The aim of the present note is to develop a unified theory of mean values of automorphic L-functions, albeit what we render here is less explicit and practical than we have desired. This is an outcome of the investigation commenced with the parts XII and XIV, where a framework was laid on the basis of the theory of automorphic representations, and a general approach to the mean values was envis...

Twenty-five years ago R. Langlands proposed [L] a “fantastic generalization” of Artin-Hasse reciprocity law in the classical class field theory. He conjectured the existence of a correspondence between automorphic irreducible infinite-dimensional representations of a reductive groupG over a global number field on the one hand, and (roughly speaking) finite dimensionsional representations of the...

This essay will explain the relationship between classical automorphic forms and representations of GL 2 (R). The classical theory of automorphic forms, in spite of initial appearances, is about the group GL 2 , not SL 2. The classical theory is concerned with functions on the upper half plane, which is acted on by fractional linear transformations in GL pos 2 (R), and it happens that the inter...

We prove the vanishing of the geometric Bloch–Kato Selmer group for the adjoint representation of a Galois representation associated to regular algebraic polarized cuspidal automorphic representations under an assumption on the residual image. Using this, we deduce that the localization and completion of a certain universal deformation ring for the residual representation at the characteristic ...

In Carayol’s note [4], a geometric construction of the Galois representations associated to Hilbert modular forms and the compatibility with the local Langlands correspondence are discussed. In loc. cit., the compatibility is established in the case = p where the Galois representation is an -adic representation and p is the prime divided by the prime p of the totally real field where the restri...

We extend the strong multiplicity one theorem of Jacquet, Piatetski-Shapiro and Shalika. Let π be a unitary, cuspidal, automorphic representation of GLn(AK). Let S be a set of finite places of K, such that the sum ∑ v∈S Nv −2/(n+1) is convergent. Then π is uniquely determined by the collection of the local components {πv | v 6∈ S, v finite} of π. Combining this theorem with base change, it is p...

Let m be an integer such that m≥7 and m≡0,1,7mod8. We construct strictly compatible systems of representations Γ ℚ →Spin (ℚ ¯ l )→ spinGL N ) are potentially automorphic motivic. As application, we prove instances the inverse Galois problem for