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The authors determine the number of level 1, polarized, algebraic regular, cuspidal automorphic representations of mathrm{GL}_n over mathbb Q of any given infinitesimal character, for essentially all n leq 8. For this, they compute the dimensions of spaces of level 1 automorphic forms for certain semisimple mathbb Z-forms of the compact groups mathrm{SO}_7, mathrm{SO}_8, mathrm{SO}_9 (and {mathrm G}_2) and determine Arthur's endoscopic partition of these spaces in all cases. They also give applications to the 121 even lattices of rank 25 and determinant 2 found by Borcherds, to level one self-dual automorphic representations of mathrm{GL}_n with trivial infinitesimal character, and to vector valued Siegel modular forms of genus 3. A part of the authors' results are conditional to certain expected results in the theory of twisted endoscopy.

Forms (Mathematics) --- Cusp forms (Mathematics)

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Let pi be the automorphic representation of extrm{GSp}_4(mathbb{A}) generated by a full level cuspidal Siegel eigenform that is not a Saito-Kurokawa lift, and au be an arbitrary cuspidal, automorphic representation of extrm{GL}_2(mathbb{A}). Using Furusawa's integral representation for extrm{GSp}_4imesextrm{GL}_2 combined with a pullback formula involving the unitary group extrm{GU}(3,3), the authors prove that the L-functions L(s,piimesau) are "nice". The converse theorem of Cogdell and Piatetski-Shapiro then implies that such representations pi have a functorial lifting to a cuspidal representation of extrm{GL}_4(mathbb{A}). Combined with the exterior-square lifting of Kim, this also leads to a functorial lifting of pi to a cuspidal representation of extrm{GL}_5(mathbb{A}). As an application, the authors obtain analytic properties of various L-functions related to full level Siegel cusp forms. They also obtain special value results for extrm{GSp}_4imesextrm{GL}_1 and extrm{GSp}_4imesextrm{GL}_2.

Cusp forms (Mathematics) --- Siegel domains. --- Modular groups.

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Cusp forms (Mathematics) --- Hecke operators. --- Eigenvalues. --- Siegel domains. --- Hecke, Opérateurs de --- Valeurs propres --- Siegel, Domaines de

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Ordered algebraic structures --- Cusp forms (Mathematics) --- Siegel domains. --- Modular groups. --- Formes paraboliques (Mathématiques) --- Siegel, Domaines de --- Groupes modulaires --- Nombres, Théorie des. --- Formes paraboliques (Mathématiques)

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Cusp forms (Mathematics) --- Hecke operators. --- Eigenvalues. --- Siegel domains. --- Number theory -- Discontinuous groups and automorphic forms [See also 11R39, 11S37, 14Gxx, 14Kxx, 22E50, 22E55, 30F35, 32Nxx] {For relations with quadratic forms, see 11E45} -- Siegel modular groups --- Number theory -- Discontinuous groups and automorphic forms [See also 11R39, 11S37, 14Gxx, 14Kxx, 22E50, 22E55, 30F35, 32Nxx] {For relations with quadratic forms, see 11E45} -- Hecke-Petersson operato --- Number theory -- Discontinuous groups and automorphic forms [See also 11R39, 11S37, 14Gxx, 14Kxx, 22E50, 22E55, 30F35, 32Nxx] {For relations with quadratic forms, see 11E45} -- Modular correspondences --- Number theory -- Discontinuous groups and automorphic forms [See also 11R39, 11S37, 14Gxx, 14Kxx, 22E50, 22E55, 30F35, 32Nxx] {For relations with quadratic forms, see 11E45} -- Holomorphic modular for --- Number theory -- Arithmetic algebraic geometry (Diophantine geometry) [See also 11Dxx, 14Gxx, 14Kxx] -- Varieties over finite and local fields [See also 14G15, 14G20]. --- Algebraic geometry -- Arithmetic problems. Diophantine geometry [See also 11Dxx, 11Gxx] -- Rigid analytic geometry. --- Algebraic geometry -- (Co)homology theory [See also 13Dxx] -- $p$-adic cohomology, crystalline cohomology. --- Algebraic geometry -- (Co)homology theory [See also 13Dxx] -- Étale and other Grothendieck topologies and (co)homologies. --- Number theory -- Discontinuous groups and automorphic forms [See also 11R39, 11S37, 14Gxx, 14Kxx, 22E50, 22E55, 30F35, 32Nxx] {For relations with quadratic forms, see 11E45} -- Representation-theoreti