Let E F be a quadratic extension of number fields. We will denote by _ the non-trivial element of the Galois group of E F and will often write _(z)=z . We will denote by U1 the unitary group in 1 variable. We assume that every Archimedean place of F splits in E. We let H0 be the unitary group for the 3_3 identity matrix. Recall that a cuspidal automorphic representation 6 of GL(3, EA) is said t...

Let π be a regular, algebraic, essentially self-dual cuspidal automorphic representation of GLn(AF ), where F is a totally real field and n is at most 5. We show that for all primes l, the l-adic Galois representations associated to π are irreducible, and for all but finitely many primes l, the mod l Galois representations associated to π are also irreducible. We also show that the Lie algebras...

We obtain second integral moments of automorphic L–functions on adele groups GL2 over arbitrary number fields, by a spectral decomposition using the structure and representation theory of adele groups GL1 and GL2. This requires complete reformulation of the notion of Poincaré series, replacing the collection of classical Poincaré series over GL2(Q) or GL2(Q(i)) with a single, coherent, global o...

• Decomposition by central characters • Square-integrable cuspforms • Smoothness of cuspforms • Eigen-cuspforms and automorphic representation • Dirichlet series versus zeta and L-functions • L-functions defined via local data • Factoring unitary representations of adele groups • Spherical representations and Satake parameters • Local data, L-groups, higher L-functions References and historical...

We prove a Lie-theoretic result for type A root systems. This allows us to determine the unipotent orbits attached to degenerate Eisenstein series on general linear groups, confirming a conjecture of David Ginzburg. In particular, this also shows that any unipotent orbit of general linear groups does occur as the unipotent orbit attached to a specific automorphic representation. The proof of th...

Let E/F be a quadratic extension of number fields. Suppose that every real place of F splits in E and let H be the unitary group in 3 variables. Suppose that Π is an automorphic cuspidal representation of GL(3, EA). We prove that there is a form φ in the space of Π such that the integral of φ over H(F )\H(FA) is non zero. Our proof is based on earlier results and the notion, discussed in this p...

The Fundamental Lemma is a somewhat obscure combinatorial identity introduced by Robert P. Langlands in 1979 as an ingredient in the theory of automorphic representations. After many years of deep contributions by mathematicians working in representation theory, number theory, algebraic geometry, and algebraic topology, a proof of the Fundamental Lemma was recently completed by Ngô Bao Châu in ...

Let Y be an arithmetic hyperbolic 3-manifold. We establish a link between quantum unique ergodicity for sections of automorphic vector bundles on Y and subconvexity for the triple product L-function, which extends a result of Watson in the case of hyperbolic 2-manifolds. The proof combines the representation theoretic microlocal lift for bundles developed by Bunke and Olbrich with the triple pr...

We work with the structure consisting of all computably enumerable (c.e.) sets ordered by set inclusion. The question we will partially address is which c.e. sets are autormorphic to low (or low2) sets. Using work of R. Miller [9], we can see that every set with semilow complement is ∆3 automorphic to a low set. While it remains open whether every set with semilow complement is effectively auto...