Hecke operators on Drinfeld cusp forms

Abstract

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.

On the Atkin Ut-operator for Γ1(t)-invariant Drinfeld cusp forms

“…Warning. To shorten some notations and simplify some exponents we normalize Hecke operators multiplying by the factor t k−m (as in [21]): this will erase any reference to the type m in the final formulas. Moreover the slope for newforms will transform into k 2 (from the previous m − k 2 ).…”

“…To check the diagonalizability of the operator U t , we fix the same basis used in [21]. Recall that by [13, Corollary 5.7] we have g(Γ 1 (t)) = 0 and, applying Theorem 2.2, we obtain…”

“…In this context, the lack of an adequate analogous of Petersson inner product leaves open the question about the diagonalizability of the Hecke operators. For the analogue of the T p some partial answers were given by Li and Meemark in [21] and Böckle and Pink in [3]: in contrast to the characteristic 0 case these operators are not always diagonalizable. We shall focus on the diagonalizability of the operator U t on forms of level t and, in particular, on the slopes of its eigenforms.…”

“…In this paper we deal with the action of U t ∶= T (t) on S 1 k,m (Γ) (the space of Γinvariant cusp forms of weight k and type m) for the congruence groups Γ = Γ 1 (t), Γ(t) of level t.In this context the question about the diagonalizability of the Hecke operators is still open, due mainly to the lack of an adequate analogous of Petersson inner product. For the operators T p with p ≠ (t) generated by a polynomial of degree 1, Li and Meemark in [8] checked diagonalizability for k ⩽ q + 3, i.e., until they found the first example of a non diagonalizable matrix (in even characteristic) together with the presence of an inseparable eigenvalue (see [8], in particular, the Remark in page 1951). Böckle and Pink computed the structure of double cusp forms for Γ 1 (t) for some fixed k and q ([3, Section 15]) and for those of weight 4 they determined all eigenvalues ([3, Proposition 15.6]).Using the Bruhat-Tits tree T as a combinatorial counterpart for Ω, Teitelbaum in [10] provided a reinterpretation of cusp forms as Γ-invariant harmonic cocycles.…”

On the action of Hecke operators on Drinfeld modular forms

“…. , i s ) run over the set of (s + 1)-tuples (s arbitrary) satisfying i 0 + · · · + i s = m, i 0 + i 1 q + · · · + i s q s = n. According to [3, (3.8)] formula (7) shows that [12] shows that for 1 ≤ j ≤ q:…”