AI 中文总结
研究半整数权尖形式无平方因子傅里叶系数的符号分布，证明特定尖形式正负此类系数数量均\(\gg_{f,\varepsilon}X^{4/7 - \varepsilon}\)，并应用于同余数问题，表明相关尖形式的无平方因子傅里叶系数无限次变号。
AI 中文摘要
我们研究半整数权尖形式的无平方因子傅里叶系数的符号分布。对于满足本征形式条件且具有非零尖点志村提升的半整数权尖形式\(f\)，我们证明了直到\(X\)的正负无平方因子傅里叶系数的数量均为\(\gg_{f,\varepsilon}X^{4/7 - \varepsilon}\)。作为同余数问题的应用，考虑权为\(3/2\)的尖形式，其志村提升是与\(E:y^2 = x^3 - x\)相关的权为\(2\)的新形式。我们证明在小于等于\(X\)的\(\gg_{\varepsilon}X^{4/7 - \varepsilon}\)个奇无平方因子整数\(t\)中，其无平方因子傅里叶系数出现每种符号。特别地，无平方因子傅里叶系数无限次变号。
英文摘要
We study the sign distribution of square-free Fourier coefficients of half-integral weight cusp forms. We prove that, for half-integral weight cusp forms \(f\) satisfying the eigenform conditions and having a nonzero cuspidal Shimura lift, the numbers of positive and negative square-free Fourier coefficients up to \(X\) are both \(\gg_{f,\varepsilon}X^{4/7-\varepsilon}\). As an application to the congruent-number problem, we consider the weight \(3/2\) cusp form whose Shimura lift is the weight \(2\) newform attached to $ E:y^2=x^3-x$. We prove that each each sign occurs among its square-free Fourier coefficients at \(\gg_{\varepsilon}X^{4/7-\varepsilon}\) odd square-free integers \(t\leq X\). In particular, the square-free Fourier coefficients change sign infinitely often.
