In this paper, we obtain two analogues of the Sturm bound for modular forms in the function field setting. In the case of mixed characteristic, we prove that any harmonic cochain is uniquely determined by an explicit finite number of its first Fourier coefficients where our bound is much smaller than the ones in the literature. A similar bound is derived for generators of the Hecke algebra on harmonic cochains. As an application, we present a computational criterion for checking whether two elliptic curves over the rational function field ${\mathbb{F}}_q(\theta )$ with same conductor are isogenous. In the case of equal characteristic, we also prove that any Drinfeld modular form is uniquely determined by an explicit finite number of its first coefficients in the t-expansion.

在本文中，我们在函数字段设置中获得了模形式的 Sturm 界的两个类似物。在混合特征的情况下，我们证明了任何谐波共链都是由其第一个傅里叶系数的显式有限数量唯一确定的，其中我们的界限远小于文献中的界限。对于谐波 cochains 上的 Hecke 代数的生成器，也得出了类似的界限。作为一个应用程序，我们提出了一个计算标准，用于检查有理函数场上的两条椭圆曲线是否${\mathbb{F}}_q(\theta )$具有相同的导体是同源的。在等特征的情况下，我们还证明了任何 Drinfeld 模形式都是由其在t展开中的第一个系数的显式有限个数唯一确定的。