~$43$}
1076
Let~$L$ be the fixed field of $\ker(\proj(\rho_g))$. We have just
1077
shown that $\Gal(L/\Q)$ is isomorphic to $A_5$.
1078
By a root field for~$L$, we mean
1079
a non-Galois extension of $\Q$ of degree~5 whose Galois closure is~$L$.
1080
\begin{proposition}
1081
The discriminant of a root
1082
field for~$L$ divides $(43\cdot 8)^2=344^2$, and
1083
in particular,~$L$ must be mentioned in Table~1
1084
of \cite[pg 122]{freyetal}.\end{proposition}
1085
\begin{proof}
1086
The analysis of the local behavior of~$\rho_f$ at~$43$ given in
1087
Proposition~\ref{prop:1376-g}
1088
shows that the inertia group at~$43$ in $\Gal(L/\Q)$ has order~$3$. Using
1089
Table~3.1 of~\cite{buhler:thesis}, we see that if
1090
$\Gal(L/\Q)\isom A_5$
1091
then it must be ``type~$2$'' at 43, and hence the discriminant of a root
1092
field of~$L$, that is, of a non-Galois extension of~$\Q$ of degree~$5$
1093
whose Galois closure is~$L$, must be $43^2$ at~$43$.
1094
1095
At~$2$ the behavior of~$\rho$ is more subtle and we shall not analyze
1096
it fully. But we can say that, because~$\rho$ has arisen from
1097
a form of level $1376=2^5.43$, we must be either of type~$5$
1098
or one of types~$14$--$17$. In particular, the discriminant at~$2$ of a root
1099
field for~$L$ will be at most~$2^6$.
1100
1101
Finally,~$L$ is unramified at all other primes, because~$\rho$ is.
1102
Hence the discriminant of a root field for~$L$, assuming that
1103
$\Gal(L/\Q)\cong A_5$, divides $(43.8)^2=344^2$.
1104
\end{proof}
1105
1106
We know that~$L$ is an icosahedral extension of~$\Q$ with
1107
discriminant dividing $43^2\cdot 2^6$. Table~1 of \cite[pg 122]{freyetal}
1108
contains all icosahedral extensions, such that the discriminant
1109
of a root field is bounded by $2083^2$. The table
1110
must contain~$L$; there is only one icosahedral extension with
1111
discriminant dividing $43^2\cdot 2^6$, so $L=K$.
1112
1113
\subsection{Obtaining a classical weight one form}
1114
We have shown that a twist of the icosahedral
1115
representation $\rho:\GQ\ra\GL(2,\C)$,
1116
nobtained by lifting $\GQ\ra \Gal(K/\Q)\ncisom A_5$,
1117
has a mod~$5$ reduction $\rho_g:\GQ\ra \GL_2(\F_5)$ that
1118
is modular. Since~$\rho$ ramifies at only finitely many primes,
1119
and~$\rho$ is unramified at~$5$ with distinct eigenvalues,
1120
\cite{buzzard-taylor} implies that~$\rho$ arises from
1121
a classical weight~$1$ newform.
1122
1123
1124
1125
\section{More examples}
1126
The data necessary to deduce modularity of each of our eight
1127
icosahedral examples is summarized in
1128
Tables~\ref{table:more1}--\ref{table:more4}.
1129
1130
The notation in Table~\ref{table:more1} is as follows.
1131
The first column contains the conductor.
1132
The second column contains a $5$-tuple $[a_4,a_3,a_2,a_1,a_0]$ such
1133
that the $A_5$-extension is the splitting field of the polynomial
1134
$h=x^5+a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0$.
1135
The column labeled $\ord(\Frob_5)$ contains the order of the image
1136
of $\Frob_5$ in $A_5$. The next column, which is labeled ``$p$ with
1137
$a_p=0$'', contains the first few~$p$ such that $a_p$ is easily seen
1138
to equal~$0$ by considering the splitting of~$h$ mod~$p$.
1139
The $\eps$ column contains the character of the representation, where the
1140
notation is as follows. Write $(\Z/N\Z)^*$ as a product of cyclic groups
1141
corresponding to the prime divisors of~$N$ in ascending order, and then
1142
the tuples give the orders of the images of these cyclic factors; when
1143
$8\mid N$, there are two cyclic factors corresponding to the prime~$2$.
1144
Finally, the last column records the dimension of $S_5(\Gamma_1(N),\eps)$.
1145
1146
The notation in Table~\ref{table:more2} is as follows. The first column
1147
contains the conductor. The second column contains an eigenform that
1148
was found by first intersecting the kernels of the Hecke operators
1149
$T_p$ with~$p$ as in Table~\ref{table:more1}, and then locating an
1150
eigenform.
1151
In each case, a companion form was found, by computing $a_p(f)$ for
1152
$p\leq$ bound, where bound is the bound from Corollary~\ref{cor:bound}.
1153
1154
Table~\ref{table:more3} shows that the fixed field
1155
of the image of each $\proj(\rho_g)$ is icosahedral.
1156
The first column contains the
1157
conductor~$N$. The second column contains a twist~$g$ of~$f$ such that
1158
$a_p(g)\in\F_5$ for all $p\nmid 5N$. The third column contains
1159
a $\Frob_p$ such that $\proj(\rho_g(\Frob_p))$ has order~$3$,
1160
along with the characteristic polynomial of $\rho_g(\Frob_p)$.
1161
As in the proof of Proposition~\ref{prop:image_is_A5},
1162
the other two boxes give data that allows us to deduce
1163
that the fixed field of the image of $\proj(\rho_g)$ is icosahedral.
1164
The case $5373$ must be treated separately, because there are
1165
three possibilities $M_1$, $M_2$, and $M_3$
1166
for the cubic field~$M$ of the analogue of
1167
Proposition~\ref{prop:image_is_A5}.
1168
For $M_1$ we find a prime~$p$ such that
1169
$$(p^2\!\!\!\mod 9, \,\,\,p^{66}\!\!\!\mod 199)\not\in\{(1,1),(4,1),(7,1)\}$$
1170
with $\rho_g(\Frob_p)$ of order not divisible by~$3$;
1171
for this, $p=2$ suffices, since the characteristic polynomial
1172
of $\rho_g(\Frob_2)$ is $(x+2)^2$
1173
and $(p^2\!\!\!\mod 9, \,\,\,p^{66}\!\!\!\mod 199) = (4,106)$.
1174
For $M_2$ we find a prime~$p$ such that
1175
$$(p^2\!\!\!\mod 9, \,\,\,p^{66}\!\!\!\mod 199)\not\in\{(1,1),(4,92),(7,106)\}$$
1176
with $\rho_g(\Frob_p)$ of order not divisible by~$3$;
1177
again, $p=2$ suffices.
1178
For $M_3$ we find a prime~$p$ such that
1179
$$(p^2\!\!\!\mod 9, \,\,\,p^{66}\!\!\!\mod 199)\not\in\{(1,1),(4,106),(7,92)\}$$
1180
with $\rho_g(\Frob_p)$ of order not divisible by~$3$;
1181
here, $p=13$ suffices, as the characteristic polynomial
1182
of $\rho_g(\Frob_p)$ is $(x+4)^2$ and
1183
$(p^2\!\!\!\mod 9, \,\,\,p^{66}\!\!\!\mod 199) = (7,106)$.
1184
1185
1186
1187
Table~\ref{table:more4} gives upper bounds on the ramification of the
1188
fixed field of the image of $\proj(\rho_g)$. These bounds
1189
were deduced using Table~3.1
1190
of~\cite{buhler:thesis} by restricting the possible ``types'' using
1191
information about the character $\eps$. Note that though
1192
the bounds are not sharp, e.g., the discriminant of
1193
the representation of conductor $2416$ is $2^4\cdot 151^2$, they
1194
are all less than $2083^2$, so the corresponding
1195
field must appear in Table~2 of~\cite{freyetal}.
1196
1197
\begin{table}
1198
\caption{\label{table:more1}Data on icosahedral representations mod~$5$}
1199
\begin{center}
1200
\begin{tabular}{|clclll|}\hline
1201
$N$&\hspace{2em}$h$&$\hspace{-1.5em}\ord(\Frob_5)$&$p$ with $a_p=0$&
1202
\hspace{1em}$\eps$ &$\hspace{-1.5em}\dim S_5(N,\eps)$\\\hline
1203
{\bf 1376}&$[2,6,8,10,8]$ & \hspace{-1.5em}$2$&$19,31,97$&$[2,1,3]$&$696$\\
1204
{\bf 2416}&$[0,-2,2,5,6]$ & \hspace{-1.5em}$2$&$53,97,127$&$[2,1,3]$&$1210$\\
1205
{\bf 3184}&$[5,8,-20,-21,-5]$& \hspace{-1.5em}$2$&$31,89,97$&$[2,1,3]$&$1594$\\
1206
{\bf 3556}&$[3,9,-6,-4,-40]$&\hspace{-1.5em}$3$&$19,29,89$&$[1,2,3]$&$2042$\\
1207
{\bf 3756}&$[0,-3,10,30,-18]$&\hspace{-1.5em}$3$&$17,61,67$&$[1,2,3]$&$2506$\\
1208
{\bf 4108}&$[4,3,9,4,5]$& \hspace{-1.5em}$3$&$17,23,31,89$&$[1,3,2]$&$2234$\\
1209
{\bf 4288}&$[4,5,8,3,2]$& \hspace{-1.5em}$3$&$19,23,47$&$[1,2,3]$&$2164$\\
1210
{\bf 5373}&$[2,1,7,23,-11]$& \hspace{-1.5em}$2$&$7,23,37,79,89$&$[2,3]$&$2394$\\
1211
\hline\end{tabular}
1212
\end{center}
1213
\end{table}
1214
1215
\begin{table}
1216
\caption{\label{table:more2}The newform $f$ and the companion form bound}
1217
\begin{center}
1218
\begin{tabular}{|cll|}\hline
1219
$N$&\hspace{7em}$f$ &\hspace{-.3em}bound \\\hline
1220
{\bf 1376}&
1221
$q + \alp^{16}q^3 + \alp^{22}q^5 + \alp^{14}q^7
1222
+ \alp^{14}q^9 + 4q^{11}+ \alp^{14}q^{13} + \cdots $
1223
& $968$\\
1224
{\bf 2416}&
1225
$q +3q^3 + \alp^{22}q^5 + \alp^{16}q^7 + \alp^{4} q^{11}
1226
+ \alp^2 q^{13} + \alp^{16}q^{15} + \cdots$
1227
& $1672$ \\
1228
{\bf 3184}&
1229
$q + \alp^{16}q^3 + 3q^5 + \alp^{22}q^7 + \alp^{14}q^9 + 3q^{11}
1230
+ \alp^{22}q^{13} + \cdots$
1231
& $2200$\\
1232
{\bf 3556}&
1233
$q + \alp^{16}q^{3} + \alp^{14}q^5 + \alp^{10}q^7 + \alp^{14}q^9
1234
+ \alp^{2}q^{11} + \alp^{22}q^{13} + \cdots$
1235
& $1408$ \\
1236
{\bf 3756}&
1237
$ q + \alp^{14}q^3 + \alp^{14}q^5 + 3q^7 + \alp^4q^9 + \alp^{16}q^{11} + \alp^{10}q^{13} + \cdots$
1238
& $1727$ \\
1239
{\bf 4108}&
1240
$ q + \alp^{16}q^{3} + \alp^{11}q^{5} + \alp^{20}q^{7} + \alp^{14}q^{9} + \alp^{10}q^{11} + 4q^{13} + \cdots $
1241
& $1540$\\
1242
1243
{\bf 4288}&
1244
$q + 3q^3 + \alp^{14}q^{5} + \alp^{20}q^{7} + 3q^9 + \alp^{20}q^{11} + \alp^{16}q^{13} + \cdots $
1245
& $2992$ \\
1246
1247
1248
{\bf 5373}&
1249
$q + \alp^{16}q^{2} + \alp^{14}q^{4} + 4q^5 + 3q^8 + \alp^{4}q^{10} + 2q^{11}+\cdots $
1250
& $3300$ \\
1251
\hline\end{tabular}
1252
\end{center}
1253
\end{table}
1254
1255
1256
\begin{table}
1257
\caption{\label{table:more3}Verification that the image of $\proj(\rho_g)$ is $A_5$}
1258
\begin{center}
1259
Find a Frobenius element with projective order $3$.\vspace{1ex}\\
1260
\begin{tabular}{|c|l|ll|}\hline
1261
$N$ & \hspace{1em} $g$ & proj. order $3$&\hspace{.7em}charpoly \\\hline
1262
{\bf 1376} & $f\tensor \eps_{43}$
1263
& $\quad\Frob_7$ & $x^2-2x-1$
1264
\\
1265
{\bf 2416}& $f\tensor \eps_{151}$
1266
& $\quad\Frob_{19}$ & $x^2+2x-1$
1267
\\
1268
{\bf 3184}& $f\tensor \eps_{199}$
1269
& $\quad\Frob_7$ & $x^2+3x+4$
1270
\\
1271
{\bf 3556}& $f\tensor \eps_{127}$
1272
& $\quad\Frob_{13}$ & $x^2+3x+4$
1273
\\
1274
{\bf 3756}&$f\tensor\eps_{313}$
1275
& $\quad\Frob_{23}$ & $x^2 + 2x + 4$
1276
\\
1277
{\bf 4108} & $f\tensor\eps_{13}$
1278
& $\quad\Frob_{29}$ & $x^2+3x+4$
1279
\\
1280
{\bf 4288}& $f\tensor\eps_{67}$
1281
& $\quad\Frob_{11}$ & $x^2+x+1$
1282
\\
1283
1284
{\bf 5373}& $f\tensor\eps_{199}$
1285
& $\quad\Frob_{11}$ & $x^2+3x+4$
1286
\\
1287
\hline\end{tabular}\vspace{3ex}
1288
1289
Not $S_3$: For all $t\in T$, find unramified $p$ s.t.\ $t\not\equiv \Box \mod p$ and $a_p(g)\neq 0$. \vspace{1ex}\\
1290
\begin{tabular}{|c|l|l|}\hline
1291
$N$ & $\qquad\quad{}T$ & $\qquad{}p$ \\\hline
1292
{\bf 1376}
1293
& $\{-1,-2\}$ & $3$, $7$\\
1294
{\bf 2416}
1295
& $ \{-1, -2\}$ & $3$, $7$ \\
1296
{\bf 3184}
1297
& $ \{-1, -2\}$ & $3$, $7$ \\
1298
{\bf 3556}
1299
& $ \{-1, -2, -7, -14\}$ & $3$, $13$, $3$, $11$\\
1300
{\bf 3756}
1301
& $ \{-1,-2,-3,-6\}$ & $7$, $7$, $11$, $13$\\
1302
{\bf 4108}
1303
& $ \{-1, -2, -79, -158\} $ &$3$, $7$, $3$, $7$ \\
1304
{\bf 4288}
1305
& $ \{-1,-2\}$ & $3$, $7$ \\
1306
{\bf 5373}
1307
& $ \{-3\}$ & $11$\\
1308
\hline\end{tabular}\vspace{3ex}\\
1309
\comment{
1310
function FindS3(t, aplist, N)
1311
P:=[p : p in [2..97] |IsPrime(p)];
1312
for i in [1..#aplist] do
1313
p := P[i];
1314
if (5*N mod p ne 0) and not IsSquare(GF(p)!t) and aplist[i] ne 0 then
1315
return p;
1316
end if;
1317
end for;
1318
end function;
1319
}
1320
1321
1322
Not $A_4$: Unramified~$p$, not cube mod $\ell$,
1323
order of $\rho_g(\Frob_p)$ not divisible by $3$.
1324
\vspace{1ex}\\
1325
\begin{tabular}{|c|c|cl|}\hline
1326
$N$ & $\ell$ & $p$&\hspace{.7em}charpoly$(\rho_g(\Frob_p))$ \\\hline
1327
{\bf 1376}
1328
& $43$ & $3$ & $\qquad(x+2)^2$ \\
1329
{\bf 2416}
1330
& $151$ & $7$ & $\qquad(x+2)^2$ \\
1331
{\bf 3184}
1332
& $199$ & $3$ & $\qquad(x+2)^2$ \\
1333
{\bf 3556}
1334
& $127$ & $3$ & $\qquad(x+2)^2$\\
1335
{\bf 3756}
1336
& $313$ & $11$ & $\qquad(x+2)^2$\\
1337
{\bf 4108}
1338
& $13$ & $3$ & $\qquad(x+2)^2$\\
1339
{\bf 4288}
1340
& $67$ & $7$ & $\qquad(x+3)^2$\\
1341
{\bf 5373}
1342
& --- & &(see text)\\
1343
\hline\end{tabular}\vspace{3ex}\\
1344
\comment{
1345
function IsCubeMod(p, ell) // is p a cube in F_ell
1346
R<x>:=PolynomialRing(GF(ell));
1347
return not IsIrreducible(x^3-p);
1348
end function;
1349
1350
procedure FindA4(ell, aplist, N, e1, e2)
1351
P:=[p : p in [2..97] |IsPrime(p)];
1352
for i in [1..#aplist] do
1353
p := P[i];
1354
if (5*N mod p ne 0) then
1355
// if (5*N mod p ne 0) and not IsCubeMod(p,ell) then
1356
t := GF(5)!(Evaluate(e2,p)*aplist[i]);
1357
d := GF(5)!Evaluate(e1,p);
1358
R<x> := PolynomialRing(GF(5));
1359
f := x^2 - t*x + d;
1360
"p =",p;
1361
"f =",f;
1362
"factor(f) =",Factorization(f);
1363
end if;
1364
end for;
1365
end procedure;
1366
}
1367
1368
\end{center}
1369
\end{table}
1370
1371
1372
\begin{table}
1373
\caption{\label{table:more4}Bounding the discrimant of the fixed field
1374
of $\proj(\rho_g)$}
1375
\begin{center}
1376
\begin{tabular}{|cl|}\hline
1377
$N$ & Bound on discriminant\\
1378
{\bf 1376}& $\qquad2^6\cdot 43^2$\\
1379
{\bf 2416}& $\qquad 2^6\cdot 151^2$\\
1380
{\bf 3184}& $\qquad 2^6\cdot 199^2$\\
1381
{\bf 3556}& $\qquad 2^2\cdot 7^2 \cdot 127^2$\\
1382
{\bf 3756}& $\qquad 2^2\cdot 3^2 \cdot 313^2$\\
1383
{\bf 4108}& $\qquad 2^2\cdot 13^2 \cdot 79^2$\\
1384
{\bf 4288}& $\qquad 2^6\cdot 67^2$\\
1385
{\bf 5373}& $\qquad 3^4\cdot 199^2$\\
1386
\hline\end{tabular}
1387
\end{center}
1388
\end{table}
1389
1390
\section{Computing mod~$p$ modular forms}
1391
\subsection{Higher weight modular symbols}
1392
\label{sec:modsym}
1393
The second author developed software that computes the space of
1394
weight~$k$ modular symbols $\sS_k(N,\eps)$, for $k\geq 2$ and
1395
arbitrary~$\eps$.
1396
See~\cite{merel:1585} for the standard facts about higher weight
1397
modular symbols, and~\cite{stein:phd} for a description of
1398
how to compute with them.
1399
1400
Let $K=\Q(\eps)$ be the field generated by the values of~$\eps$.
1401
The cuspidal modular symbols $\sS_k(N,\eps)$ are a
1402
finite dimensional vector space over~$K$, which is generated by all
1403
linear combinations of higher weight modular symbols
1404
$$X^i Y^{k-2-i}\{\alp,\beta\}$$
1405
that lie in the kernel of an appropriate boundary map. There is an
1406
involution~$*$ that acts on $\sS_k(N,\eps)$, and
1407
$\sS_k(N,\eps)^+\tensor_K\C$ is isomorphic, as a module over the Hecke
1408
algebra, to the space $S_k(N,\eps;\C)$ of cusp forms.
1409
1410
Fix $k=5$. In each case considered in this paper,
1411
there is a prime ideal~$\lambda$
1412
of the ring of integers $\mathcal{O}$ of~$K$
1413
such that $\mathcal{O}/\lambda\isom \F_{25}$.
1414
Let~$\cL$ be the $\mathcal{O}$-module generated by all modular
1415
symbols of the form $X^iY^{3-i}\{\alp,\beta\}$,
1416
and let
1417
$$\sS_5(N,\eps;\F_{25})=(\cL\tensor_{\mathcal{O}}\F_{25})\cap \sS_5(N,\eps).$$
1418
This is the space that we computed.
1419
The Hecke algebra acts on $\sS_5(N,\eps;\F_{25})$, so when
1420
we find an eigenform we find a maximal ideal of the Hecke algebra.
1421
1422
As an extra check on our computation of
1423
$\sS_5(N,\eps;\F_{25})$, we computed the dimension
1424
of $S_5(N,\eps;\C)$ using both the formula of~\cite{cohen-oesterle}
1425
and the Hijikata trace formula (see~\cite{hijikata:trace})
1426
applied to the identity Hecke operator.
1427
1428
1429
\comment{%it's all in my thesis and it's not that relevant.
1430
The Manin symbols are
1431
$[i, (c,d)]$ where $0\leq i\leq k-2=3$ and
1432
$(c,d)$ vary over points in the projective plane.
1433
The Manin symbol $[i,(c',d')]$ corresponds to the
1434
modular symbol $(g.X^iY^{3-i})\{g(0),g(\infty)\}$
1435
where $g=\abcd{a}{b}{c}{d}\in\SL_2(\Z)$ is a matrix whose lower
1436
two entries are congruent to $(c',d')$ modulo $N$,
1437
and $g.X^iY^{3-i} := (dX-bY)^i(-cX+aY)^{3-i}$.
1438
Let $\sigma=\abcd{0}{-1}{1}{0}$, $\tau={0}{-1}{1}{-1}$
1439
and for $\gamma\in\SL_2(\Z)$, let
1440
$[i,(c,d)]\gamma = [\gamma.X^iY^{3-i}, (c,d)\gamma]$.
1441
Since there are only finitely many
1442
Manin symbols, we can
1443
compute $\sS_5(N,\eps)$ as the quotient of the $\F$-vector
1444
space generated by Manin symbols modulo
1445
the following relations:
1446
\begin{align*}
1447
{[i,(c,d)] + [i,(c,d)]\sigma} &= 0\\
1448
{[i,(c,d)] + [i,(c,d)]\tau + [i,(c,d)]\tau^2} &= 0\\
1449
{[i,(n c,n d)]}&=\eps(n)[i,(c,d)]
1450
\qquad n\in (\Z/N\Z)^*
1451
\end{align*}
1452
The quotient was computed by using a fast ``hashing'' function
1453
to quotient out by the $2$-term relations. The quotient
1454
by the $3$-term relations was then computed using sparse
1455
Gauss elimination. One important subtlety is that, e.g., $\sigma$
1456
and~$\tau$ do not commute so, after modding out by
1457
the~$\sigma$ relations, it is important to mod out by~$3$
1458
term relations coming both from~$\tau$ and~$\sigma\tau$.
1459
1460
The main result of~\cite{merel:1585} gives
1461
a way to compute the action of $T_p$ directly
1462
on the Manin symbols.
1463
Suppose $f\in\sS_5(N,\eps;\F_{25})$ is an eigenvector; to
1464
naively compute the action of~$T_p$ on~$f$ requires computing
1465
the action of~$T_p$ on each Manin symbol involved in~$f$,
1466
and then summing the result. This requires roughly
1467
$\dim\sS$ times as long as computing~$T_p$ on a single
1468
Manin symbol.
1469
In order to quickly compute a large number of
1470
Hecke eigenvalues we use the following projection trick.
1471
Let $\vphi\in\Hom(\sS_5(N,\eps;\F_{25}),\F_{25})$ be a (left) eigenvector for all
1472
Hecke operators~$T_p$ having the same eigenvalues as~$f$.
1473
Choose a Manin symbol $x=[i,(c,d)]$ such
1474
that $\vphi(x)\neq 0$. Since~$x$ is of a very simple form,
1475
it is easy to compute~$T_p(x)$ quickly. We have
1476
$\vphi(T_p(x)) = (T_p(\vphi))(x) = a_p \vphi(x)$,
1477
so since $\vphi(x)\neq 0$ we divide and find
1478
$a_p = \vphi(T_p(x))/\vphi(x)$.
1479
In fact, we use a generalization of this trick to
1480
quickly compute the action of~$T_p$ on any Hecke stable subspace
1481
$V\subset \Hom(\sS(N,\eps;\F_{25}),\F_{25})$.
1482
}
1483
1484
1485
\subsection{Complexity}
1486
We implemented the modular symbols algorithms mentioned above
1487
in \magma{} (see \cite{magma}) because of its robust support
1488
for linear algebra over small finite fields.
1489
1490
The following table gives a flavor of the complexity of the
1491
machine computations appearing in this paper.
1492
The table indicates how much
1493
CPU time on a Sun Ultra E450 was required to compute all data
1494
for the given level,
1495
including the matrices $T_p$ on the $2$-dimensional spaces,
1496
for $p<2000$. For example, the total time for level $N=1376$
1497
was~$6$ minutes and~$58$ seconds.
1498
\begin{center}
1499
\begin{tabular}{|cr|}\hline
1500
\vspace{-2ex}&\\
1501
N & time (minutes)\\
1502
\vspace{-2ex}&\\
1503
1376& 6:58\hspace{2.5em}\mbox{}\\
1504
2416& 10:42\hspace{2.5em}\mbox{}\\
1505
3184& 14:16\hspace{2.5em}\mbox{}\\
1506
3556& 19:55\hspace{2.5em}\mbox{}\\
1507
3756& 27:47\hspace{2.5em}\mbox{}\\
1508
4108& 23:11\hspace{2.5em}\mbox{}\\
1509
4288& 15:18\hspace{2.5em}\mbox{}\\
1510
5376& 24:49\hspace{2.5em}\mbox{}\\\hline
1511
\end{tabular}
1512
\end{center}
1513
1514
\subsection{Acknowledgment}
1515
Some of the computing equipment was purchased
1516
by the second author using a UC Berkeley Vice Chancellor Research Grant.
1517
Additional computer runs were
1518
made on the Sun Ultra E450 of the Computational Algebra Group at
1519
the University of Sydney. Allan Steel was very helpful in optimizing our
1520
code.
1521
1522
\comment{\bibliographystyle{amsplain}
1523
\bibliography{biblio}}
1524
1525
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
1526
\begin{thebibliography}{10}
1527
1528
\bibitem{artin:conjecture}
1529
E.~Artin, \emph{{\"U}ber eine neue {A}rt von {L}-reihen}, Abh. Math. Sem. in
1530
Univ. Hamburg \textbf{3} (1923/1924), no. 1, 89--108.
1531
1532
\bibitem{buhler:thesis}
1533
J.\thinspace{}P. Buhler, \emph{Icosahedral \protect{G}alois representations},
1534
Springer-Verlag, Berlin, 1978, Lecture Notes in Mathematics, Vol. 654.
1535
1536
\bibitem{bdsbt}
1537
K.~Buzzard, M.~Dickinson, N.~Shepherd-Barron, and R.~Taylor, \emph{On
1538
icosahedral {A}rtin representations}, in preparation.
1539
1540
1541
\bibitem{buzzard-taylor}
1542
K.~Buzzard and R.~Taylor, \emph{Companion forms and weight one forms}, Ann. of
1543
Math. (2) \textbf{149} (1999), no.~3, 905--919.
1544
1545
\bibitem{cohen-oesterle}
1546
H.~Cohen and J.~Oesterl{\'e}, \emph{Dimensions des espaces de formes
1547
modulaires}, (1977), 69--78. Lecture Notes in Math., Vol. 627.
1548
1549
\bibitem{magma}
1550
W.~Bosma, J.~Cannon, and C.~Playoust, \emph{The {M}agma algebra system {I}:
1551
{T}he user language}, J. Symb. Comp. \textbf{24} (1997), no.~3-4, 235--265,
1552
\\\protect{\sf http://www.maths.usyd.edu.au:8000/u/magma/}.
1553
1554
\bibitem{deligne-serre}
1555
P.~Deligne and J-P. Serre, \emph{Formes modulaires de poids $1$}, Ann. Sci.
1556
\'Ecole Norm. Sup. (4) \textbf{7} (1974), 507--530 (1975).
1557
1558
\bibitem{freyetal}
1559
G.~Frey (ed.), \emph{On {A}rtin's conjecture for odd \protect{$2$}-dimensional
1560
representations}, Springer-Verlag, Berlin, 1994.
1561
1562
\bibitem{gross:tameness}
1563
B.\thinspace{}H. Gross, \emph{A tameness criterion for \protect{G}alois
1564
representations associated to modular forms (mod \protect{$p$})}, Duke Math.
1565
J. \textbf{61} (1990), no.~2, 445--517.
1566
1567
\bibitem{hijikata:trace}
1568
H.~Hijikata, \emph{Explicit formula of the traces of \protect{H}ecke operators
1569
for \protect{$\Gamma_0(N)$}}, J. Math. Soc. Japan \textbf{26} (1974), no.~1,
1570
56--82.
1571
1572
\bibitem{langlands:basechange}
1573
R.\thinspace{}P. Langlands, \emph{Base change for \protect{${\rm {G}{L}}(2)$}},
1574
Princeton University Press, Princeton, N.J., 1980.
1575
1576
\bibitem{merel:1585}
1577
L.~Merel, \emph{Universal \protect{F}ourier expansions of modular forms}, On
1578
{A}rtin's conjecture for odd \protect{$2$}-dimensional representations
1579
(Berlin), Springer, 1994, pp.~59--94. Lecture Notes in Math., Vol. 1585.
1580
1581
\bibitem{miyake}
1582
T.~Miyake, \emph{Modular forms}, Springer-Verlag, Berlin, 1989, Translated from
1583
the Japanese by Yoshitaka Maeda.
1584
1585
\bibitem{shimura:intro}
1586
G.~Shimura, \emph{Introduction to the arithmetic theory of automorphic
1587
functions}, Princeton University Press, Princeton, NJ, 1994, Reprint of the
1588
1971 original, Kan Memorial Lectures, 1.
1589
1590
\bibitem{stein:phd}
1591
W.\thinspace{}A. Stein, \emph{Explicit approaches to modular abelian
1592
varieties}, U.\thinspace{}C. Berkeley Ph.D. thesis (2000).
1593
1594
\bibitem{sturm:cong}
1595
J.~Sturm, \emph{On the congruence of modular forms}, Number theory (New York,
1596
1984--1985), Springer, Berlin, 1987, pp.~275--280.
1597
Lecture Notes in Math., Vol. 1240.
1598
1599
\bibitem{taylor:artin2}
1600
R.~Taylor, \emph{On icosahedral {A}rtin representations II},
1601
in preparation.
1602
1603
\bibitem{tunnell:artin}
1604
J.~Tunnell, \emph{Artin's conjecture for representations of octahedral type},
1605
Bull. Amer. Math. Soc. (N.S.) \textbf{5} (1981), no.~2, 173--175.
1606
1607
\end{thebibliography}
1608
1609
1610
\end{document}
1611
1612
1613
1614
1615
1616
1617
***
1618
1619
[8458981, 509]
1620
1621
1622
// Cohen-Oesterle Dimension computations in MAGMA:
1623
1624
> G<a2,b2,c> := DirichletGroup(1376,CyclotomicField(EulerPhi(1376)));
1625
> eps:=a2*(c^(Order(c) div 3));
1626
> Order(eps);
1627
6
1628
> DimensionCuspForms(eps,5);
1629
1630
1631
1632
> G<a2,b2,c> := DirichletGroup(2416,CyclotomicField(EulerPhi(2416)));
1633
> eps:=a2*(c^(Order(c) div 3));
1634
> Order(eps);
1635
6
1636
> DimensionCuspForms(eps,5);
1637
1210
1638
1639
> G<a2,b2,c> := DirichletGroup(3184,CyclotomicField(EulerPhi(3184)));
1640
> eps:=a2*(c^(Order(c) div 3));
1641
> Order(eps);
1642
6
1643
> DimensionCuspForms(eps,5);
1644
1594
1645
1646
1647
> G<a,b,c> := DirichletGroup(3556,CyclotomicField(EulerPhi(3556)));
1648
> eps:=(b^(Order(b) div 2))*(c^(Order(c) div 3));
1649
> Order(eps);
1650
6
1651
> DimensionCuspForms(eps,5);
1652
2042;
1653
1654
1655
> G<a,b,c> := DirichletGroup(3756,CyclotomicField(EulerPhi(3756)));
1656
> eps:=(b^(Order(b) div 2))*(c^(Order(c) div 3));
1657
> Order(eps);
1658
6
1659
> DimensionCuspForms(eps,5);
1660
1661
1662
> G<a,b,c> := DirichletGroup(4108,CyclotomicField(EulerPhi(4108)));
1663
> eps:=(b^(Order(b) div 3))*(c^(Order(c) div 2));
1664
> DimensionCuspForms(eps,5);
1665
1666
> G<a,b,c> := DirichletGroup(4288,CyclotomicField(EulerPhi(4288)));
1667
> eps:=(b^(Order(b) div 2))*(c^(Order(c) div 2));
1668
> DimensionCuspForms(eps,5);
1669
1670
1671
> G<a,b> := DirichletGroup(5373,CyclotomicField(EulerPhi(5373)));
1672
> eps:=(a^(Order(a) div 2))*(b^(Order(b) div 3));
1673
> DimensionCuspForms(eps,5);
1674
1675
1676
///////////////////////////////////////////
1677
1678
procedure powfrob(p, e1, e2, aplist)
1679
Primes := [p : p in [2..97] |IsPrime(p)];
1680
n := Index(Primes,p);
1681
a := GF(5)!(Evaluate(e2,p)*aplist[n]);
1682
b := GF(5)!Evaluate(e1,p);
1683
R<x>:=PolynomialRing(GF(5));
1684
Q<y> := quo<R|x^2-a*x+b>;
1685
x^2 - a*x + b;
1686
for i in [1..24] do
1687
f := MinimalPolynomial(y^i);
1688
if Degree(f) le 1 then
1689
printf "rho_g(Frob_%o)^%o satisfies %o.\n", p, i, f;
1690
end if;
1691
end for;
1692
end procedure;
1693
1694
1695
1696
> // 1376
1697
> h := x^5+2*x^4+6*x^3+8*x^2+10*x+8;
1698
> N := 2^5*43;
1699
> F<alp> := GF(25);
1700
> G<a2,b2,c>:=DirichletGroup(N, F);
1701
> eps := a2 * (c^(Order(c) div 3));
1702
> aplist := [0,alp^16,alp^22,alp^14,4,alp^14,alp^14,0, alp^16,alp^8,0,alp^10,1,alp^10,1,alp^22,4,alp^14,alp^4,alp^20,alp^2,alp^20,alp^4,alp^10];
1703
> qEigenform(aplist,eps,5);
1704
//q + alp^16*q^3 + alp^22*q^5 + alp^14*q^7 + alp^14*q^9 + 4*q^11 + alp^14*q^13 + alp^14*q^15 + alp^14*q^17 + O(q^20)
1705
1706
1707
1708
> // 2416
1709
> h := x^5-2*x^3+2*x^2+5*x+6;
1710
> N := 2^4*151;
1711
> k:=5;
1712
> F<alp> := GF(25);
1713
> G<a2,b2,c>:=DirichletGroup(N, F);
1714
> eps := a2 * (c^(Order(c) div 3));
1715
> aplist:=[0,3,alp^22,alp^16, alp^4, alp^2, alp^22,3,alp^22,3,alp^16,alp^22,2,alp^8,alp^8,0,1,alp^8,3,alp^8,2,2,2,alp^20,0,2,alp^16,1];
1716
> qEigenform(aplist,eps,k);
1717
// q + 3*q^3 + alp^22*q^5 + alp^16*q^7 + alp^4*q^11 + alp^2*q^13 + alp^16*q^15 + alp^22*q^17 + 3*q^19 + O(q^20)
1718
1719
> // 3184
1720
> h := x^5+5*x^4+8*x^3-20*x^2-21*x-5;
1721
> N:=2^4*199;
1722
> F<alp> := GF(25);
1723
> G<a2,b2,c>:=DirichletGroup(N, F);
1724
> eps := a2 * (c^(Order(c) div 3));
1725
> aplist:=[0,alp^16,3,alp^22,3,alp^22,3,alp^16];
1726
> qEigenform(aplist,eps,5);
1727
// q + alp^16*q^3 + 3*q^5 + alp^22*q^7 + alp^14*q^9 + 3*q^11 + alp^22*q^13 + alp^10*q^15 + 3*q^17 + alp^16*q^19 + O(q^20)
1728
1729
> // 3556
1730
> h := x^5+3*x^4+9*x^3-6*x^2-4*x-40;
1731
> N := 2^2*7*127;
1732
> F<alp> := GF(25);
1733
> G<a,b,c>:=DirichletGroup(N, F);
1734
> eps := b^(Order(b) div 2) * (c^(Order(c) div 3));
1735
> aplist := [0,alp^16,alp^14,alp^10,alp^2,alp^22,alp^14,0, alp^10,0,alp^16,alp^20];
1736
1737
1738
> // 3756
1739
> h := x^5-3*x^3+10*x^2+30*x-18;
1740
> N := 2^2*3*313;
1741
> F<alp> := GF(25);
1742
> G<a,b,c>:=DirichletGroup(N, F);
1743
> eps := b^(Order(b) div 2) * (c^(Order(c) div 3));
1744
> aplist:=[0,alp^14,alp^14,3,alp^16,alp^10,0,3,3,alp^2,alp^22,alp^22,alp^20,alp^16,alp^4,4,alp^8,0];
1745
> qEigenform(aplist,eps,5);
1746
// q + alp^14*q^3 + alp^14*q^5 + 3*q^7 + alp^4*q^9 + alp^16*q^11 + alp^10*q^13 + alp^4*q^15 + 3*q^19 + alp^8*q^21 + 3*q^23 + alp^4*q^25 + 3*q^27 + alp^2*q^29 + alp^22*q^31 + 2*q^33 + alp^8*q^35 + alp^22*q^37 + q^39 + alp^20*q^41 + alp^16*q^43 + 3*q^45 + alp^4*q^47 + 3*q^49 + 4*q^53 + 2*q^55 + alp^8*q^57 + alp^8*q^59 + O(q^62)
1747
1748
1749
// 4108
1750
> h := x^5+4*x^4+3*x^3+9*x^2+4*x+5;
1751
> N := 2^2*13*79;
1752
> F<alp> := GF(25);
1753
> G<a,b,c>:=DirichletGroup(N, F);
1754
> eps := b^(Order(b) div 3) * (c^(Order(c) div 2));
1755
> aplist := [0,alp^16,alp^11, alp^20,alp^10,4,0,alp^14,0, alp^22, 0, alp^22,alp^10,alp^2,3,4,alp^14,alp^2,alp^10];
1756
> qEigenform(aplist,eps,5);
1757
//q + alp^16*q^3 + alp^11*q^5 + alp^20*q^7 + alp^14*q^9 + alp^10*q^11
1758
// + 4*q^13 + alp^3*q^15 + alp^14*q^19 + 4*q^21 + O(q^24)
1759
1760
1761
// 4288
1762
> h := x^5+4*x^4+5*x^3+8*x^2+3*x+2;
1763
> N := 2^6*67;
1764
> F<alp> := GF(25);
1765
> G<a2,b2,c>:=DirichletGroup(N, F);
1766
> eps := a2*b2^(Order(b2) div 2)*(c^(Order(c) div 3));
1767
> aplist := [0,3,alp^14, alp^20, alp^20, alp^16, alp^16,0,0, alp^14,alp^20,alp^4,alp^8,2,0,4,1,alp^16,alp^4,alp^20,0,alp^20,0,2];
1768
> qEigenform(aplist,eps,5);
1769
//q + 3*q^3 + alp^14*q^5 + alp^20*q^7 + 3*q^9 + alp^20*q^11 + alp^16*q^13 + alp^8*q^15 + alp^16*q^17 + alp^14*q^21 + O(q^24)
1770
1771
1772
// 5373
1773
> h := x^5+2*x^4+x^3+7*x^2+23*x-11;
1774
> N := 3^3*199;
1775
> F<alp> := GF(25);
1776
> G<a,b>:=DirichletGroup(N, F);
1777
> eps := a^(Order(a) div 2) * (b^(Order(b) div 3));
1778
> aplist := [alp^16, 0, 4, 0, 2, alp^22, 1, alp^16, 0];
1779
> qEigenform(aplist,eps,5);
1780
//q + alp^16*q^2 + alp^14*q^4 + 4*q^5 + 3*q^8 + alp^4*q^10 + 2*q^11 + alp^22*q^13 + q^17 + alp^16*q^19 + alp^2*q^20 + alp^22*q^22 + O(q^24)
1781
1782
1783