2020/03/02 by Cécile Armana, Armana, Cécile, Fu-Tsun Wei +2
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.2003.00815
Minor changes. To appear in Journal of Number Theory
openalex publication_date 2020/03/02 · openalex created_date 2020/03/06 · arxiv created 2020/08/25 · arxiv updated 2020/08/26 · openalex updated_date 2026/07/28
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 \mathbbFq(θ) 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.