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\title{An Algorithm for Massive Realizations of Projective Linear
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Groups as Galois Groups over $\mathbb{Q}$ via Modular Representations}
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\author{Luis Dieulefait\thanks{supported by TMR -
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Marie Curie Fellowship ERB4001GT974451} \ and Nuria Vila\thanks{
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partially
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supported by DGES grant PB96-0970-C02-01}\\
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Dept. d'Algebra i Geometria, Universitat de Barcelona\\
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Gran Via de les Corts Catalanes 585\\
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08007 - Barcelona\\
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Spain}
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\begin{document}
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\maketitle
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%\tableofcontents
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\vspace{6mm}
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%\begin{abstract}
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%\end{abstract}
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\section{Introduction}
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The images of the $\ell$-adic Galois representations attached to a
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newform $f = \sum_{n=1}^{\infty} a_n q^n $ of weight 2 on
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$\Gamma_0 (N)$ are known to be ``as large as possible" for almost
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every $\ell$, with the sole condition that $f$ doesn't have complex
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multiplication (CM), thanks to the work of Momose and Ribet (cf. [Mo 81] ,
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[Ri 85] ).
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Actually, if $f$ has any
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inner twist, this is only true for the restriction of these
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representations to a certain open subgroup of
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$G_{\mathbb{Q}} = Gal(\mathbb{\overline{Q}} /\mathbb{Q})$ defined in terms
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of the
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twisting characters. Even in this case, thanks to a theorem of Papier,
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something can be said about the image of $G_{\mathbb{Q}}$ (see [Ri 85] and
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[D]).
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In this paper we will restrict ourselves to newforms without inner
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twists and without CM. We will give an algorithm for an explicit
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determination
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of the
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finitely many primes $\ell$ (seeming to be) exceptional, i.e., such
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that the image of the corresponding representation is not ``as large as
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possible" .
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As a consequence, we will be able to realize many families of
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projective linear groups over finite fields as Galois groups over
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$\mathbb{Q}$.
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We have
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concentrated our efforts on newforms $f$ such that
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$\mathbb{Q}_f$, the number field generated
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by its coefficients $a_n$, is an abelian extension of $\mathbb{Q}$, to
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get a simple characterisation of primes with a prescribed residue class
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degree.
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The examples are designed to realize, for high density sets of primes,
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the following groups: $PSL(2,\mathbb{F}_{\ell^2}), \; \;
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PGL(2,\mathbb{F}_{\ell^3})
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\;$ and $PSL(2,\mathbb{F}_{\ell^4})$ as Galois groups over
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$\mathbb{Q}$. In particular, the groups in the
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exponent $2$ case will be realized as Galois groups over
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$\mathbb{Q}$ for every $3 < \ell < 5000000$, and the ones in the
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exponent $3$ case for every $3 < \ell < 500000$.\\
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\section{The Algorithm}
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Let $f \in S_2(N)=S_2(\Gamma_0(N))$ be a newform, i.e., a normalized
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eigenform for the whole
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Hecke algebra. We assume from now on that $f$ does
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not have CM nor inner twists.
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Let $\mathbb{Q}_f$ be the number field generated
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by its coefficients $a_n $ and $\mathcal{O}$ its ring of integers. For
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every prime $\ell$ put: $\mathcal{O}_{\ell} = \mathcal{O}
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\otimes_{\mathbb{Z}} \mathbb{Z}_\ell $ and $\mathbb{Q}_{f, \ell} =
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\mathbb{Q}_f \otimes_{\mathbb{Q}} \mathbb{Q}_\ell $
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By Deligne's theorem, there exists a continuous Galois representation:
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$$\rho_{\ell} : Gal( \overline{\mathbb{Q}} / \mathbb{Q}) \rightarrow
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GL(2, \mathcal{O}_{\ell} ) \subseteq GL(2, \mathbb{Q}_{f, \ell} )$$
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unramified outside $\ell N$ satisfying, for every prime $p \nmid \ell
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N$ : $$ trace \; \rho_{\ell} (Frob \; p) = a_p , \qquad \qquad det \;
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\rho_{\ell} (Frob \; p) = p $$ Let $G_{\mathbb{Q}}=Gal(
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\overline{\mathbb{Q}} /
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\mathbb{Q})$ and $G_\ell = \rho_{\ell}(G_{\mathbb{Q}})$, closed subgroup
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of $GL(2,
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\mathcal{O}_{\ell} ) $. From the decompositions: $\mathbb{Q}_{f, \ell}
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= \prod_{\lambda \mid \ell} \mathbb{Q}_{f, \lambda}$ and $
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\mathcal{O}_{ \ell} = \prod_{\lambda \mid \ell} \mathcal{O}_{\lambda}$
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we get a decomposition of $\rho_{\ell}$ as direct sum of
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representations : $$\rho_{\lambda} : G_{\mathbb{Q}} \rightarrow GL(2,
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\mathcal{O}_{\lambda} ) \subseteq GL(2, \mathbb{Q}_{f, \lambda} )$$
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Let $\lambda$ be a prime in $\mathbb{Q}_{f} $. Consider the reduction
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$\overline{\rho}_{\lambda}$ of $\rho_{\lambda}$ , obtained by composing
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$\rho_{\lambda}$ with the reduction map: $GL(2, \mathcal{O}_{\lambda} )
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\rightarrow GL(2, \mathbb{F}_{\lambda})$, where $\mathbb{F}_{\lambda}$
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is the residue field of $\lambda$. Let $\ell = \ell (\lambda)$ be the
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rational prime such that $\lambda \mid \ell$ and let
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$\overline{G}_{\lambda}$ be the image of $\overline{\rho}_{\lambda}$.
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In order to compute the images of the representations
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$\overline{\rho}_{\lambda}$ we will use the following result
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of Ribet [Ri 85]:
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146
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\begin{teo}
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\label{teo:irreymult}:
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For almost every $\lambda$ we have:
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151
a) The representation $\overline{\rho}_{\lambda}$ is an
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irreducible 2-dimensional representation over
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$\mathbb{F}_{\lambda}$
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b) The order of the group $\overline{G}_\lambda=
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\overline{\rho}_{\lambda}(G_{\mathbb{Q}})$ is divisible by $\ell$
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\end{teo}
159
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In the next section we will give an algorithm to compute
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a set of primes containing those that do not verify these two conditions.
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The following result of Ribet [Ri 85] completes the determination of
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the images:
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\begin{teo}
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\label{teo:hlal}: Let $ A_{\ell} = \{x \in GL(2,
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\mathcal{O}_{\ell}) \quad / \; det(x) \in \mathbb{Z}_{\ell}^* \} $ \\
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The equality $G_{\ell} = A_{\ell}$ holds for almost every prime. In fact,
169
this equality holds whenever the
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following conditions are satisfied :\\
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0) $\ell$ does not ramify in $\mathbb{Q}_f / \mathbb{Q} $ \\
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1) The determinant map $G_{\ell} \rightarrow
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\mathbb{Z}_{\ell}^*$ is surjective \\
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2) $\ell \geq 5$ \\
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3) $G_{\ell}$
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contains an element $x_{\ell} $ such that: $(trace \; x_{\ell})^2 $
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generates $\mathcal{O}_{\ell}$ as a $\mathbb{Z}_{\ell} $-algebra \\
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4) for each
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$\lambda \mid \ell$, the group $\overline{\rho}_{\lambda}(G_{\mathbb{Q}})$
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is an
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irreducible subgroup of $GL(2, \mathbb{F}_{\lambda})$ whose order is
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divisible by $\ell$.
183
\end{teo}
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To see that these conditions hold for almost every prime, theorem
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\ref{teo:irreymult} is applied (to deal with condition (4)) together
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with the following facts:\\ -there exists an integer $v$ such that $v$
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is relatively prime to
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$N$ and $a_v^2$ generates $\mathbb{Q}_f$ over
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$\mathbb{Q}$ (condition (3))\\ -the determinant of $\rho_{\ell}$
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coincides on $G_{\mathbb{Q}}$ with the $\ell$-adic cyclotomic character
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$\chi_{\ell}$ (condition (1)). \\
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This result allows us to realize projective linear groups over finite
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fields
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as Galois groups over $\mathbb{Q}$ in the
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following way: Let $\ell$ be a prime verifying the conditions
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of theorem \ref{teo:hlal} and $\lambda$ a prime in $\mathcal{O}$ over
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$\ell$.
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Then $\overline{G}_{\lambda} \subseteq GL(2, \mathbb{F}_{\lambda})$
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and $P(\overline{G}_{\lambda} )$, its image in $PGL(2,
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\mathbb{F}_\lambda)$,
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is (cf. [Re-Vi 95]):\\
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$ PSL(2, \mathbb{F}_\lambda) \quad \mbox{if} \;
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[\mathbb{F}_\lambda : \mathbb{F}_\ell] \; \mbox{is even},\\
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PGL(2, \mathbb{F}_\lambda) \quad \mbox{otherwise}. $
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\subsection{The Exceptional Primes for Theorem 2.1 }
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\subsubsection{Reducible Representations}
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Let us suppose that for a prime $\lambda \in \mathcal{O}$ dividing
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$\ell$, $\overline{\rho}_{\lambda}(G_{\mathbb{Q}})$ is a reducible
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subgroup of
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$GL(2, \mathbb{F}_{\lambda})$. Then
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$$ \overline{\rho}_{\lambda} \cong \pmatrix { \phi_1& *\cr
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0& \phi_2\cr } $$
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with $\phi_i = \epsilon_i \overline{\chi}_{\ell}^{m_i}$, for $i=1,2$,
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$\epsilon_1, \epsilon_2$
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Dirichlet characters unramified outside $N$ with image in
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$\mathbb{F}_{\lambda}$ and $\overline{\chi}_{\ell}$ the cyclotomic
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character $mod \; \ell$. \\
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By Deligne's theorem, for every prime $p \nmid \ell N$ we have: $$a_p
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= trace \; \rho_{\lambda} (Frob \; p) \equiv^{mod \; \lambda}
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\phi_1(Frob \; p) +
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\phi_2(Frob \;p) = \epsilon_1(p) \; p^{m_1} + \epsilon_2(p) \; p^{m_2}
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$$ $$ p = det \; \rho_{\lambda} (Frob \; p) \equiv^{mod \; \lambda}
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\phi_1(Frob \; p) \;
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\phi_2(Frob \;p) = \epsilon_1(p) \; \epsilon_2(p) \; p^{m_1 + m_2}$$
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>From the last formula we see that: $m_1 + m_2 \equiv 1
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\pmod{\ell -1}$ and
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$\epsilon_1(p) \epsilon_2(p) = 1$ for every $p \nmid N$, so that we can
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choose
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$0 \leq m_1 < m_2 < \ell - 1$.
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Generalizing results of [Sw 73] to the case of general level,
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Faltings and Jordan proved in [Fa-Jo 95]
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the following result, that we will state for general weight, i.e.,
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when the representation comes from a newform $f \in S_k (N)$:
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\begin{teo}
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\label{teo:Faltings}: Suppose that the representation
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$\overline{\rho}_{\lambda}$ is reducible.
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Then if $\ell > k, \ell \nmid N$,
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$\overline{\rho}_{\lambda} = \epsilon_1 \oplus
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\epsilon_2 \overline{\chi}_{\ell}$, with the characters $\epsilon_i$
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unramified outside $N$.
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\end{teo}
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Carayol and Liv ([Ca 89] , [Li 89]) have given bounds for the
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conductors of modular Galois representations. Using these results,
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together with the ones of Faltings and Jordan, we have (see [Fa-Jo 95]) :
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\begin{cor}
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\label{teo:ceroyuno}: Let $f$ be a newform of weight $2$ and level $N$
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and
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$\lambda \mid \ell$ a prime in $\mathcal{O}$ such that
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$\overline{\rho}_{\lambda}$ is reducible. Then if $\ell > 2 , \ell
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\nmid N ,$
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we have, for every $p \nmid \ell N$: $$a_p \equiv \epsilon(p) + p \;
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\epsilon^{-1}(p) \pmod{\lambda}$$ with $\epsilon$ a character
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unramified outside $N$ whose conductor $c$ verifies: $c^2 \mid N$.
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\end{cor}
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In the algorithm, we will only use the following two cases of this
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corollary:\\
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a) if $p \neq \ell, p \equiv 1 \pmod{c} $ ,
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then: $a_p \equiv 1+p \pmod{\lambda}$. \\
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b) if $p \neq \ell, p \equiv -1 \pmod{c} $ ,
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then: $a_p \equiv \pm (1+p) \pmod{\lambda}$. \\
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This congruences can't be equalities, because: $\arrowvert a_p
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\arrowvert \leq 2 \sqrt{p}$ . So that only finitely many $\lambda$ can
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satisfy them.\\
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In the case of prime level $N$, a better criterion for reducibility was
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given by Mazur in [Ma 77]. Here the condition $\ell \nmid N$ can be
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removed and the reducible primes are the Eisenstein primes, defined as
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follows (see [Ma 77]):
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\begin{defin}:
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Let $\mathbf{T}$ be the Hecke ring generated by the Hecke operators
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acting on $J_0(N)$.\\
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The Eisenstein ideal $\frak{\Im }\subseteq \mathbf{T}$ is the one
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generated by the elements: $1+l-\mathbf{T}_l\ (l\neq N\ ),\ 1+\omega .$
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An
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Eisenstein prime is a prime ideal $\beta \subseteq \mathbf{T}$ in the
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support of $\frak{%
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\Im .}$ Let $n=num(\frac{N-1}{12})$. We have a one-to-one correspondence
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between Eisenstein primes $\beta $ and prime factors $p$ of $n,$ given
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by:
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\[
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{\rm Prime\quad factors\quad of\quad }n\leftrightarrow
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{\rm Eisenstein\quad primes}
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\]
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\[
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p\leftrightarrow (\frak{\Im ,}p)
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\]
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Besides, for every Eisenstein prime $\beta ,$ $\mathbf{T/}\beta
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=\mathbf{F}%
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_p \;$. In particular, whenever the inertia is nontrivial
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the involved prime will not be Eisenstein.\\
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\end{defin}
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\begin{lema}
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\label{teo:Eisenstein}
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Let $f$ be a newform of weight $2$ and prime level $N$. Let
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$\lambda \mid \ell$, $\ell \geq 5$, be a prime in $\mathcal{O}$ such
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that
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$\overline{\rho}_{\lambda}$ is reducible.\\
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Then: $\ell \mid N-1$
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\end{lema}
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\subsubsection{The second condition}
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We now turn to the second condition in theorem \ref{teo:irreymult}. \\
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Let $\lambda$ and $\ell$ be as before and suppose that the order of
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$\overline{G}_{\lambda}$ is not divisible by $\ell$.
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The reducible case being already treated, we will also assume that
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$\overline{\rho}_{\lambda}$ is irreducible. We consider
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$P(\overline{G}_{\lambda})$, its image in $PGL(2,\mathbb{F}_{\lambda})
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$.
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The only possibilities for this image are to be:\\
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1) cyclic\\
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2) dihedral or\\
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3) isomorphic to one of
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the following special groups : $A_4 , S_4 , A_5 $ \\
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In case 1),
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$\overline{\rho}_{\lambda}$ would be reducible, contradicting our
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assumption.\\
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In case 2) there exists a Cartan subgroup
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$C_{\lambda}$ of $GL(2, \mathbb{F}_{\lambda}) $ such that
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$\overline{G}_{\lambda}$ is contained in the normaliser $N_{\lambda}$
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of $C_{\lambda}$, but not in $C_{\lambda }$. \\
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Let $\varphi_{\lambda}
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: G_{\mathbb{Q}} \rightarrow \{ \pm 1 \}$ be the composition: $$
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G_{\mathbb{Q}} \rightarrow
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\overline{G}_{\lambda} \subseteq N_{\lambda} \twoheadrightarrow
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N_{\lambda}/C_{\lambda} \cong \{ \pm 1 \} $$ The kernel of
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$\varphi_{\lambda}$ is then an open subgroup of $G_{\mathbb{Q}}$ of index
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$2$, so
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its fixed field $K_{\lambda}$ is a quadratic field unramified outside
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$\ell N$. \\
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We now consider the restriction of $\overline{\rho}_{\lambda}$ to $I$,
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an inertia subgroup of $G_{\mathbb{Q}}$ for $\ell$. Its
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semisimplification can be
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described in terms of fundamental characters as in [Se 87]. Using the
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fact that the
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weight of $f$ is $2$, it follows (cf. [Ri 97]) that the image of $I$ in
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$PGL(2, \mathbb{F}_\lambda)$ is cyclic of order $\ell \pm 1$. Then,
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this is a cyclic subgroup of the dihedral group
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$P(\overline{G}_\lambda)$ with order, if $\ell>3$, greater than $2$, and
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therefore
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must be contained in its center . This implies that the quadratic field
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$K_{\lambda}$
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does not ramify at $\ell$, if $\ell >3$. This is enough to make the
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dihedral case
355
computable, but there is another observation that will simplify the
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computations if $N$ has any prime factor $q$ such that $q \parallel N$.
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In this case, the ramification subgroup of $\overline{G}_{\lambda}$ for
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the prime $q$ is unipotent, if $q$ divides the conductor of
359
$ \overline{\rho}_{\lambda}$, and trivial otherwise.
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This, together with the assumption that
361
the order of $\overline{G}_{\lambda}$ is prime to $\ell$, implies that
362
these ramification groups are in fact trivial for all these primes.\\
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We conclude that the field $K_{\lambda}$ can only ramify at those prime
364
factors $r$ of $N$ (if any) such that $r^2 \mid N$. Let $N'$ be the
365
product of these prime factors. If we call $\alpha$ the quadratic
366
character corresponding to $K_{\lambda}$ we have proved the following
367
368
\begin{lema}
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\label{teo:dihedral}: Suppose that $\overline{\rho}_{\lambda}$
370
falls in case (2). Then if
371
$\ell > 3$, we have: $$ a_p \equiv \alpha (p) \; a_p
372
\pmod{\lambda} $$ for every $p \nmid \ell N$; where $\alpha$ is a
373
quadratic character unramified outside $N'$.\\
374
In particular, if $N$ is
375
squarefree, no $\lambda$ verifies the hypothesis of this lemma.\\
376
\end{lema}
377
378
In case 3) it is known (see [Ri 85] ) that for every $p
379
\nmid \ell N$\\ $$ a_p^2 \equiv 0, \; p, \; 2p \; \;or \; \; 4p
380
\pmod{\lambda},
381
\quad or \; \; a_p^4-3 p a_p^2 + p^2 \equiv 0 \pmod{\lambda}
382
\quad \quad (2.1)$$ In [Ri 85] it is proved that this can hold
383
only for finitely many $\ell$. Actually, we will need to apply (2.1) only
384
to the prime $\ell =5$ because in the case of weight $2$ and irreducible
385
$ \overline{\rho}_{\lambda}$, using again
386
the description of the semisemplification of the restriction of
387
$\overline{\rho}_{\lambda}$ to an inertia subgroup for the prime $\ell$
388
in terms of fundamental characters, it follows that there is an element
389
of order $\ell \pm 1$ in the projective image of this
390
inertia subgroup, and therefore in $P( \overline{G}_{\lambda})$, and
391
this rules out case 3) if $\ell \geq 7$ (cf. [Se 72], [Ri 97]).\\
392
393
It is proved in [Ri 97] that in the squarefree level case, 3) can not
394
hold not
395
even for $5$, so that in this case the following result holds:
396
397
\begin{teo}
398
\label{semistable}:
399
Let $f$ be a newform of weight $2$ and level $N$ squarefree.
400
Let $\lambda \mid \ell$, $\ell \geq 5$, be a prime in $\mathcal{O}$ such
401
that
402
$\overline{\rho}_{\lambda}$ is irreducible. Then the image of
403
$\overline{\rho}_\lambda$ has order multiple of $\ell$.
404
\end{teo}
405
406
407
\subsection{Description of the algorithm}
408
At this point we can give a complete description of the algorithm to
409
compute the exceptional primes in theorem \ref{teo:hlal}. Suppose that
410
we have a newform $f \in S_2(N)$ not having CM nor inner twists, and
411
we consider the representation $\rho_\ell$ for a prime $\ell \geq
412
5$ . The conditions of theorem \ref{teo:hlal} are treated as follows :\\
413
414
\begin{itemize}
415
\item Step (i): It is enough, in order to verify conditions 0) and 3),
416
to find a
417
coefficient $a_p$ with $p \nmid \ell N$ such that: $a _{p}^{2}$
418
(and a fortriori $a_{p}$) generates $ \mathbb{Q} _{f}$ over $
419
\mathbb{Q}$ and $\ell \nmid disc( a _{p} ^{2} )$.
420
\item Step (ii) :
421
Find the maximal $c$
422
such that $c ^{2} \mid N$, and a coefficient
423
$a _{p}$ with $p \equiv 1 \pmod{c} $
424
such that
425
$$p \nmid \ell N \; \mbox{and} \; a _{p} - (1+p) \not\equiv 0
426
\pmod{\lambda} $$
427
for
428
every $\lambda \mid \ell$. Then, if $\ell \nmid N$, the reducible case is
429
excluded.\\
430
To verify the congruences, it's enough to compute the norm of $a _{p} -
431
(1+p)$ and
432
check that $\ell \nmid \mathcal{N}(a _{p}-(1+p))$.\\
433
Similarly, take a prime $p \equiv -1 \pmod{c}$ with
434
$$p \nmid \ell N \; \mbox{and} \;
435
a _{p} \not\equiv \pm (1+p) \pmod{\lambda}$$
436
It suffices to
437
verify that $\ell \nmid \mathcal{N}(a _{p} \pm (1+p))$.\\
438
439
In the prime level case reducible primes $\ell \geq 5$
440
are those $\ell \mid N - 1$. In particular, the
441
prime $N$ (untractable with the other approach ) is not reducible.\\
442
443
444
\bf{Remark}: \rm The remaining conditions have not to be verified in the
445
case of
446
squarefree level. In this case, all exceptional primes $\ell \geq 5$
447
are detected with steps (i) and (ii) .
448
\item Step (iii) : For every quadratic character $\alpha$
449
ramifying only at the primes $q$
450
such that $q ^{2} \mid N$, find a coefficient $a_{p}$ with:
451
$$p \nmid N \ell, \; \; a _{p} \neq 0,
452
\; \; \alpha (p) = -1 \; \mbox{and} \; a_{p} \not\equiv - a_{p}
453
\pmod{\lambda}$$
454
for every $\lambda \mid \ell$. Then, the dihedral case is excluded.\\
455
To verify the congruences, it suffices to check that $ \ell \nmid
456
\mathcal{N}
457
(a_{p})$.
458
\item Step (iv): Take a prime $p \nmid 5N$ and compute the resultants
459
of $Q$, the minimal polynomial of
460
$a_p \;$, and the polynomials $P_i \; , \; \; i=1,...,5$ in formula
461
(2.1):
462
$$x^2, \quad x^2-p \; , \quad x^2 -2p \; , \quad x^2-4p \; , \quad
463
x^4-3px^2+p^2$$
464
If
465
$$5 \nmid Res(P_i , Q) , \; \; i=1,...,5 \;$$
466
it follows that none of the
467
$P_i$ has $a_p$ as a root $mod \; \widehat{5}$, for every $ \widehat{5}
468
\mid
469
5$. Then, the case of special images is excluded.
470
471
472
\end{itemize}
473
474
\vskip 1cm
475
We conclude the section with a brief explanation of how, for the newforms
476
in
477
the examples, we have determined the corresponding number fields and
478
checked that the newforms have no CM nor inner twists.
479
480
\medskip
481
482
Determination of the number field $ \mathbb{Q}_{f}$:
483
Let $f$ be a newform in $S_{2} (N)$. If $f$ has a coefficient $a_{p}, \;
484
p \nmid
485
N$, which is a simple root of the characteristic polynomial of the
486
Hecke operator $T_{p}$ acting on $ S_{2}^{new} (N)$, then it follows
487
that $ \mathbb{Q}(a_{p}) = \mathbb{Q}_{f}$.
488
489
\medskip
490
491
CM-Inner Twists:
492
It is known that a newform $f \in S_{2} (N)$ doesn't have CM nor inner
493
twists if
494
the level is squarefree. It is also known that if there exists a prime
495
$q$ strictly dividing the level $N$, i.e., $ \; q \parallel N$, then in
496
$S_{2}
497
(N)$ there is no newform with CM.
498
In the remaining cases, to avoid CM one has to check, for every
499
character $\alpha$ ramifying only at primes dividing the level $N$, that
500
there exists a coefficient $a_{p}$ with $\alpha (p) = -1, \quad
501
a_{p} \neq 0 .$
502
Finally, to avoid inner twists in the non-squarefree level case, one
503
has to find a coefficient $a_p$ with $ \mathbb{Q}(a_p^2) =
504
\mathbb{Q}_{f}$, as in step (i) of the algorithm . \\
505
506
507
\section{Examples- $\mathbb{Q}_f$ quadratic or abelian quartic }
508
All the examples of newforms have been computed with an algorithm
509
implemented by W. Stein based on ideas of J. Cremona.\\
510
The algorithm described in the previous section and all other
511
computations have been done with PARI GP.\\
512
\subsection{Applying the algorithm}
513
514
515
In this section we look for newforms $f$ of weight $2$ such that the
516
field $\mathbb{Q}_f$ is quadratic or quartic and abelian.\\
517
We made a table of all weight $2$ newforms with level up to $640$
518
and we found $10$ with fields of these types (all different fields). We
519
verified that these newforms have no CM nor inner twists.\\
520
Then, we applied the algorithm to find
521
the exceptional primes for the representations attached to each one of
522
them.
523
The results are summarised in the following table:
524
525
526
527
528
529
\newpage
530
531
\begin{tabular}{cccc}
532
& [Level / $\; \mathbb{Q}_f $] && [Step (i) : $p ,
533
\; a_p$] \\
534
1 & $23 / \; x^2-5 $ && $2, \; x^2+x-1$
535
\\
536
2 & $29 / \; x^2 -2 $ && $2, \;
537
x^2+2x-1$ \\
538
3 & $410/ \; x^2-3$ && $ 3, \;
539
x^2-2x-2$ \\
540
4 & $410/ \; x^2-17$ && $7, \; x^2-2x-16$ $//$ $11,
541
\; x^2-2x-16$ \\
542
5 & $414/ \; x^2-7 $ && $5, \; x^2+2x-6$ $//$
543
$11, \; x^2-2x-6$ \\
544
6 & $496/ \; x^2-33$ && $5, \; x^2-3x-6$ $//$ $
545
13, \; x^2-2x-32$ \\
546
7 & $418/ \; x^2-13$ && $3, \;
547
x^2+3x-1$ \\
548
8 & $546/ \; x^2-57$ && $5, \;x^2+x-14$ $//$
549
$11,\;x^2-3x-12$ \\
550
9 & $226/ \; F_{20}$ && $3, \; x^4-2x^3-6x^2+12x-4$ $//$ $5,
551
\; x^4-4x^3-4x^2+16x-4 $ \\
552
10 & $358/ \; V_{17}$ && $3, \; x^4+2x^3-7x^2-8x-1$ $//$ $5,
553
\; x^4+7x^3+12x^2-3x-13$ \\
554
\end{tabular}\\
555
$ $\\
556
557
558
559
\begin{tabular}{cccccc}
560
& [Step (i) : $\ell$] & [Step (ii):$p,\; a_p$] & [Step (ii) :
561
$\ell$] & [Step (iii) : $p , \; a_p$ ] \\
562
1 & $5$ & ---- & $11$ &
563
---- \\
564
2 & empty & ---- & $7$ &
565
---- \\
566
3 & empty & $ 3$ & $5,41$ &
567
---- \\
568
4 & $17$ & $ 3 , \; 2$ & $5,41$ &
569
---- \\
570
5 & $7$ & $7, \; 2$ & $23$ &
571
$5$ and $11$ \\
572
6 & $11$ & $3, \; -2$ & $31$ &
573
$3$ or $//$ $5 $ and $13$ \\
574
7 & $13$ & $3$ & $11,19$ &
575
---- \\
576
8 & $19$ & $5$ $//$ $11$ & $7,13$ &
577
---- \\
578
9 & $5$ & from $3 $ to $17$ & $113, 19$ &
579
---- \\
580
10 & $17$ & $3$ $//$ $5$ & $179$ &
581
---- \\
582
\end{tabular}\\
583
$ $\\
584
585
586
\begin{tabular}{cccccc}
587
& [Step (iii): $\ell$] & [Step (iv) : $ p , \; a_p$] & [Step (iv)
588
:$5$?] & [Results] \\
589
1 & ---- & ---- & ----
590
& $5,11$ \\
591
2 & ---- & ---- & ---- & $7$
592
\\
593
3 & ---- & ---- & ---- &
594
$5,41$ \\
595
4 & ---- & ---- & ---- & $5,41$
596
\\
597
5 & empty & $11$ & NO &
598
$7,23$ \\
599
6 & empty & $13$ & NO &
600
$11,31$ \\
601
7 & ---- & ---- & ---- &
602
$11,13,19$ \\
603
8 & ---- & ---- & ---- & $7,13,19$
604
\\
605
9 & ---- & ---- & ---- & $5,19,113$
606
\\
607
10 & ---- & ---- & ---- & $17,179$
608
\\
609
\end{tabular}
610
611
$$ \mbox{Table 1}$$
612
613
614
615
\newpage
616
617
Explanation of Table 1:\\
618
In this table we exhibit $10$ cusp forms and the result of the
619
algorithm of the previous section applied to them. The first eight
620
are newforms with quadratic number field, and the last two
621
examples correspond to the abelian quartic field case: $\; F_{20}$
622
denotes the maximal real subfield of the cyclotomic field of $20$-th
623
roots of unity, and $V_{17}$ a totally real subfield of the cyclotomic
624
field of $17$-th roots of unity.\\
625
The minimal polynomials of the coefficients
626
used at each step are included, but the value is not repeated if it
627
has already appeared in a previous column. In every odd column, the
628
exceptional primes found at the corresponding step are listed, and they
629
are put together in the last column. \\
630
We remark that here exceptional primes are those for which the algorithm
631
was not able to prove that the image is ``as big as possible" , but that
632
it may be so for some of them. For instance all primes appearing in the
633
fifth column are primes dividing the level, except for the Eisenstein
634
primes $11$ and $7$ in the first and second row and the prime $19$
635
appearing in the ninth row. In this last case, having computed the
636
coefficients from $a_3$ to $a_{17}$, the corresponding representation
637
seems to be reducible.\\
638
639
\subsection{ Galois realizations of groups $PSL(2, \mathbb{F}_{\ell^2})$}
640
For each of the examples in Table 1, we consider primes with
641
residue class degree $2$. The condition for this is easily given in
642
terms of congruences , or using Legendre 's symbols. Thus , from Table
643
1
644
and the discussion following theorem \ref{teo:hlal} we conclude that for
645
the following sets of primes the groups $PSL(2, \mathbb{F}_{\ell^2})$
646
are Galois groups over $\mathbb{Q} \;$: $\; \ell \geq 5$ and\\
647
$\ell \equiv 2,3 \pmod{5} \quad$ (already proved in [Me 88])\\
648
$\ell \not\equiv \pm 1 \pmod8 $\\
649
$ ( \frac{3}{\ell}) = -1 \quad \quad \ell \neq 5,41$\\
650
$\ell \equiv \pm 3 , \pm 5 , \pm 6 , \pm 7 \pmod{17} \quad \quad
651
\ell \neq 5,41 $\\
652
$ (\frac{7}{\ell}) = -1 \quad \quad \ell \neq 23$\\
653
$ (\frac{33}{\ell}) = -1 $\\
654
$ (\frac{13}{\ell}) = -1 \quad \quad \ell \neq 11,19$\\
655
$ (\frac{57}{\ell}) = -1 \quad \quad \ell \neq 13$\\
656
$ \ell \equiv 9,11 \pmod{ 20} $\\
657
$ \ell \equiv \pm 2 ,\pm 8 \pmod{17} \quad \quad \ell \neq 179 $\\
658
659
Observe that the primes listed as exceptional at some of the rows:\\
660
$5, 41, 23, 11, 19, 13, 179$, are covered by some other row. After some
661
other elementary simplifications, we obtain the following result:
662
663
\begin{teo}
664
\label{teo:muchos2}:
665
The group $PSL(2,\mathbb{F}_{\ell^2})$ is a Galois group over
666
$\mathbb{Q}$ whenever $\ell$ is a prime greater than $3$ satisfying one
667
of the following conditions:\\
668
$\ell \not\equiv \pm 1 \pmod{120}$\\
669
$\ell \not\equiv \pm 1, \pm 4 \pmod{17}$ \\
670
$( \frac{7}{\ell}) = -1$ or $ ( \frac{11}{\ell}) = -1$ or $
671
(\frac{13}{\ell})=-1$
672
or $( \frac{19}{\ell}) = -1 $
673
\end{teo}
674
In [Re-Vi 95], using the Galois representations
675
attached to cusp
676
forms of weight $k$ for $SL(2,\mathbb{Z})$, other projective linear
677
groups
678
are realized. In particular, for
679
$k=24,28,30,32,34,38$ , there is a cusp form $f$ with a quadratic
680
$\mathbb{Q}_f$ and it is proved that, except for a few computed
681
exceptions, for every prime $\ell$ inert in some of these quadratic
682
fields the group $PSL(2, \mathbb{F}_{\ell^2})$ is a Galois group over
683
$\mathbb{Q}$. \\
684
We computed the primes $\ell \geq 5$ up to $5000000$ not covered by
685
this
686
result of [Re-Vi 95] nor by theorem \ref{teo:muchos2} and we found only
687
$6$ :
688
$$620759, \quad 878641, \quad 1782959, \quad 3747241, \quad 3871921,
689
\quad 4490639 $$
690
To cover these primes, we consider three newforms of
691
prime level:\\
692
693
\begin{tabular}{cc}
694
Level & $a_2$ \\
695
41 & $x^3+x^2-5x-1$ \\
696
59 & $x^5-9x^3+2x^2+16x-8$ \\
697
79 & $x^5-6x^3+8x-1$ \\
698
\end{tabular}\\
699
700
In each case the coefficient $a_2$ generates $\mathbb{Q}_f$.\\
701
The $6$ primes not yet covered have the property that in some
702
of these three $\mathbb{Q}_f$ there is a prime above them with residue
703
class degree $2$. Applying the algorithm, it is easy to see that none
704
of them is exceptional.
705
\begin{teo}
706
\label{teo:masivo2}:
707
For every prime $5 \leq \ell < 5000000$, $PSL(2, \mathbb{F}_{\ell^2})$
708
is a
709
Galois group over $\mathbb{Q}$.
710
\end{teo}
711
712
\subsection{ Galois realizations of groups $PSL(2, \mathbb{F}_{\ell^4})$}
713
Let us consider the inert primes in the two examples
714
of
715
quartic abelian number field: $F_{20}$ and $V_{17}$, which are given by
716
the conditions: $\ell \equiv 2,3 \pmod5$ and
717
$\ell \equiv \pm 3 , \pm 5 , \pm 6 , \pm 7 \pmod{17} \;$,
718
respectively. Among the exceptional primes obtained in these two
719
examples, the only one satisfying these conditions is $113$ in the
720
first example. But $113 \equiv -6 \pmod{17}$, so that it is
721
covered by the second example.
722
\begin{teo}
723
\label{algunos4}: For every prime $\ell \geq 5$ satisfying one of the
724
following conditions:\\
725
$ \ell \equiv 2,3 \pmod5$\\
726
$\ell \equiv \pm 3 , \pm 5 , \pm 6 , \pm 7 \pmod{17} \;$\\
727
$ \mbox{ } \quad \; PSL(2, \mathbb{F}_{\ell^4})$ is a Galois group over
728
$\mathbb{Q}$.
729
\end{teo}
730
731
\section{Three examples designed to realize $PGL(2, \mathbb{F}_{\ell^3})$
732
for many $\ell$}
733
734
In this section we consider three examples. The first two are examples
735
of newforms whose corresponding number field is abelian cubic, namely:
736
$\mathbb{Q}_f = F_7 $ and
737
$\mathbb{Q}_f = F_9 $, where for
738
every $n$,
739
$F_n = \mathbb{Q}(\zeta_n + \zeta_n^{-1})$, $\zeta_n$ a primitive
740
$n$-root
741
of unity.\\
742
The third example is the only example in all this paper involving
743
relatively big numbers. It consists of a newform whose corresponding
744
number field has degree $27$, and is a cubic extension of $F_{27}$.\\
745
Using the algorithm, we will compute all exceptional primes in each
746
case, and we will apply the result to the realization of groups
747
$PGL(2, \mathbb{F}_{\ell^3})$ as Galois groups over $\mathbb{Q}$.\\
748
The following result of Brumer helped us finding these examples (see [Br
749
96]):
750
751
\begin{teo}
752
\label{teo:Brummer}: Let $f \in S_2(N)$ be a newform not having CM .
753
Suppose that $
754
p^{r_p}\parallel N$ . Let $s_p=$ $\left\lceil \frac{r_p}2-1-\frac
755
1{p-1}\right\rceil $ and $\zeta$ a primitive $p^{s_p}$-root of unity .
756
Then $\mathbb{Q}_f\supseteq $ $\mathbb{Q}(\zeta +\zeta ^{-1})$ if $p>2$
757
(resp.
758
$\mathbb{Q}(\zeta^2 +\zeta ^{-2})$ if $p=2$)\\
759
\end{teo}
760
761
So that we looked for newforms without CM with corresponding number
762
field $F_7$ and $F_9$ in the spaces $S_2^{new}(343)$ and
763
$S_2^{new}(243)$, respectively, and we found them.\\
764
We will not use the one of level $343$ because there is also a newform
765
$f \in S_2(97)$ with $\mathbb{Q}_f = F_7$ and it is easier to apply the
766
algorithm to this one, because it has prime level.\\
767
768
\subsection{Applying the algorithm}
769
\begin{itemize}
770
\item
771
First example: There is a newform $f \in S_2(97)$ with $\mathbb{Q}_f =
772
F_7$. We will only need the following two coefficients, given by
773
their minimal polynomials:
774
$$ a_2:\quad x^3+4x^2+3x-1 \quad \mbox{and} \quad a_{17}: \quad
775
x^3+3x^2-4x-13$$
776
Running the algorithm, we find, using both $a_2$ and $a_{17}$, that $7$
777
is the only exceptional prime at step (i). There is no
778
exceptional (Eisenstein) prime at step (ii). The level being prime,
779
we conclude that $7$ is the only exceptional prime $\ell \geq 5$ .\\
780
\item
781
Second example: There is a newform $f \in S_2(243)$ having not CM nor
782
inner twists, with $\mathbb{Q}_f = F_9 $. \\
783
We apply the algorithm to find the exceptional primes $\ell \geq 5$. At
784
step (i), we use the coefficient : $\; a_2 : \quad x^3-3x^2+3$ to see
785
that there is no exceptional prime. At step (ii), we first observe that
786
we have to take $c = 9$ because $N=243=3^5$. Then we use the
787
coefficients:
788
$$ a_{19} : \quad x^3+3x^2-24x+1 \quad \quad a_{53}: \quad
789
x^3-18x^2+81x-81 $$
790
and $a_{37}$, which is Galois conjugate to $a_{19}$, to see that there
791
is no exceptional prime. Finally, we use again $a_2$ to see that there
792
is no exceptional prime at step (iii) and that $5$ is not exceptional
793
at step (iv). So that, in this example, there is no prime $\ell \geq 5$
794
exceptional.\\
795
\item
796
Third example: In the space $S_2^{new}(2187)$ we haven't found a
797
newform $f$ with $\mathbb{Q}_f = F_{27}$ but we have found one with
798
$\mathbb{Q}_f $ of degree $27$ being a cubic extension of $F_{27}$.
799
This newform has no CM nor inner twists.\\
800
The field $\mathbb{Q}_f$ is NOT a Galois field. It is generated by the
801
coefficient $a_2$, with minimal polynomial:
802
$$A_2(x)=
803
x^{27}+9x^{26}-222x^{24}-459x^{23}+2133x^{22}+7362x^{21}-9045x^{20}-55485x^{19}
804
$$
805
$$+4047x^{18}+241677x^{17}+128898x^{16}-643257x^{15} -609714x^{14} +
806
1040283x^{13} $$
807
$$+1377729x^{12}-957987x^{11}-1758753x^{10}
808
+410742x^9+1285227x^8-5184x^7 $$
809
$-520830x^6 -53136x^5+106434x^4+14094x^3-8262x^2-972x+27 $\\
810
811
To apply step (i) of the algorithm we will use also the coefficient
812
$a_5$, given by the polynomial:
813
$$ A_5(x)=
814
x^{27}+18x^{26}+81x^{25}-399x^{24}-4347x^{23}-4077x^{22}+72270x^{21}+216621x^{20}$$
815
$$-480411x^{19}-2883570x^{18}-56565x^{17}+19123614x^{16}+20821662x^{15}-67078422x^{14}$$
816
$$-137250288x^{13}+98869950x^{12}+423746154x^{11}+82935981x^{10}-662094576x^9$$
817
$$-503302086x^8+423203940x^7+625539429x^6+37164501x^5-254848680x^4$$
818
$-101328894x^3+21227994x^2+18844893x+2847447$\\
819
820
Instead of factoring the discriminants of $a_2^2$ and $a_5^2$, we
821
compute its greatest common divisor (and check that $5 \nmid
822
disc(a_2^2)$).
823
This gives the following
824
exceptional primes: $811$ and $7655551041527$. We also computed the
825
discriminant of the field $\mathbb{Q}_f$ and we found that these two
826
primes ramify.\\
827
828
Now we turn to step (ii). We have $c=27$ because the level is $
829
2187=3^7$. We use the following two coefficients: $a_{53}$ given
830
by:\\
831
832
$x^{27}+90x^{26}+3321x^{25}+58986x^{24}+286659x^{23}-8402130x^{22}-173468061x^{21}$\\
833
$-997212708x^{20}+8609338368x^{19}+174882898125x^{18}+879497766396x^{17}$\\
834
$-3839798343339x^{16}-72574129126800x^{15}-322522524201888x^{14}+507275006736879x^{13}$\\
835
$+11881100841555393x^{12}+50880136840412436x^{11}+32267290657623723x^{10}$\\
836
$-583118269440460587x^9-2803147835818522302x^8-6056528831138196135x^7$\\
837
$-5328052040331553149x^6+4040855350703314974x^5+14364803283290718444x^4$\\
838
$+11649146213084963013x^3+205986024736020486x^2-3863232567933946101x$\\
839
$-1195010794014230997$\\
840
841
and $a_{109}$, given by :\\
842
843
$x^{27}-1161x^{25}+1143x^{24}+552258x^{23}-943704x^{22}-141492546x^{21}+288666909x^{20}$\\
844
$+21785693796x^{19}-42415128438x^{18}-2149133334543x^{17}+3249157564620x^{16}$\\
845
$+140451320065959x^{15}-118337820691194x^{14}-6119920386937629x^{13}$\\
846
$+286365050094708x^{12}+174146647974621273x^{11}+137088225897636096x^{10}$\\
847
$-3067205587844711523x^9-4979500588408020255x^8+29948677258338999450x^7$\\
848
$+76790301210845625861x^6-119349417294626396391x^5-535760443536103781838x^4$\\
849
$-167198875027467682545x^3+1273594079757375456489x^2+1827054356350586351796x$\\
850
$+769405689432892210627$\\
851
852
We found no exceptional prime at step (ii).\\
853
We finally check, using only $a_2$, that there are no exceptional
854
primes at step (iii) and that $5$ is not exceptional at step (iv).
855
We conclude that the only exceptional primes are $811$ and
856
$7655551041527$.\\
857
\end{itemize}
858
859
\subsection{ Galois realizations}
860
We now want to apply these three examples to the realization of groups
861
$PGL(2, \mathbb{F}_{\ell^3})$ as Galois groups over $\mathbb{Q}$. To do
862
this we restrict, in each of the examples,
863
to those primes $\ell \geq 5$ unramified in
864
$\mathbb{Q}_f$ such that there exists a place $\lambda \in \mathbb{Q}_f$
865
above them having residue class degree $3$ .\\
866
In the first two examples, this is equivalent to say that $\ell$ is
867
inert, and this gives: $\; \ell \not\equiv \pm 1,0 \pmod7$ and
868
$\; \ell \not \equiv \pm 1 \pmod9 $. \\
869
In the third example, these primes
870
can be characterized in terms of the decomposition of the polynomials
871
$A_2$ and $A_5$.
872
The fact that $[\mathbb{Q}_f : F_{27}] =3$ implies
873
that all these primes $\ell$ verify: $\ell \equiv \pm 1 \pmod9$,
874
i.e., they are in the complement of the set of primes covered by
875
the second example. \\
876
We have already shown that in our examples the only exceptional primes
877
are those that ramify in $\mathbb{Q}_f$. Thus , we have:
878
\begin{teo}
879
\label{teo:muchos3}:
880
For every prime $\ell \geq 5$ satisfying one of the following
881
conditions:\\
882
$\ell \not\equiv \pm 1 \pmod7$\\
883
$\ell \not\equiv \pm 1 \pmod9$ \\
884
$ \ell \nmid disc(A_2)$ and $A_2$ has a cubic factor when reduced $mod \;
885
\ell$\\
886
$ \ell \nmid disc(A_5)$ and $A_5$ has a cubic factor when reduced $mod \;
887
\ell$\\
888
$\mbox{ } \quad \; PGL(2, \mathbb{F}_{\ell^3})$ is a Galois group over
889
$\mathbb{Q}$.
890
\end{teo}
891
892
As in the previous section, we want to combine this result with the
893
one obtained in [Re-Vi 95] using cusp
894
forms $f$ for $SL(2,\mathbb{Z})$ with cubic
895
$\mathbb{Q}_f$ and weight $k =36, \; 40, \; 42, \; 44, \; 46 $ and
896
$50$. They computed for these cases the few exceptional inert primes.\\
897
We computed the primes $\ell \geq 5$ up to $500000$ such that the
898
groups
899
$PGL(2,\mathbb{F}_{\ell^3})$ are not covered by
900
this
901
result of [Re-Vi 95] nor by theorem \ref{teo:muchos3} and we found only
902
$12$ :\\
903
$11087, \; 97649, \; 176597, \; 202987, \; 237691, \; 297793$\\
904
$358273, \; 368803, \; 394631, \; 407287, \; 408437 $ and $496817$\\
905
To cover these primes, we introduce four newforms of
906
squarefree level :\\
907
908
\begin{tabular}{cc}
909
Level & $a_2$ \\
910
71 & $ x^3+x^2-4x-3$ \\
911
87 & $ x^3-2x^2-4x+7$ \\
912
91 & $ x^3-x^2-4x+2$ \\
913
97 & $ x^4-3x^3-x^2+6x-1$ \\
914
915
\end{tabular}\\
916
917
918
In each case the coefficient $a_2$ generates $\mathbb{Q}_f$.\\
919
The $12$ primes not yet covered have the property that in some
920
of these four $\mathbb{Q}_f$ there is a prime above them with residue
921
class degree $3$. Applying the algorithm, it is easy to see that none
922
of them is exceptional. Then, we have:
923
\begin{teo}
924
925
\label{teo:masivo3}:
926
For every prime $5 \leq \ell < 500000$, $PGL(2, \mathbb{F}_{\ell^3})$
927
is a Galois group over $\mathbb{Q}$.
928
\end{teo}
929
930
931
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932
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939
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960
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961
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979
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982
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983
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984
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985
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987
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991
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992
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994
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995
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998
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999
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1000
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1002
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1005
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1006
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1008
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1009
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1010
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1011
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1013
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1014
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1015
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1018
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1019
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1020
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1021
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1024
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1025
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1026
1027
\end{document}
1028
1029
1030
.\\
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