Functoriality and the Inverse Galois problem II: groups of type B n and G 2

Chandrashekhar Khare[1]; Michael Larsen[2]; Gordan Savin[3]

  • [1] Department of Mathematics, UCLA, Los Angeles CA 90095-1555, U.S.A.
  • [2] Department of Mathematics, Indiana University, Bloomington, IN 47405, U.S.A.
  • [3] Department of Mathematics, University of Utah, 155 South 1400 East, Room 233, Salt Lake City, UT 84112-0090, U.S.A

Annales de la faculté des sciences de Toulouse Mathématiques (2010)

  • Volume: 19, Issue: 1, page 37-70
  • ISSN: 0240-2963

Abstract

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This paper contains an application of Langlands’ functoriality principle to the following classical problem: which finite groups, in particular which simple groups appear as Galois groups over ℚ ? Let ℓ be a prime and t a positive integer. We show that that the finite simple groups of Lie type B n ( ℓ k ) = 3 D S O 2 n + 1 ( 𝔽 ℓ k ) d e r if ℓ ≡ 3 , 5 ( mod 8 ) and G 2 ( ℓ k ) appear as Galois groups over ℚ , for some k divisible by t . In particular, for each of the two Lie types and fixed ℓ we construct infinitely many Galois groups but we do not have a precise control of k .

How to cite

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Khare, Chandrashekhar, Larsen, Michael, and Savin, Gordan. "Functoriality and the Inverse Galois problem II: groups of type $B_n$ and $G_2$." Annales de la faculté des sciences de Toulouse Mathématiques 19.1 (2010): 37-70. <http://eudml.org/doc/115867>.

@article{Khare2010,
abstract = {This paper contains an application of Langlands’ functoriality principle to the following classical problem: which finite groups, in particular which simple groups appear as Galois groups over $\mathbb\{Q\}$? Let $\ell $ be a prime and $t$ a positive integer. We show that that the finite simple groups of Lie type $B_\{n\}(\ell ^\{k\})=3DSO_\{2n+1\}(\{\mathbb\{F\}\}_\{\ell ^\{k\}\})^\{der\}$ if $\ell \equiv 3,5\hspace\{4.44443pt\}(\@mod \; 8)$ and $G_\{2\}(\ell ^\{k\})$ appear as Galois groups over $\mathbb\{Q\}$, for some $k$ divisible by $t$. In particular, for each of the two Lie types and fixed $\ell $ we construct infinitely many Galois groups but we do not have a precise control of $k$.},
affiliation = {Department of Mathematics, UCLA, Los Angeles CA 90095-1555, U.S.A.; Department of Mathematics, Indiana University, Bloomington, IN 47405, U.S.A.; Department of Mathematics, University of Utah, 155 South 1400 East, Room 233, Salt Lake City, UT 84112-0090, U.S.A},
author = {Khare, Chandrashekhar, Larsen, Michael, Savin, Gordan},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {inverse Galois problem; Langlands functoriality},
language = {eng},
month = {1},
number = {1},
pages = {37-70},
publisher = {Université Paul Sabatier, Toulouse},
title = {Functoriality and the Inverse Galois problem II: groups of type $B_n$ and $G_2$},
url = {http://eudml.org/doc/115867},
volume = {19},
year = {2010},
}

TY - JOUR
AU - Khare, Chandrashekhar
AU - Larsen, Michael
AU - Savin, Gordan
TI - Functoriality and the Inverse Galois problem II: groups of type $B_n$ and $G_2$
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2010/1//
PB - Université Paul Sabatier, Toulouse
VL - 19
IS - 1
SP - 37
EP - 70
AB - This paper contains an application of Langlands’ functoriality principle to the following classical problem: which finite groups, in particular which simple groups appear as Galois groups over $\mathbb{Q}$? Let $\ell $ be a prime and $t$ a positive integer. We show that that the finite simple groups of Lie type $B_{n}(\ell ^{k})=3DSO_{2n+1}({\mathbb{F}}_{\ell ^{k}})^{der}$ if $\ell \equiv 3,5\hspace{4.44443pt}(\@mod \; 8)$ and $G_{2}(\ell ^{k})$ appear as Galois groups over $\mathbb{Q}$, for some $k$ divisible by $t$. In particular, for each of the two Lie types and fixed $\ell $ we construct infinitely many Galois groups but we do not have a precise control of $k$.
LA - eng
KW - inverse Galois problem; Langlands functoriality
UR - http://eudml.org/doc/115867
ER -

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