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39
40\title{An Algorithm for Massive Realizations of Projective Linear
41Groups as Galois Groups over $\mathbb{Q}$ via Modular Representations}
42
43\author{Luis Dieulefait\thanks{supported by TMR -
44 Marie Curie Fellowship ERB4001GT974451} \ and Nuria Vila\thanks{
45partially
46 supported by DGES grant PB96-0970-C02-01}\\
47Dept. d'Algebra i Geometria, Universitat de Barcelona\\
48Gran Via de les Corts Catalanes 585\\
4908007 - Barcelona\\
50Spain}
51\begin{document}
52
53\maketitle
54
55%\tableofcontents
56
57\vspace{6mm}
58
59%\begin{abstract}
60
61%\end{abstract}
62
63
64\section{Introduction}
65
66The images of the $\ell$-adic Galois representations attached to a
67newform $f = \sum_{n=1}^{\infty} a_n q^n$ of weight 2  on
68$\Gamma_0 (N)$ are known to be as large as possible" for almost
69every $\ell$, with the sole condition that $f$ doesn't have complex
70multiplication (CM), thanks to the work of Momose and Ribet (cf. [Mo 81] ,
71[Ri 85] ).
72 Actually, if $f$ has any
73inner twist, this is only true for the restriction of these
74representations to a certain open subgroup of
75$G_{\mathbb{Q}} = Gal(\mathbb{\overline{Q}} /\mathbb{Q})$ defined in terms
76of the
77twisting characters. Even in this case, thanks to a theorem of Papier,
78something can be said about the image of $G_{\mathbb{Q}}$ (see [Ri 85] and
79[D]).
80
81In this paper we will restrict ourselves to newforms without inner
82twists and without CM. We will give an algorithm for an explicit
83determination
84 of the
85finitely many primes $\ell$ (seeming to be) exceptional, i.e., such
86that the image of the corresponding representation is not as large as
87possible" .
88As a consequence, we will be able to realize many families of
89projective linear groups over finite fields as Galois groups over
90$\mathbb{Q}$.
91
92We have
93concentrated our efforts on newforms $f$ such that
94$\mathbb{Q}_f$, the number field generated
95by its coefficients $a_n$, is an abelian extension of $\mathbb{Q}$, to
96get a simple characterisation  of primes with a prescribed residue class
97degree.
98The examples are designed to realize, for high density sets of primes,
99the following groups: $PSL(2,\mathbb{F}_{\ell^2}), \; \; 100PGL(2,\mathbb{F}_{\ell^3}) 101 \;$ and $PSL(2,\mathbb{F}_{\ell^4})$ as Galois groups over
102 $\mathbb{Q}$. In particular, the groups in the
103  exponent $2$ case will be realized as  Galois groups over
104  $\mathbb{Q}$ for every $3 < \ell < 5000000$, and the ones in the
105  exponent $3$ case for every $3 < \ell < 500000$.\\
106
107\section{The Algorithm}
108Let $f \in S_2(N)=S_2(\Gamma_0(N))$ be a newform, i.e., a normalized
109eigenform for the whole
110Hecke algebra. We assume from now on that $f$ does
111not have CM nor inner twists.
112Let $\mathbb{Q}_f$ be the number field generated
113by its coefficients $a_n$ and $\mathcal{O}$ its ring of integers. For
114every prime $\ell$ put: $\mathcal{O}_{\ell} = \mathcal{O} 115\otimes_{\mathbb{Z}} \mathbb{Z}_\ell$ and $\mathbb{Q}_{f, \ell} = 116\mathbb{Q}_f \otimes_{\mathbb{Q}} \mathbb{Q}_\ell$
117
118By Deligne's theorem, there exists a continuous Galois representation:
119
120$$\rho_{\ell} : Gal( \overline{\mathbb{Q}} / \mathbb{Q}) \rightarrow 121GL(2, \mathcal{O}_{\ell} ) \subseteq GL(2, \mathbb{Q}_{f, \ell} )$$
122
123unramified outside $\ell N$ satisfying, for every prime $p \nmid \ell 124N$ : $$trace \; \rho_{\ell} (Frob \; p) = a_p , \qquad \qquad det \; 125\rho_{\ell} (Frob \; p) = p$$ Let $G_{\mathbb{Q}}=Gal( 126\overline{\mathbb{Q}} / 127\mathbb{Q})$ and $G_\ell = \rho_{\ell}(G_{\mathbb{Q}})$, closed subgroup
128of $GL(2, 129\mathcal{O}_{\ell} )$. From the decompositions: $\mathbb{Q}_{f, \ell} 130= \prod_{\lambda \mid \ell} \mathbb{Q}_{f, \lambda}$ and $131\mathcal{O}_{ \ell} = \prod_{\lambda \mid \ell} \mathcal{O}_{\lambda}$
132we get a decomposition of $\rho_{\ell}$ as direct sum of
133representations : $$\rho_{\lambda} : G_{\mathbb{Q}} \rightarrow GL(2, 134\mathcal{O}_{\lambda} ) \subseteq GL(2, \mathbb{Q}_{f, \lambda} )$$
135Let $\lambda$ be a prime in  $\mathbb{Q}_{f}$. Consider the reduction
136$\overline{\rho}_{\lambda}$ of $\rho_{\lambda}$ , obtained by composing
137$\rho_{\lambda}$ with the reduction map: $GL(2, \mathcal{O}_{\lambda} ) 138\rightarrow GL(2, \mathbb{F}_{\lambda})$, where $\mathbb{F}_{\lambda}$
139is the residue field of $\lambda$. Let $\ell = \ell (\lambda)$ be the
140rational prime such that $\lambda \mid \ell$ and let
141$\overline{G}_{\lambda}$ be the image of $\overline{\rho}_{\lambda}$.
142In order to compute the images of the representations
143 $\overline{\rho}_{\lambda}$ we will use the following result
144 of Ribet [Ri 85]:
145
146
147\begin{teo}
148\label{teo:irreymult}:
149 For almost every  $\lambda$ we have:
150
151a) The representation  $\overline{\rho}_{\lambda}$ is an
152irreducible 2-dimensional representation over
153$\mathbb{F}_{\lambda}$
154
155b) The order of the group $\overline{G}_\lambda= 156\overline{\rho}_{\lambda}(G_{\mathbb{Q}})$ is divisible by $\ell$
157
158\end{teo}
159
160In the next section we will give an algorithm to compute
161a set of primes containing those that do not verify these two conditions.
162The following result of Ribet [Ri 85] completes the determination of
163the images:
164
165\begin{teo}
166\label{teo:hlal}: Let $A_{\ell} = \{x \in GL(2, 167\mathcal{O}_{\ell}) \quad / \; det(x) \in \mathbb{Z}_{\ell}^* \}$ \\
168The equality $G_{\ell} = A_{\ell}$ holds for almost every prime. In fact,
169this equality holds whenever the
170following conditions are satisfied :\\
1710) $\ell$ does not ramify in $\mathbb{Q}_f / \mathbb{Q}$ \\
172 1) The determinant map $G_{\ell} \rightarrow 173\mathbb{Z}_{\ell}^*$ is surjective \\
174 2) $\ell \geq 5$ \\
175  3) $G_{\ell}$
176contains an element $x_{\ell}$ such that: $(trace \; x_{\ell})^2$
177generates $\mathcal{O}_{\ell}$ as a $\mathbb{Z}_{\ell}$-algebra \\
1784) for each
179$\lambda \mid \ell$, the group $\overline{\rho}_{\lambda}(G_{\mathbb{Q}})$
180is an
181irreducible subgroup of $GL(2, \mathbb{F}_{\lambda})$ whose order is
182divisible by $\ell$.
183\end{teo}
184
185
186To see that these conditions hold for almost every prime, theorem
187\ref{teo:irreymult} is applied (to deal with condition (4)) together
188with the following facts:\\ -there exists an integer $v$ such that $v$
189is relatively prime to
190 $N$ and $a_v^2$ generates $\mathbb{Q}_f$ over
191$\mathbb{Q}$ (condition (3))\\ -the determinant of $\rho_{\ell}$
192coincides on $G_{\mathbb{Q}}$ with the $\ell$-adic cyclotomic character
193$\chi_{\ell}$ (condition (1)).           \\
194
195This result allows us to realize projective linear groups  over finite
196fields
197as Galois groups over $\mathbb{Q}$ in the
198following way: Let $\ell$ be a prime verifying the conditions
199of theorem \ref{teo:hlal} and $\lambda$ a prime in $\mathcal{O}$ over
200$\ell$.
201 Then $\overline{G}_{\lambda} \subseteq GL(2, \mathbb{F}_{\lambda})$
202and $P(\overline{G}_{\lambda} )$, its image in $PGL(2, 203\mathbb{F}_\lambda)$,
204 is (cf. [Re-Vi 95]):\\
205     $PSL(2, \mathbb{F}_\lambda) \quad \mbox{if} \; 206 [\mathbb{F}_\lambda : \mathbb{F}_\ell] \; \mbox{is even},\\ 207 PGL(2, \mathbb{F}_\lambda) \quad \mbox{otherwise}.$
208
209
210\subsection{The Exceptional Primes for Theorem 2.1 }
211
212\subsubsection{Reducible Representations}
213Let us suppose that for a prime $\lambda \in \mathcal{O}$ dividing
214$\ell$, $\overline{\rho}_{\lambda}(G_{\mathbb{Q}})$ is a reducible
215subgroup of
216 $GL(2, \mathbb{F}_{\lambda})$. Then
217$$\overline{\rho}_{\lambda} \cong \pmatrix { \phi_1& *\cr 218 0& \phi_2\cr }$$
219with $\phi_i = \epsilon_i \overline{\chi}_{\ell}^{m_i}$, for $i=1,2$,
220  $\epsilon_1, \epsilon_2$
221Dirichlet characters unramified outside $N$ with image in
222 $\mathbb{F}_{\lambda}$ and $\overline{\chi}_{\ell}$ the cyclotomic
223character $mod \; \ell$. \\
224By Deligne's theorem, for every prime $p \nmid \ell N$ we have: $$a_p 225= trace \; \rho_{\lambda} (Frob \; p) \equiv^{mod \; \lambda} 226 \phi_1(Frob \; p) + 227\phi_2(Frob \;p) = \epsilon_1(p) \; p^{m_1} + \epsilon_2(p) \; p^{m_2} 228$$ $$p = det \; \rho_{\lambda} (Frob \; p) \equiv^{mod \; \lambda} 229 \phi_1(Frob \; p) \; 230\phi_2(Frob \;p) = \epsilon_1(p) \; \epsilon_2(p) \; p^{m_1 + m_2}$$
231>From the last formula we see that: $m_1 + m_2 \equiv 1 232 \pmod{\ell -1}$ and
233$\epsilon_1(p) \epsilon_2(p) = 1$ for every $p \nmid N$, so that we can
234choose
235 $0 \leq m_1 < m_2 < \ell - 1$.
236Generalizing results of [Sw 73] to the case of general level,
237 Faltings and Jordan proved in [Fa-Jo 95]
238the following result, that we will state for general weight, i.e.,
239 when the representation comes from a newform $f \in S_k (N)$:
240\begin{teo}
241\label{teo:Faltings}: Suppose that the representation
242$\overline{\rho}_{\lambda}$ is reducible.
243 Then if $\ell > k, \ell \nmid N$,
244  $\overline{\rho}_{\lambda} = \epsilon_1 \oplus 245\epsilon_2 \overline{\chi}_{\ell}$, with the characters $\epsilon_i$
246unramified outside $N$.
247\end{teo}
248Carayol and Liv ([Ca 89] , [Li 89]) have given bounds for the
249conductors of modular Galois representations. Using these results,
250together with the ones of Faltings and Jordan, we have (see [Fa-Jo 95]) :
251
252\begin{cor}
253\label{teo:ceroyuno}: Let $f$ be a newform of weight $2$ and level $N$
254and
255 $\lambda \mid \ell$ a prime in $\mathcal{O}$ such that
256  $\overline{\rho}_{\lambda}$ is reducible. Then if $\ell > 2 , \ell 257\nmid N ,$
258we have, for every $p \nmid \ell N$: $$a_p \equiv \epsilon(p) + p \; 259\epsilon^{-1}(p) \pmod{\lambda}$$ with  $\epsilon$ a character
260unramified outside $N$ whose conductor $c$ verifies: $c^2 \mid N$.
261\end{cor}
262In the algorithm, we will only use the following two cases of this
263corollary:\\
264  a) if $p \neq \ell, p \equiv 1 \pmod{c}$ ,
265 then: $a_p \equiv 1+p \pmod{\lambda}$. \\
266 b) if  $p \neq \ell, p \equiv -1 \pmod{c}$ ,
267 then: $a_p \equiv \pm (1+p) \pmod{\lambda}$. \\
268
269This congruences can't be  equalities, because: $\arrowvert a_p 270\arrowvert \leq 2 \sqrt{p}$ . So that only finitely many $\lambda$ can
271satisfy them.\\
272
273In the case of prime level $N$, a better criterion for reducibility was
274given by Mazur in [Ma 77]. Here the condition $\ell \nmid N$ can be
275removed and the reducible primes are the Eisenstein primes, defined as
276follows (see [Ma 77]):
277\begin{defin}:
278Let $\mathbf{T}$ be the Hecke ring generated by the Hecke operators
279acting on $J_0(N)$.\\
280The Eisenstein ideal $\frak{\Im }\subseteq \mathbf{T}$ is the one
281generated by the elements: $1+l-\mathbf{T}_l\ (l\neq N\ ),\ 1+\omega .$
282An
283Eisenstein prime is a prime ideal $\beta \subseteq \mathbf{T}$ in the
284support of $\frak{% 285\Im .}$ Let  $n=num(\frac{N-1}{12})$. We have a one-to-one correspondence
286between Eisenstein primes $\beta$ and prime factors $p$ of $n,$ given
287by:
288$289{\rm Prime\quad factors\quad of\quad }n\leftrightarrow 290{\rm Eisenstein\quad primes} 291$
292$293p\leftrightarrow (\frak{\Im ,}p) 294$
295
296Besides, for every Eisenstein prime $\beta ,$ $\mathbf{T/}\beta 297=\mathbf{F}% 298_p \;$. In particular, whenever the inertia is nontrivial
299 the involved prime will not be Eisenstein.\\
300\end{defin}
301\begin{lema}
302\label{teo:Eisenstein}
303Let $f$ be a newform of weight $2$ and prime level $N$. Let
304 $\lambda \mid \ell$, $\ell \geq 5$, be a prime in $\mathcal{O}$ such
305that
306  $\overline{\rho}_{\lambda}$ is reducible.\\
307  Then: $\ell \mid N-1$
308  \end{lema}
309
310\subsubsection{The second condition}
311We now turn to the second condition in theorem \ref{teo:irreymult}. \\
312Let $\lambda$ and $\ell$ be as before and suppose that the order of
313$\overline{G}_{\lambda}$ is not divisible by $\ell$.
314The reducible case being already treated, we will also assume that
315$\overline{\rho}_{\lambda}$ is irreducible. We consider
316$P(\overline{G}_{\lambda})$, its  image in $PGL(2,\mathbb{F}_{\lambda}) 317$.
318The only possibilities for this image are to be:\\
319 1) cyclic\\
320          2) dihedral or\\
321            3) isomorphic to one of
322the following special groups : $A_4 , S_4 , A_5$  \\
323
324 In case 1),
325$\overline{\rho}_{\lambda}$ would be reducible, contradicting our
326assumption.\\
327
328 In  case 2) there exists a Cartan subgroup
329$C_{\lambda}$ of $GL(2, \mathbb{F}_{\lambda})$ such that
330$\overline{G}_{\lambda}$ is contained in the normaliser $N_{\lambda}$
331of $C_{\lambda}$, but not in $C_{\lambda }$. \\
332Let $\varphi_{\lambda} 333: G_{\mathbb{Q}} \rightarrow \{ \pm 1 \}$ be the composition: $$334G_{\mathbb{Q}} \rightarrow 335\overline{G}_{\lambda} \subseteq N_{\lambda} \twoheadrightarrow 336N_{\lambda}/C_{\lambda} \cong \{ \pm 1 \}$$ The kernel of
337$\varphi_{\lambda}$ is then an open subgroup of $G_{\mathbb{Q}}$ of index
338$2$, so
339its fixed field $K_{\lambda}$ is a quadratic field unramified outside
340$\ell N$. \\
341We now consider the restriction of $\overline{\rho}_{\lambda}$ to $I$,
342an inertia subgroup of $G_{\mathbb{Q}}$ for $\ell$. Its
343semisimplification can be
344described in terms of fundamental characters as in [Se 87]. Using the
345 fact that the
346weight of $f$ is $2$, it follows (cf. [Ri 97]) that the image of $I$ in
347$PGL(2, \mathbb{F}_\lambda)$ is cyclic of order $\ell \pm 1$. Then,
348this is a cyclic subgroup of the dihedral group
349$P(\overline{G}_\lambda)$ with order, if $\ell>3$, greater than $2$,  and
350therefore
351must be contained in its center . This implies that the quadratic field
352$K_{\lambda}$
353does not ramify at $\ell$, if $\ell >3$. This is enough to make the
354dihedral case
355computable, but there is another observation that will simplify the
356computations if $N$ has any prime factor $q$ such that $q \parallel N$.
357In this case, the ramification subgroup of $\overline{G}_{\lambda}$ for
358the prime $q$ is unipotent, if $q$ divides the conductor of
359$\overline{\rho}_{\lambda}$, and trivial otherwise.
360 This, together with the assumption that
361the order of $\overline{G}_{\lambda}$ is prime to $\ell$, implies that
362these ramification groups are in fact trivial for all these primes.\\
363We conclude that the field $K_{\lambda}$ can only ramify at those prime
364factors $r$ of $N$ (if any) such that $r^2 \mid N$. Let $N'$ be the
365product of these prime factors. If we call $\alpha$ the quadratic
366character corresponding to $K_{\lambda}$ we have proved the following
367
368\begin{lema}
369\label{teo:dihedral}: Suppose that $\overline{\rho}_{\lambda}$
370falls 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
373quadratic character unramified outside $N'$.\\
374In particular, if $N$ is
375squarefree, no $\lambda$ verifies the hypothesis of this lemma.\\
376\end{lema}
377
378In  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
383only for finitely many $\ell$. Actually, we will need to apply (2.1) only
384to the prime $\ell =5$ because in the case of weight $2$ and irreducible
385$\overline{\rho}_{\lambda}$, using again
386the description of the semisemplification of the restriction of
387$\overline{\rho}_{\lambda}$ to an inertia subgroup for the prime $\ell$
388in terms of fundamental characters, it follows that there is an element
389of order $\ell \pm 1$ in the projective image of this
390inertia subgroup, and therefore in $P( \overline{G}_{\lambda})$, and
391this rules out case 3) if $\ell \geq 7$ (cf. [Se 72], [Ri 97]).\\
392
393It is proved in [Ri 97] that in the squarefree level case, 3) can not
394hold not
395even for $5$, so that in this case the following result holds:
396
397\begin{teo}
398\label{semistable}:
399Let $f$ 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
401that
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}
408At this point we can give a complete description of the algorithm to
409compute the exceptional primes in theorem \ref{teo:hlal}. Suppose that
410we have a newform $f \in S_2(N)$ not having CM nor inner twists, and
411we consider the representation $\rho_\ell$ for a prime  $\ell \geq 4125$ . 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),
416to find a
417coefficient $a_p$ 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 $c$
422such that $c ^{2} \mid N$, and a coefficient
423$a _{p}$ with $p \equiv 1 \pmod{c}$
424such that
425$$p \nmid \ell N \; \mbox{and} \; a _{p} - (1+p) \not\equiv 0 426\pmod{\lambda}$$
427 for
428every $\lambda \mid \ell$. Then, if $\ell \nmid N$, the reducible case is
429excluded.\\
430To verify the congruences, it's enough to compute the norm of $a _{p} - 431(1+p)$ and
432check that $\ell \nmid \mathcal{N}(a _{p}-(1+p))$.\\
433Similarly, 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
437verify that $\ell \nmid \mathcal{N}(a _{p} \pm (1+p))$.\\
438
439In the prime level case reducible primes $\ell \geq 5$
440are those $\ell \mid N - 1$. In particular, the
441prime $N$ (untractable with the other approach ) is not reducible.\\
442
443
444\bf{Remark}: \rm The remaining conditions have not to be verified in the
445case of
446squarefree level. In this case, all exceptional primes $\ell \geq 5$
447are detected with steps (i) and (ii) .
448  \item Step (iii) : For every quadratic character $\alpha$
449  ramifying only at the primes $q$
450such that $q ^{2} \mid N$, find a coefficient $a_{p}$ with:
451$$p \nmid N \ell, \; \; a _{p} \neq 0, 452\; \; \alpha (p) = -1 \; \mbox{and} \; a_{p} \not\equiv - a_{p} 453\pmod{\lambda}$$
454for every $\lambda \mid \ell$. Then, the dihedral case is excluded.\\
455To 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
459of $Q$, the minimal polynomial of
460$a_p \;$, and the polynomials $P_i \; , \; \; i=1,...,5$  in formula
461(2.1):
462$$x^2, \quad x^2-p \; , \quad x^2 -2p \; , \quad x^2-4p \; , \quad 463x^4-3px^2+p^2$$
464If
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 $\widehat{5} 468\mid 4695$. Then, the case of special images is excluded.
470
471
472\end{itemize}
473
474\vskip 1cm
475We conclude the section with a brief explanation of how, for the newforms
476in
477the examples, we have determined the corresponding number fields and
478checked that the newforms have no CM nor inner twists.
479
480\medskip
481
482Determination of the number field $\mathbb{Q}_{f}$:
483Let $f$ be a newform in $S_{2} (N)$. If $f$ has a coefficient $a_{p}, \; 484p \nmid 485N$, which is a simple root of the characteristic polynomial of the
486Hecke operator $T_{p}$ acting on $S_{2}^{new} (N)$, then it follows
487that $\mathbb{Q}(a_{p}) = \mathbb{Q}_{f}$.
488
489\medskip
490
491CM-Inner Twists:
492It is known that a newform $f \in S_{2} (N)$ doesn't have CM nor inner
493twists if
494the 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.
498In the remaining cases, to avoid CM one has to check, for every
499character $\alpha$ ramifying only at primes dividing the level $N$, that
500there exists a coefficient $a_{p}$ with $\alpha (p) = -1, \quad 501a_{p} \neq 0 .$
502Finally, to avoid inner twists in the non-squarefree level case, one
503has 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 }
508All the examples of newforms have been computed with an algorithm
509implemented by W. Stein based on ideas of J. Cremona.\\
510The algorithm described in the previous section and all other
511computations have been done with PARI GP.\\
512\subsection{Applying the algorithm}
513
514
515In this section we look for newforms $f$ of weight $2$ such that the
516field $\mathbb{Q}_f$ is quadratic or quartic and abelian.\\
517We made a table of all weight $2$ newforms with level up to $640$
518and we found $10$ with fields of these types (all different fields). We
519verified that these newforms have no CM nor inner twists.\\
520 Then, we applied the algorithm to find
521the exceptional primes for the representations attached to each one of
522them.
523The 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  &   $29 / \; x^2 -2$      &&                       $2, \; 537x^2+2x-1$        \\
538   3  &   $410/ \; x^2-3$       &&                       $3, \; 539x^2-2x-2$      \\
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  &    $496/ \; x^2-33$      &&           $5, \; x^2-3x-6$ $//$  $54513, \; x^2-2x-32$    \\
546   7  &    $418/ \; x^2-13$      &&                       $3, \; 547x^2+3x-1$                \\
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
617Explanation of Table 1:\\
618In this table we exhibit $10$ cusp forms and the result of the
619algorithm of the previous section applied to them. The first eight
620are newforms with quadratic number field, and the last two
621examples correspond to the abelian quartic field case: $\; F_{20}$
622denotes the maximal real subfield of the cyclotomic field of $20$-th
623roots of unity, and $V_{17}$ a totally real subfield of the cyclotomic
624field of $17$-th roots of unity.\\
625 The minimal polynomials of the coefficients
626used at each step are included, but the value is not repeated if it
627has already appeared in a previous column. In every odd column, the
628exceptional primes found at the corresponding step are listed, and they
629are put together in the last column. \\
630We remark that here exceptional primes are those for which the algorithm
631was not able to prove that the image is as big as possible" , but that
632it may be so for some of them. For instance all primes appearing in the
633fifth column are primes dividing the level, except for the Eisenstein
634primes $11$ and $7$ in the first and second row and the prime $19$
635appearing in the ninth row. In this last case, having computed the
636coefficients from $a_3$ to $a_{17}$, the corresponding representation
637seems to be reducible.\\
638
639\subsection{ Galois realizations of groups $PSL(2, \mathbb{F}_{\ell^2})$}
640For each of the examples in Table 1, we consider primes with
641residue 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
644and the discussion following theorem \ref{teo:hlal} we conclude that for
645the following sets of primes the groups $PSL(2, \mathbb{F}_{\ell^2})$
646are 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
659Observe 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
661other elementary simplifications, we obtain the following result:
662
663\begin{teo}
664\label{teo:muchos2}:
665The 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
667of 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 $671(\frac{13}{\ell})=-1$
672or $( \frac{19}{\ell}) = -1$
673\end{teo}
674In [Re-Vi 95], using the Galois representations
675 attached to cusp
676forms of weight $k$ for $SL(2,\mathbb{Z})$, other projective linear
677groups
678are 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
681exceptions, for every prime $\ell$ inert in some of these quadratic
682fields the group $PSL(2, \mathbb{F}_{\ell^2})$ is a Galois group over
683$\mathbb{Q}$. \\
684We computed the primes $\ell \geq 5$ up to $5000000$ not covered by
685this
686result of [Re-Vi 95] nor by theorem \ref{teo:muchos2} and we found only
687$6$ :
688$$620759, \quad 878641, \quad 1782959, \quad 3747241, \quad 3871921, 689\quad 4490639$$
690To cover these primes, we consider three newforms of
691prime 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
700In each case the coefficient $a_2$ generates $\mathbb{Q}_f$.\\
701The $6$  primes not yet covered have the property that in some
702of these three $\mathbb{Q}_f$ there is a prime above them with residue
703class degree $2$. Applying  the algorithm, it is easy to see that none
704of them is exceptional.
705\begin{teo}
706\label{teo:masivo2}:
707For every prime $5 \leq \ell < 5000000$, $PSL(2, \mathbb{F}_{\ell^2})$
708is a
709Galois group over $\mathbb{Q}$.
710\end{teo}
711
712\subsection{ Galois realizations of groups $PSL(2, \mathbb{F}_{\ell^4})$}
713Let us consider the inert primes in the two examples
714of
715quartic abelian number field: $F_{20}$ and $V_{17}$, which are given by
716the conditions: $\ell \equiv 2,3 \pmod5$ and
717$\ell \equiv \pm 3 , \pm 5 , \pm 6 , \pm 7 \pmod{17} \;$,
718respectively. Among the exceptional primes obtained in these two
719examples, the only one satisfying these conditions is $113$ in the
720first example. But $113 \equiv -6 \pmod{17}$, so that it is
721covered by the second example.
722\begin{teo}
723\label{algunos4}: For every prime $\ell \geq 5$ satisfying one of the
724following 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
734In this section we consider three examples. The first two are examples
735of newforms whose corresponding number field is abelian cubic, namely:
736$\mathbb{Q}_f = F_7$ and
737$\mathbb{Q}_f = F_9$, where for
738every $n$,
739$F_n = \mathbb{Q}(\zeta_n + \zeta_n^{-1})$, $\zeta_n$ a primitive
740$n$-root
741 of unity.\\
742The third example is the only example in all this paper involving
743relatively big numbers. It consists of a newform whose corresponding
744number field has degree $27$, and is a cubic extension of $F_{27}$.\\
745Using the algorithm, we will compute all exceptional primes in each
746case, and we will apply the result to the realization of groups
747$PGL(2, \mathbb{F}_{\ell^3})$ as Galois groups over $\mathbb{Q}$.\\
748The following result of Brumer helped us finding these examples (see [Br
74996]):
750
751\begin{teo}
752\label{teo:Brummer}: Let $f \in S_2(N)$ be a newform not having CM .
753 Suppose that $754p^{r_p}\parallel N$ . Let $s_p=$ $\left\lceil \frac{r_p}2-1-\frac 7551{p-1}\right\rceil$   and $\zeta$ a primitive $p^{s_p}$-root of unity .
756Then $\mathbb{Q}_f\supseteq$ $\mathbb{Q}(\zeta +\zeta ^{-1})$ if $p>2$
757(resp.
758$\mathbb{Q}(\zeta^2 +\zeta ^{-2})$ if $p=2$)\\
759\end{teo}
760
761So that we looked for  newforms without CM with corresponding number
762field $F_7$ and $F_9$ in the spaces $S_2^{new}(343)$ and
763$S_2^{new}(243)$, respectively, and we found them.\\
764We 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
766algorithm to this one, because it has prime level.\\
767
768\subsection{Applying the algorithm}
769\begin{itemize}
770  \item
771First example: There is a newform $f \in S_2(97)$ with $\mathbb{Q}_f = 772F_7$. We will only need the following two coefficients, given by
773their minimal polynomials:
774$$a_2:\quad x^3+4x^2+3x-1 \quad \mbox{and} \quad a_{17}: \quad 775x^3+3x^2-4x-13$$
776Running the algorithm, we find, using both $a_2$ and $a_{17}$, that $7$
777is the only exceptional prime at step (i). There is no
778exceptional (Eisenstein) prime at step (ii). The level being prime,
779we conclude that $7$ is the only exceptional prime $\ell \geq 5$  .\\
780\item
781Second example: There is a newform $f \in S_2(243)$ having not CM nor
782inner twists, with $\mathbb{Q}_f = F_9$. \\
783We apply the algorithm to find the exceptional primes $\ell \geq 5$. At
784step (i), we use the coefficient : $\; a_2 : \quad x^3-3x^2+3$ to see
785that there is no exceptional prime. At step (ii), we first observe that
786we have to take $c = 9$ because $N=243=3^5$. Then we use the
787coefficients:
788$$a_{19} : \quad x^3+3x^2-24x+1 \quad \quad a_{53}: \quad 789x^3-18x^2+81x-81$$
790and $a_{37}$, which is Galois conjugate to $a_{19}$, to see that there
791is no exceptional prime. Finally, we use again $a_2$ to see that there
792is no exceptional prime at step (iii) and that $5$ is not exceptional
793at step (iv). So that, in this example, there is no prime $\ell \geq 5$
794exceptional.\\
795\item
796Third example: In the space $S_2^{new}(2187)$ we haven't found a
797newform $f$ with $\mathbb{Q}_f = F_{27}$ but we have found one with
798$\mathbb{Q}_f$ of degree $27$ being a cubic extension of $F_{27}$.
799This 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 $$A_2(x)= 803x^{27}+9x^{26}-222x^{24}-459x^{23}+2133x^{22}+7362x^{21}-9045x^{20}-55485x^{19} 804$$
805$$+4047x^{18}+241677x^{17}+128898x^{16}-643257x^{15} -609714x^{14} + 8061040283x^{13}$$
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
811To apply step (i) of the algorithm we will use also the coefficient
812$a_5$, given by the polynomial:
813$$A_5(x)= 814x^{27}+18x^{26}+81x^{25}-399x^{24}-4347x^{23}-4077x^{22}+72270x^{21}+216621x^{20}$$
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
820Instead of factoring the discriminants of $a_2^2$ and $a_5^2$, we
821compute its greatest common divisor (and check that $5 \nmid 822disc(a_2^2)$).
823 This gives the following
824exceptional primes: $811$ and $7655551041527$. We also computed the
825discriminant of the field $\mathbb{Q}_f$ and we found that these two
826primes ramify.\\
827
828Now we turn to step (ii). We have $c=27$ because the level is $8292187=3^7$. We use the following two coefficients: $a_{53}$ given
830by:\\
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
841and $a_{109}$, 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
852We found no exceptional prime at step (ii).\\
853We finally check, using only $a_2$, that there are no exceptional
854primes at step (iii) and that $5$ is not exceptional at step (iv).
855We conclude that the only exceptional primes are $811$ and
856$7655551041527$.\\
857\end{itemize}
858
859\subsection{ Galois realizations}
860We 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
862this 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 $3$ .\\
866In the first two examples, this is equivalent to say that $\ell$ is
867inert, and this gives: $\; \ell \not\equiv \pm 1,0 \pmod7$ and
868$\; \ell \not \equiv \pm 1 \pmod9$. \\
869In the third example, these primes
870can 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
873that 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
875the second example.   \\
876We have already shown that in our examples the only exceptional primes
877are those that ramify in $\mathbb{Q}_f$. Thus , we have:
878\begin{teo}
879\label{teo:muchos3}:
880For every prime $\ell \geq 5$ satisfying one of the following
881conditions:\\
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
892As in the previous section, we want to combine this result with the
893one obtained in [Re-Vi 95] using  cusp
894forms $f$  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.\\
897We computed the primes $\ell \geq 5$ up to $500000$ such that the
898groups
899$PGL(2,\mathbb{F}_{\ell^3})$ are not covered by
900this
901result 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 $496817$\\
905To cover these primes, we introduce four newforms of
906squarefree 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
918In each case the coefficient $a_2$ generates $\mathbb{Q}_f$.\\
919The $12$  primes not yet covered have the property that in some
920of these four $\mathbb{Q}_f$ there is a prime above them with residue
921class degree $3$. Applying  the algorithm, it is easy to see that none
922of them is exceptional. Then, we have:
923\begin{teo}
924
925\label{teo:masivo3}:
926For every prime $5 \leq \ell < 500000$, $PGL(2, \mathbb{F}_{\ell^3})$
927is a Galois group over $\mathbb{Q}$.
928\end{teo}
929
930
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