Icosahedral Fibres of the Symmetric Cube and Algebraicity

For any number field F, call a cusp form π = π_∞⊗πf on GL(2)/F special icosahedral, or just s-icosahedral for short, if π is not solvable polyhedral, and for a suitable “conjugate” cusp form π' on GL(2)/F, sym^3(π) is isomorphic to sym^3(π'), and the symmetric fifth power L-series of π equals the Rankin-Selberg L-function L(s, sym^2(π') × π) (up to a finite number of Euler factors). Then the point of this Note is to obtain the following result: Let π be s-icosahedral (of trivial central character). Then π f is algebraic without local components of Steinberg type, π ∞ is of Galois type, and π_v is tempered every-where. Moreover, if π' is also of trivial central character, it is s-icosahedral, and the field of rationality Q(πf) (of πf) is K := Q[√5], with π' _f being the Galois conjugate of πf under the non-trivial automorphism of K. There is an analogue in the case of non-trivial central character ω, with the conclusion that π is algebraic when ω is, and when ω has finite order, Q(πf) is contained in a cyclotomic field

Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2)

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

Recovering Cusp forms on GL(2) from Symmetric Cubes

Suppose $\pi$, $\pi'$ are cusp forms on GL$(2)$, not of solvable polyhedral type, such that they have the same symmetric cubes. Then we show that either $\pi$, $\pi'$ are twist equivalent, or else a certain degree $36$ $L$-function associated to the pair has a pole at $s=1$. If we further assume that the symmetric fifth power of $\pi$ is automorphic, then in the latter case, $\pi$ is icosahedral in a suitable sense, agreeing with the usual notion when there is an associated Galois representation.Comment: 10 page

Remarks on the symmetric powers of cusp forms on GL(2)

In this paper we prove the following conditional result: Let F be a number field, and pi a cusp form on GL(2)/F which is not solvable polyhedral. Assume that all the symmetric powers sym^m(pi) are modular, i.e., define automorphic forms on GL(m+1)/F. If sym^6(pi) is cuspidal, then all the symmetric powers are cuspidal, for all m. Moreover, sym^6(pi) is Eisenteinian iff sym^5(pi) is an abelian twist of the functorial product of pi with the symmetric square of a cusp form pi' on GL(2)/F.Comment: A sentence has been modified in the Introduction. It has nothing to do with the main result of the pape

Arithmetic Quotients of the Complex Ball and a Conjecture of Lang

We prove that various arithmetic quotients of the unit ball in $\mathbb{C}^n$ are Mordellic, in the sense that they have only finitely many rational points over any finitely generated field extension of $\mathbb{Q}$. In the previously known case of compact hyperbolic complex surfaces, we give a new proof using their Albanese in conjunction with some key results of Faltings, but without appealing to the Shafarevich conjecture. In higher dimension, our methods allow us to solve an alternative of Ullmo and Yafaev. Our strongest result uses in addition Rogawski's theory and establishes the Mordellicity of the Baily-Borel compactifications of Picard modular surfaces of some precise levels related to the discriminant of the imaginary quadratic fields.Comment: 21 pages, final versio