The Ramanujan Journal ( IF 0.7 ) Pub Date : 2021-02-26 , DOI: 10.1007/s11139-021-00401-2 Rishabh Agnihotri , Kalyan Chakraborty
We prove that given any \(\epsilon >0\), a non-zero adelic Hilbert cusp form \({\mathbf {f}}\) of weight \(k=(k_1,k_2,\ldots ,k_n)\in ({\mathbb {Z}}_+)^n\) and square-free level \(\mathfrak {n}\) with Fourier coefficients \(C_{{\mathbf {f}}}(\mathfrak {m})\), there exists a square-free integral ideal \(\mathfrak {m}\) with \(N(\mathfrak {m})\ll k_0^{3n+\epsilon }N(\mathfrak {n})^{\frac{6n^2+1}{2}+\epsilon }\) such that \(C_{{\mathbf {f}}}(\mathfrak {m})\ne 0\). The implied constant depends on \(\epsilon , F\).
中文翻译:
关于某些希尔伯特模态形式的傅立叶系数
我们证明了给定的任何\(\小量> 0 \) ,非零adelic希尔伯特尖形式\({\ mathbf {F}} \)重量的\(K =(K_1,K_2,\ ldots,k_n)\ in({\ mathbb {Z}} _ +)^ n \)和无平方水平\(\ mathfrak {n} \)中具有傅立叶系数\(C _ {{\ mathbf {f}}}(\ mathfrak {m })\),存在一个无平方积分理想\(\ mathfrak {m} \),其中\(N(\ mathfrak {m})\ ll k_0 ^ {3n + \ epsilon} N(\ mathfrak {n}) ^ {\ frac {6n ^ 2 + 1} {2} + \ epsilon} \)使得\(C _ {{\ mathbf {f}}}(\ mathfrak {m})\ ne 0 \)。隐含常数取决于\(\ epsilon,F \)。