This note shows how to construct a representation of the automorphic extension G of the simple group S = O 8 (5) by a symmetric group on three points, together with an embedding of the normalizer H of an O 8 (2) type subgroup of O + 8 (5). As an application, it is shown that the permutation representation of G on the cosets of H has a base of length two. This question arose in [BGS11].

Let π be a regular algebraic cuspidal automorphic representation of GL2 over an imaginary quadratic number field K, and let l be a prime number. Assuming the central character of π is invariant under the non-trivial automorphism of K, it is shown that there is a continuous irreducible l-adic representation ρ of Gal(K/K) such that L(s, ρv) = L(s, πv) whenever v is a prime of K outside an explici...

This note shows how to construct a representation of the automorphic extension G of the simple group S = O 8 (5) by a symmetric group on three points, together with an embedding of the normalizer H of an O 8 (2) type subgroup of O + 8 (5). As an application, it is shown that the permutation representation of G on the cosets of H has a base of length two. This question arose in [BGS].

1.1. The problem. Let f be a cuspidal newform for Γ1(N) with weight k ≥ 2 and character ε. There are well-established methods for computing such forms using modular symbols, see [Ste07]. Let πf be the corresponding automorphic representation of the adèle group GL2(AQ). (As there are numerous possible normalisations, we shall recall the construction in section 2.1 below.) This representation dec...

Let π = ⊗ p≤∞ πp be a cuspidal automorphic representation of GL(n) over Q. The adelic representation π is composed of π∞, an archimedean representation of GL(n,R), and nonarchimedean representations πp of GL(n,Qp) for each prime p. For all places p < ∞ outside of a finite set S, πp is unramified and parameterized by principal series parameters {αp,j}. Letting Ap ∈ GL(n,C) denote the diagonal ma...

Automorphic Lie Algebras are interesting because of their fundamental nature and their role in our understanding of symmetry. Particularly crucial is their description and classification as it allows us to understand and apply them in different contexts, from mathematics to physical sciences. While the problem of classification of Automorphic Lie Algebras with dihedral symmetry was already cons...

Let π $\pi$ be a cuspidal automorphic representation of GSp 4 ( A Q ) $\operatorname{GSp}_4(\mathbf {A}_\mathbf {Q})$ , whose archimedean component is holomorphic discrete series or limit representation. If not CAP endoscopic, then we show that its associated ℓ $\ell$ -adic Galois representations are irreducible and crystalline for 100 % $100\%$ primes . If, moreover, neither an induction nor s...

A loop is (right) automorphic if all its (right) inner mappings are automorphisms. Using the classification of primitive groups of small degrees, we show that there is no non-associative simple commutative automorphic loop of order less than 2, and no non-associative simple automorphic loop of order less than 2500. We obtain numerous examples of non-associative simple right automorphic loops. W...

Let π be a cuspidal automorphic representation of PGL3(A) (where A is the adele ring of F ). The functoriality conjecture of Langlands predicts that, via the inclusion γ of dual groups, there is a global L-packet L(π) of G2 associated to π. Indeed, for each place v of F , there should be a local L-packet L(πv) of G2(Fv). It is expected that L(πv) contains a unique generic representation σ(πv). ...

In this paper we obtain special value results for L-functions associated to classical and paramodular Saito-Kurokawa lifts. In particular, we consider standard L-functions associated to Saito-Kurokawa lifts as well as degree eight L-functions obtained by twisting with an automorphic form defined on GL(2). The results are obtained by combining classical and representation theoretic arguments.