On $d$-gonality of Drinfel'd modular curves and strong Uniform Boundedness Conjecture

Abstract

We give an upper bound of the level $\mathfrak{n}$ by means of $d$-gonality of the Drinfel'd modular curves $X_0(\mathfrak{n})$ mod $\mathfrak{p}$ for $\mathfrak{p} \nmid \mathfrak{n}$. As a corollary of the result, we obtain an estimation in the strong Uniform Boundedness Conjecture for Drinfel'd modules of rank 2. We also discuss some asymptotically (and practically) good bound in this connection.