Let K be an algebraic number field, and π = ⊗πv an irreducible, automorphic, cuspidal representation of GLm(AK) with analytic conductor C(π). The theorem on analytic strong multiplicity one established in this note states, essentially, that there exists a positive constant c depending on ε > 0,m, and K only, such that π can be decided completely by its local components πv with norm N(v) < c · C...

We study the possible weights of an irreducible 2-dimensional mod p representation of Gal(F/F ) which is modular in the sense of that it comes from an automorphic form on a definite quaternion algebra with centre F which is ramified at all places dividing p, where F is a totally real field. In most cases we determine the precise list of possible weights; in the remaining cases we determine the ...

Abstract We prove an asymptotic expansion of the second moment central values $\mathrm {GL}(n)\times \mathrm {GL}(n)$ Rankin–Selberg L -functions $L(1/2,\pi \otimes \pi _0)$ for a fixed cuspidal automorphic representation $\pi _0$ over family $ with analytic conductors bounded by quantity that is tending to infinity. Our proof uses integral representations -functions, period regularised Eisenst...

• The theory of admissible representations of GL(2,Qp) (or more generally, GL(2, F ) with F/Qp a finite extension). • The theory of automorphic representations of GL(2); in particular, the correspondence between Hecke eigenforms in the classical sense and automorphic representations. • The Jacquet-Langlands correspondence, relating automorphic forms on GL(2) with those on a division algebra. • ...

These notes were originally written in Fall 2010 to provide a very quick overview of some basic topics in modular forms, automorphic forms and automorphic representations. I have not made any significant changes since, or even proofread them completely (so some information may be outdated, and errors may remain), mostly just corrected some typos. If you spy any more errors, or have suggestions,...

Let us begin with some motivation and consider a continuous irreducible representation ρ with trivial determinant of the absolute Galois group GF (of a number field F ) into GL(2,C) which is icosahedral, i.e., whose image in PGL(2,C) is the alternating group A5. Then it is well known that ρ is rational over K = Q[ √ 5], but not equivalent to its Galois conjugate ρ under the non-trivial automorp...

Let K/F be a finite Galois extension of number fields. It is well known that the Tchebotarev density theorem implies that an irreducible, finitely ramified p-adic representation ρ of the absolute Galois group of K is determined (up to equivalence) by the characteristic polynomials of Frobenius elements Frv at any set of primes v of K of degree 1 over F . Here we prove an analogue for GL(n), nam...

We establish some new cases of Artin’s conjecture. Our results apply to Galois representations over Q with image S5 satisfying certain local hypotheses, the most important of which is that complex conjugation is conjugate to (12)(34). In fact, we prove the stronger claim conjectured by Langlands that these representations are automorphic. For the irreducible representations of dimensions 4 and ...

Serre has asked if there are motives M with motivic Galois group of type G2 [Se3; pg 386]. This paper is the first step in a project to construct such a motive M , of rank 7 and weight 0, over the base field Q. Let G be the anisotropic form of G2 over Q, and let π = ⊗̂vπv be an automorphic representation of the adelic group G(A). At almost all primes p, the local representation πp is unramified ...

One disclaimer is necessary: while in principle this factorization result makes it clear that representation theory is relevant to study of automorphic forms and L-functions, in practice there are other things necessary. In effect, one needs to know that the representation theory of reductive linear p-adic groups is tractable, so that conversion of other issues into representation theory is a c...