In this paper we give an example of a noncongruence subgroup whose three-dimensional space of cusp forms of weight 3 has the following properties. For each of the four residue classes of odd primes modulo 8 there is a basis whose Fourier coefficients at infinity satisfy a three-term Atkin and Swinnerton-Dyer congruence relation, which is the p-adic analogue of the three-term recursion satisfied by the coefficients of classical Hecke eigenforms. We also show that there is an automorphic L-function over Q whose local factors agree with those of the l-adic Scholl representations attached to the space of noncongruence cusp forms.

Abstract. In this paper we obtain a closed form expression of the zeta function Z(X Γ , u) of a finite quotient X Γ of the Bruhat-Tits building of PGL 3 over a nonarchimedean local field F by a discrete cocompact torsion-free subgroup Γ of PGL 3 . Analogous to a graph zeta function, Z(X Γ , u) is a rational function with two different expressions and it satisfies the Riemann hypothesis if and only if X Γ is a Ramanujan complex.

Let F be a non-archimedean local field with a finite residue field. To a 2-dimensional finite complex X Γ arising as the quotient of the Bruhat-Tits building X associated to Sp 4 (F) by a discrete torsion-free cocompact subgroup Γ of PGSp 4 (F), associate the zeta function Z(X Γ , u) which counts geodesic tailless cycles contained in the 1-skeleton of X Γ . Using a representationtheoretic approach, we obtain two closed form expressions for Z(X Γ , u) as a rational function in u. Equivalent statements for X Γ being a Ramanujan complex are given in terms of vertex, edge, and chamber adjacency operators, respectively. The zeta functions of such Ramanujan complexes are distinguished by satisfying the Riemann Hypothesis.

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In this paper, we study the Drinfeld cusp forms for Γ 1 (T ) and Γ (T ) using Teitelbaum's interpretation as harmonic cocycles. We obtain explicit eigenvalues of Hecke operators associated to degree one prime ideals acting on the cusp forms for Γ 1 (T ) of small weights and conclude that these Hecke operators are simultaneously diagonalizable. We also show that the Hecke operators are not diagonalizable in general for Γ 1 (T ) of large weights, and not for Γ (T ) even of small weights. The Hecke eigenvalues on cusp forms for Γ (T ) with small weights are determined and the eigenspaces characterized.