We show that the compactly supported cohomology of certain $\text{U}(n,n)$- or $\text{Sp}(2n)$-Shimura varieties with $\unicode[STIX]{x1D6E4}_{1}(p^{\infty })$-level vanishes above the middle degree. The only assumption is that we work over a CM field $F$ in which the prime $p$ splits completely. We also give an application to Galois representations for torsion in the cohomology of the locally symmetric spaces for $\text{GL}_{n}/F$. More precisely, we use the vanishing result for Shimura varieties to eliminate the nilpotent ideal in the construction of these Galois representations. This strengthens recent results of Scholze [On torsion in the cohomology of locally symmetric varieties, Ann. of Math. (2) 182 (2015), 945–1066; MR 3418533] and Newton–Thorne [Torsion Galois representations over CM fields and Hecke algebras in the derived category, Forum Math. Sigma 4 (2016), e21; MR 3528275].
Let F be a global field, A a central simple algebra over F and K a finite (separable or not) field extension of F with degree [K : F ] dividing the degree of A over F . An embedding of K into A over F exists implies an embedding exists locally everywhere. In this paper we give detailed discussions about when the converse (i.e. the local-global principle in question) may hold.Date: November 6, 2018.
We generalize the work of Ohta on the congruence modules attached to elliptic Eisenstein series to the setting of Hilbert modular forms. Our work involves three parts. In the first part, we construct Eisenstein series adelically and compute their constant terms by computing local integrals. In the second part, we prove a control theorem for one-variable ordinary Λ-adic Hilbert modular forms following Hida's work on the space of multivariable ordinary Λ-adic Hilbert cusp forms. In part three, we compute congruence modules related to Hilbert Eisenstein series through an analog of Ohta's methods.for all cusps c, and hence, the assertion follows.Corollary 5.5. Let the notation and the assumption be as in Lemma 5.4. Then we haveMoreover, we have H 0 (S(c, Γ 1 (np r )) /W , ω k ) ⊗ W W m ≃ H 0 (S(c, Γ 1 (np r )) /W , ω k ⊗ W W m ) for all r ∈ Z >0 .
In the present article, we study the conjecture of Sharifi on the surjectivity of the map ̟ θ . Here θ is a primitive even Dirichlet character of conductor N p, which is exceptional in the sense of Ohta. After localizing at the prime ideal p of the Iwasawa algebra related to the trivial zero of the Kupota-Leopoldt p-adic L-function Lp(s, θ −1 ω 2 ), we compute the image of ̟ θ,p in local Galois cohomology groups and prove that it is an isomorphism. Also, we prove that the residual Galois representations associated to the cohomology of modular curves are decomposable after taking the same localization. Sharifi's ConjectureThroughout this paper, we will denote by N a positive integer, denote by p ≥ 5 a prime number not dividing N φ(N ). The goal of this section is to review Sharifi's conjecture on the map ̟ following [13] and to state Theorem 1.1. We refer the reader to loc. cit. for more details on Sharifi's conjecture.
In the present article, we study the conjecture of Sharifi on the surjectivity of the map ϖ θ \varpi _{\theta } . Here θ \theta is a primitive even Dirichlet character of conductor N p Np , which is exceptional in the sense of Ohta. After localizing at the prime ideal p \mathfrak {p} of the Iwasawa algebra related to the trivial zero of the Kubota–Leopoldt p p -adic L L -function L p ( s , θ − 1 ω 2 ) L_p(s,\theta ^{-1}\omega ^2) , we compute the image of ϖ θ , p \varpi _{\theta ,\mathfrak {p}} in a local Galois cohomology group and prove that it is an isomorphism. Also, we prove that the residual Galois representations associated to the cohomology of modular curves are decomposable after taking the same localization.
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