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Chenevier's book: Automorphic forms and Even unimodular lattices
2019-05-29 10:12:19
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The main theorem of this book is classify some level one discrete autormorphic representations of $SO(24)$ interms of Arthur parameter. ## Automorphic forms of level 1 Fix a algebraic reductive group $G$ defined over $\mathbb Z$ The space $$ \mathcal A^2(G)=\mathcal L^2(G(\mathbb Q)/G(\mathbb A)/Z.G(\hat{\mathbb Z})) $$ is called automorphic forms of level 1. This is a Hilbert space with a natural action of the group $G(\mathbb R)$ and the Hecke algebra $H(G)=\otimes_{p} H_{p}(G)$. The decomosite of $\mathcal A^2(G)$ interms of the two actions gives us the level 1 automorphic forms. $$ \mathcal A^2(G)=\mathcal A^2_{disc}(G)\oplus \mathcal A^2_{cont}(G) $$ and $$\mathcal A^2_{disc}(G)=\oplus m(\pi)\pi \quad \mbox{for} \,\,{\pi \in \Pi_{disc}(G)} $$ The space $\mathcal A^2_{cusp} (G) \subset \mathcal A^2_{disc}(G)$ contains the space of cusp forms. The book study with the subsets $\Pi_{cusp}(G) \subset \Pi_{disc} (G)$ and the multiplicity $m(\pi)$
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