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Taking advantage of methods originating with quantization theory, we try to get some better hold - if not an actual construction - of Maass (non-holomorphic) cusp-forms. We work backwards, from the rather simple results to the mostly devious road used to prove them.

Functoriality and the Inverse Galois problem II: groups of type $B_{n}$ and $G_{2}$

Chandrashekhar Khare, Michael Larsen, Gordan Savin (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

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 $ℓ \equiv 3, 5 (mod 8)$ 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...

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