ترغب بنشر مسار تعليمي؟ اضغط هنا

The dualizing module and top-dimensional cohomology group of $text{GL}_n(mathcal{O})$

216   0   0.0 ( 0 )
 نشر من قبل Andrew Putman
 تاريخ النشر 2019
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

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.



قيم البحث

اقرأ أيضاً

268 - Jeremy Miller , Peter Patzt , 2019
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 $mat hcal{T}_n(mathbb{Q})$ 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 $p geq 5$.
129 - Sebastian Thomas 2009
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.
103 - Yiwen Ding 2018
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 a n 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).
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا