Potential Automorphy of Odd-dimensional Symmetric Powers of Elliptic Curves, and Applications

The present article was motivated by a question raised independently by Barry Mazur and Nick Katz. The articles [CHT,HST,T] contain a proof of the Sato-Tate conjecture for an elliptic curve E over a totally real field whose j-invariant j(E) is not an algebraic integer. The Sato-Tate conjecture for E is an assertion about the equidistribution of Frobenius angles of E, or equivalently about the number of points |E(Fp)| on E modulo p as p varies. The precise statement of the conjecture, which is supposed to hold for any elliptic curve without complex multiplication, is recalled in §5. Now suppose E and E are two elliptic curves without complex multiplication, and suppose E and E are not isogenous. The question posed by Mazur and Katz is roughly the following: are the distributions of the Frobenius angles of E and E, or equivalently of the numbers p+1−|E(Fp)| and p+1−|E (Fp)|, independent? The Sato-Tate conjecture, in the cases considered in [CHT,HST,T], is a consequence of facts proved there about L-functions of symmetric powers of the Galois representation on the Tate module Tl(E) of E, following a strategy elaborated by Serre in [S]. These facts in turn follow from one of the main theorems of [CHT,HST,T], namely that, if n is even, the (n − 1)st symmetric power of Tl(E) is potentially automorphic, in that it is associated to a cuspidal automorphic representation of GL(n) over some totally real Galois extension of the original base field. The restriction to even n is inherent in the approach to potential modularity developed in [HST], which applies only to even-dimensional representations. The necessary properties of all symmetric power L-functions follow from this result for even-dimensional symmetric powers, together with basic facts about RankinSelberg L-functions proved by Jacquet-Shalika-Piatetski-Shapiro and Shahidi. In a