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Register a new userThe dualizing module and top-dimensional cohomology group of $text{GL}_n(mathcal{O})$
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For a number ring $mathcal{O}$, Borel and Serre proved that $text{SL}_n(mathcal{O})$ is a virtual duality group whose dualizing module is the Steinberg module. They also proved that $text{GL}_n(mathcal{O})$ is a virtual duality group. In contrast to $text{SL}_n(mathcal{O})$, we prove that the dualizing module of $text{GL}_n(mathcal{O})$ is sometimes the Steinberg module, but sometimes instead is a variant that takes into account a sort of orientation. Using this, we obtain vanishing and nonvanishing theorems for the cohomology of $text{GL}_n(mathcal{O})$ in its virtual cohomological dimension.
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Let $Gamma_n(p)$ be the level-$p$ principal congruence subgroup of $text{SL}_n(mathbb{Z})$. Borel-Serre proved that the cohomology of $Gamma_n(p)$ vanishes above degree $binom{n}{2}$. We study the cohomology in this top degree $binom{n}{2}$. Let $mathcal{T}_n(mathbb{Q})$ denote the Tits building of $text{SL}_n(mathbb{Q})$. Lee-Szczarba conjectured that $H^{binom{n}{2}}(Gamma_n(p))$ is isomorphic to $widetilde{H}_{n-2}(mathcal{T}_n(mathbb{Q})/Gamma_n(p))$ and proved that this holds for $p=3$. We partially prove and partially disprove this conjecture by showing that a natural map $H^{binom{n}{2}}(Gamma_n(p)) rightarrow widetilde{H}_{n-2}(mathcal{T}_n(mathbb{Q})/Gamma_n(p))$ is always surjective, but is only injective for $p leq 5$. In particular, we completely calculate $H^{binom{n}{2}}(Gamma_n(5))$ and improve known lower bounds for the ranks of $H^{binom{n}{2}}(Gamma_n(p))$ for $p geq 5$.
We give an elementary proof of the well-known fact that the third cohomology group H^3(G, M) of a group G with coefficients in an abelian G-module M is in bijection to the set Ext^2(G, M) of equivalence classes of crossed module extensions of G with M.
Let $L$ be a finite extension of $mathbb{Q}_p$, and $rho_L$ be an $n$-dimensional semi-stable non crystalline $p$-adic representation of $mathrm{Gal}_L$ with full monodromy rank. Via a study of Breuils (simple) $mathcal{L}$-invariants, we attach to $rho_L$ a locally $mathbb{Q}_p$-analytic representation $Pi(rho_L)$ of $mathrm{GL}_n(L)$, which carries the exact information of the Fontaine-Mazur simple $mathcal{L}$-invariants of $rho_L$. When $rho_L$ comes from an automorphic representation of $G(mathbb{A}_{F^+})$ (for a unitary group $G$ over a totally real filed $F^+$ which is compact at infinite places and $mathrm{GL}_n$ at $p$-adic places), we prove under mild hypothesis that $Pi(rho_L)$ is a subrerpresentation of the associated Hecke-isotypic subspaces of the Banach spaces of $p$-adic automorphic forms on $G(mathbb{A}_{F^+})$. In other words, we prove the equality of Breuils simple $mathcal{L}$-invariants and Fontaine-Mazur simple $mathcal{L}$-invariants.
We study the arithmetic of degree $N-1$ Eisenstein cohomology classes for locally symmetric spaces associated to $mathrm{GL}_N$ over an imaginary quadratic field $k$. Under natural conditions we evaluate these classes on $(N-1)$-cycles associated to degree $N$ extensions $F/k$ as linear combinations of generalised Dedekind sums. As a consequence we prove a remarkable conjecture of Sczech and Colmez expressing critical values of $L$-functions attached to Hecke characters of $F$ as polynomials in Kronecker--Eisenstein series evaluated at torsion points on elliptic curves with multiplication by $k$. We recover in particular the algebraicity of these critical values.
In this paper we establish a new case of Langlands functoriality. More precisely, we prove that the tensor product of the compatible system of Galois representations attached to a level-1 classical modular form and the compatible system attached to an n-dimensional RACP automorphic representation of GL_n of the adeles of Q is automorphic, for any positive integer n, under some natural hypotheses (namely regularity and irreducibility).
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