Notes on Atkin-Lehner theory for Drinfeld modular forms

Abstract

In this article, we settle a part of the Conjecture by Bandini and Valentino (\cite{BV19a}) for $S_{k,l}({\Gamma }_0(T))$ when $\dim S_{k,l}({\mathrm{GL}}_2(A))\le 2$. Then, we frame this conjecture for prime, higher levels, and provide some evidence in favour of it. For any square-free level $\mathfrak{n}$, we define oldforms $S_{k,l}^{\mathrm{old}}({\Gamma }_0(\mathfrak{n}))$, newforms $S_{k,l}^{\mathrm{new}}({\Gamma }_0(\mathfrak{n}))$, and investigate their properties. These properties depend on the commutativity of the (partial) Atkin-Lehner operators with the $U_{\mathfrak{p}}$-operators. Finally, we show that the set of all $U_{\mathfrak{p}}$-operators are simultaneously diagonalizable on $S_{k,l}^{\mathrm{new}}({\Gamma }_0(\mathfrak{n}))$.

References in corpus (2)

• On the structure and slopes of Drinfeld cusp forms
• On mod $\mathfrak{p}$ congruences for Drinfeld modular forms of level $\mathfrak{pm}$