Hecke Operators on Drinfeld Cusp Forms

In this paper, we study the Drinfeld cusp forms for $\Gamma_1(T)$ and $\Gamma(T)$ 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 $\Gamma_1(T)$ 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 $\Gamma_1(T)$ of large weights, and not for $\Gamma(T)$ even of small weights. The Hecke eigenvalues on cusp forms for $\Gamma(T)$ with small weights are determined and the eigenspaces characterized.Comment: 28 pages To appear in JN

On Atkin-Swinnerton-Dyer congruence relations

In this paper we exhibit a noncongruence subgroup \G whose space of weight 3 cusp forms S_3(\G) admits a basis satisfying the Atkin-Swinnerton-Dyer congruence relations with two weight 3 newforms for certain congruence subgroups. This gives a modularity interpretation of the motive attached to S_3(\G) by A. Scholl and also verifies the Atkin-Swinnerton-Dyer congruence conjecture for this space.Comment: 25 page

Fourier coefficients of noncongruence cuspforms

Given a finite index subgroup of SL2(ℤ) with modular curve defined over ℚ, under the assumption that the space of weight k (≥2) cuspforms is one-dimensional, we show that a form in this space with Fourier coefficients in ℚ has bounded denominators if and only if it is a congruence modular form. © 2012 London Mathematical Society

Ramanujan Graphs on Cosets of PGL2(Fq)PGL_2(\mathbb{F}_q)

In this paper we study Cayley graphs on \PGL_2(\mathbb F_q) mod the unipotent subgroup, the split and nonsplit tori, respectively. Using the Kirillov models of the representations of \PGL_2(\mathbb F_q) of degree greater than one, we obtain explicit eigenvalues of these graphs and the corresponding eigenfunctions. Character sum estimates are then used to conclude that two types of the graphs are Ramanujan, while the third is almost Ramanujan. The graphs arising from the nonsplit torus were previously studied by Terras et al. We give a different approach here

On Atkin and Swinnerton-Dyer Congruence Relations (2)

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 eigen forms. We also show that there is an automorphic $L$-function over $\mathbb Q$ whose local factors agree with those of the $l$-adic Scholl representations attached to the space of noncongruence cusp forms.Comment: Last version, to appear on Math Annale