A fundamental question, first raised by Langlands, is to know whether the
Rankin-Selberg product of two (not necessarily holomorphic) cusp forms f and g
is modular, i.e., if there exists an automorphic form f box g on GL(4)/Q whose
standard L-function equals L^*(s, f x g) after removing the ramified and
archimedean factors. The first main result of this paper is to answer it in the
affirmative, in fact with the base field Q replaced by any number field F. Our
proof uses a mixture of converse theorems, base change and descent, and it also
appeals to the local regularity properties of Eisenstein series and the scalar
products of their truncations.
One of the applications of this result is that the space of cusp forms on
SL(2) has multiplicity one. Concretely this means the following: If f, g are
newforms of holomorphic or Maass type with trivial character such that the
squares of the p-th coeficients of f and g are the same at almost all primes p,
then g must be a twist of f by a quadratic Dirichlet character.Comment: 67 pages, published version, abstract added in migratio
