Notes on Atkin-Lehner theory for Drinfeld modular forms

Published 20 Dec 2021 in math.NT | (2112.10340v2)

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 $\mathrm{dim}\ S_{k,l}(\mathrm{GL}2(A))\leq 2$. 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 $S_{k,l}^{{\mathrm{new}}(\Gamma_0(\mathfrak{n}))$.}