over a field F of characteristic 0. The methods of [A3] are by comparison with representations of the group GL(N). This eventually leads to results on the representations of the full orthogonal group O(N). But it is the connected subgroup G = SO(N) of O(N) that is the ultimate object of interest. In [A3, §8.4], we were able to characterize certain representations of G from the results obtained ...

Let $F$ be a CM number field. We prove modularity lifting theorems for regular $n$-dimensional Galois representations over without any self-duality condition. deduce that all elliptic curves $E$ are potentially modular, and furthermore satisfy the Sato--Tate conjecture. As an application of different sort, we also Ramanujan Conjecture weight zero cuspidal automorphic $\mathrm{GL}_2(\mathbb{A}_F...

Let π1, π2, π3 be three cuspidal automorphic representations for the group SL(2,Z), where π1 and π2 are fixed has large analytic conductor. We prove a subconvex bound L(1∕2,π1⊗π2⊗π3) of Weyl-type quality. Allowing to an Eisenstein series, we also obtain L(1∕2+it,π1⊗π2).

In this paper, Hecke eigenvalues of several automorphic forms for congruence subgroups of SL(3; Z) are listed. To compute such tables, we describe an algorithm which combines techniques developed by Ash, Grayson and Green with the Lenstra{Lenstra{Lovv asz algorithm. With our implementation of this new algorithm we were able to handle much larger levels than those treated by Ash, Grayson and Gre...

The Langlands Program relates Galois representations and automorphic representations of reductive algebraic groups. The trace formula is a powerful tool in the study of this connection and the Langlands Functoriality Conjecture. After giving an introduction to the Langlands Program and its geometric version, which applies to curves over finite fields and over the complex field, I give a survey ...

We use the theta lift to study the multiplicity with which certain automorphic representations of cohomological type occur in a family of congruence covers of an arithmetic manifold. When the family of covers is a so-called p-adic congruence tower, we obtain sharp asymptotics for the number of representations that occur as lifts. When combined with theorems on the surjectivity of the theta lift...

Suppose E/F is a quadratic extension of number fields and GU(2, 2) is the quasi-split unitary similitude group attached to E/F . We prove functoriality from globally generic cuspidal representations of GU(2, 2)(AF ) to automorphic representations of GL6(AF ) corresponding to the twisted exterior square map on the dual side. For a split quadratic algebra E/F , the twisted exterior square map red...

We compute the arithmetic L-invariants (of Greenberg–Benois) of twists of symmetric powers of p-adic Galois representations attached to Iwahori level Hilbert modular forms (under some technical conditions). Our method uses the automorphy of symmetric powers and the study of analytic Galois representations on p-adic families of automorphic forms over symplectic and unitary groups. Combining thes...

We give a classification of the cuspidal automorphic representations attached to rational elliptic curves with non-trivial torsion point odd order. Such are parameterizable, and in this paper, we find necessary sufficient conditions on parameters determine when split or non-split multiplicative reduction occurs. Using known results additive occurs for these parametrized curves, classify terms p...