AI 中文总结
研究由二次型扭曲的椭圆曲线的秩，利用模\(L\)函数导数的矩，证明存在无穷多个\(d\)是两平方数之和使\(E^d\)秩为\(1\)，还为相关椭圆纤维化提供信息。
AI 中文摘要
设\(E\)是\(\mathbb{Q}\)上的椭圆曲线，\(E^d\)是由二次特征\(\chi_d\)扭曲得到的。我们证明存在无穷多个\(d\)是两个平方数之和，使得\(E^d\)的秩为\(1\)。此结果通过模\(L\)函数导数的矩得到，特别涵盖了Munshi工作中遗漏的低阶导数。这一结果也为椭圆纤维化\((1 + t^2)y^2 = f(x)\)（其中\(f(x)\)是三次多项式）提供了信息。
英文摘要
Let $E$ be an elliptic curve over $\mathbb{Q}$ and let $E^d$ be its twist by the quadratic character $χ_d$. We prove there are infinitely many twists $d$ which are sums of two squares such that $E^d$ has rank $1$. This result is achieved using moments of derivatives of modular $L$-functions, and particularly captures the lower derivatives which were left out in the work of Munshi. Such a result, in particular, also gives us information on the elliptic fibration $(1+t^2)y^2=f(x)$, where $f(x)$ is a cubic polynomial.
Comments21 pages. Fixed typos
