Newforms, inner twists, and the inverse Galois problem for projective linear groups

Luis V. Dieulefait

Journal de théorie des nombres de Bordeaux (2001)

  • Volume: 13, Issue: 2, page 395-411
  • ISSN: 1246-7405

Abstract

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We reformulate more explicitly the results of Momose, Ribet and Papier concerning the images of the Galois representations attached to newforms without complex multiplication, restricted to the case of weight 2 and trivial nebentypus. We compute two examples of these newforms, with a single inner twist, and we prove that for every inert prime greater than 3 the image is as large as possible. As a consequence, we prove that the groups PGL ( 2 , 𝔽 ℓ 2 ) for every prime ( ℓ ≡ 3 , 5 ( mod 8 ) , ℓ > 3 ) , and PGL ( 2 , 𝔽 ℓ 5 ) for every prime ℓ ¬ ≡ 0 ± 1 ( mod 11 ) ; ℓ > 3 ) , are Galois groups over ℚ .

How to cite

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Dieulefait, Luis V.. "Newforms, inner twists, and the inverse Galois problem for projective linear groups." Journal de théorie des nombres de Bordeaux 13.2 (2001): 395-411. <http://eudml.org/doc/248730>.

@article{Dieulefait2001,
abstract = {We reformulate more explicitly the results of Momose, Ribet and Papier concerning the images of the Galois representations attached to newforms without complex multiplication, restricted to the case of weight $2$ and trivial nebentypus. We compute two examples of these newforms, with a single inner twist, and we prove that for every inert prime greater than $3$ the image is as large as possible. As a consequence, we prove that the groups $\text\{PGL\}(2,\mathbb \{F\}_\{\ell ^2\})$ for every prime $(\ell \equiv 3,5 (\text\{mod \} 8), \ell &gt; 3)$, and $\text\{PGL\}(2,\mathbb \{F\}_\{\ell ^5\})$ for every prime $\ell \lnot \equiv 0 \pm 1 (\text\{mod \} 11); \ell &gt; 3)$, are Galois groups over $\mathbb \{Q\}$.},
author = {Dieulefait, Luis V.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {Galois representations; new forms; Galois groups},
language = {eng},
number = {2},
pages = {395-411},
publisher = {Université Bordeaux I},
title = {Newforms, inner twists, and the inverse Galois problem for projective linear groups},
url = {http://eudml.org/doc/248730},
volume = {13},
year = {2001},
}

TY - JOUR
AU - Dieulefait, Luis V.
TI - Newforms, inner twists, and the inverse Galois problem for projective linear groups
JO - Journal de théorie des nombres de Bordeaux
PY - 2001
PB - Université Bordeaux I
VL - 13
IS - 2
SP - 395
EP - 411
AB - We reformulate more explicitly the results of Momose, Ribet and Papier concerning the images of the Galois representations attached to newforms without complex multiplication, restricted to the case of weight $2$ and trivial nebentypus. We compute two examples of these newforms, with a single inner twist, and we prove that for every inert prime greater than $3$ the image is as large as possible. As a consequence, we prove that the groups $\text{PGL}(2,\mathbb {F}_{\ell ^2})$ for every prime $(\ell \equiv 3,5 (\text{mod } 8), \ell &gt; 3)$, and $\text{PGL}(2,\mathbb {F}_{\ell ^5})$ for every prime $\ell \lnot \equiv 0 \pm 1 (\text{mod } 11); \ell &gt; 3)$, are Galois groups over $\mathbb {Q}$.
LA - eng
KW - Galois representations; new forms; Galois groups
UR - http://eudml.org/doc/248730
ER -

References

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