Everywhere Unramified Automorphic Cohomology for Sl(3,z)

We conjecture that the only irreducible cuspidal automorphic representations for GL(3)/Q of cohomological type and level 1 are (up to twisting) the symmetric square lifts of classical cuspforms on GL(2)/Q of level 1. We present computational evidence for this conjecture. 1. Statement and explanation of a conjecture Arithmetic objects defined over Q and unramified everywhere are rare. For example, a theorem of Minkowski states that there are no finite extensions of Q unramified everywhere. A theorem of Fontaine [8] states that there are no abelian varieties defined over Q unramified everywhere. In this note we consider automorphic cohomology for SL(n,Z) defined over Q and unramified everywhere, n ≥ 2. Let A denote the adeles of Q. Given a Hecke-eigenclass class α in the cuspidal cohomology of SL(n,Z) with coefficients in an irreducible finite dimensional complex rational representation V of GL(n,C), there is uniquely determined an irreducible cuspidal automorphic representation πα = ⊗πα,v of GL(n,A). The infinity type πα,∞ = φ(V ) depends only on V and for every prime p, πα,p is an irreducible unramified principle series representation of GL(n,Qp) depending only on the Hecke eigenvalues of α at p. We say an irreducible cuspidal automorphic π is cohomological if π∞ = φ(V ) for some V as above. Then each cohomological π corresponds to some such Hecke-eigenclass α. See [7]. (The cuspidal cohomology is defined to be the subspace of the cohomology which can be represented by cuspidal automorphic differential forms on the appropriate symmetric space.) We ask: are there no nonzero cuspidal cohomological automorphic representations of GL(n)/Q unramified everywhere, or equivalently, isH∗ cusp(SL(n,Z), V ) = 0 for all V . The answer is well-known to be “no” for n = 2. For then the theory of classical modular forms and the Shimura-Eichler theorem combine to tell us that if V = Sym(C2), then H∗ cusp(SL(2,Z), V ) 6= 0 if g is even and either = 10 or ≥ 14. For example, Ramanujan’s ∆ gives rise to a nontrivial class inH∗ cusp(SL(2,Z),Sym (C2)). So although arithmetic objects defined over Q unramified everywhere are rare, there are some. It should be noted that the existence of cuspforms of level 1 for SL(2,Z) can be accounted for by the simple topological fact that the Euler characteristic of SL(2,Z) is non-zero, or by the representation-theoretic fact that GL(2,R) has a First author partially supported by NSA grant MDA 904-00-1-0046 and NSF grants DMS0139287 and DMS-0455240. This manuscript is submitted for publication with the understanding that the United States government is authorized to reproduce and distribute reprints.