论文标题
模块化$ l $ series in Level $ p^2 $的中央衍生物的不变
Nonvanishing of Central Derivatives of Modular $L$-series in Level $p^2$
论文作者
论文摘要
已知与模块化形式相关的L功能的二次扭曲可以满足功能方程，这可能是偶数或奇怪的。由于总和Zagier而引起的结果明确计算了L功能或其导数的中心值。在质量上，当功能方程式甚至是官方方程时，米歇尔和拉马克里什南都使用平均方法证明了总Zagier公式的几个后果，包括非散布结果。本研究涉及由原始水平的新形式引起的l功能，该级别必然具有奇怪的功能方程。这样的L功能的中心值为零；总Zagier公式计算其衍生物的中心值。使用Michel-Ramakrishnan平均方法，我们计算了这些衍生物在不同L功能上的平均值。特别是，我们表明，在适当的条件下，在对称中心的l功能中只有一个简单的零功能。证明要求我们将OldForms的贡献从级别的$ p $绑定到。
A quadratic twist of the L-function associated with a modular form is known to satisfy a functional equation, which may be even or odd. A result due to Gross and Zagier explicitly computes the central value of the L-function or its derivative. In prime level when the functional equation is even, Michel and Ramakrishnan have used an averaging method to prove several consequences of the Gross-Zagier formulae, including a non-vanishing result. The present research concerns L-functions arising from newforms in prime-squared level, which necessarily have odd functional equations. Such an L-function has a central value of zero; the Gross-Zagier formulae compute the central value of its derivative. Using the Michel-Ramakrishnan averaging method, we compute the average value of these derivatives over different L-functions. In particular, we show that under suitable conditions there exists an L-function with only a simple zero at the center of symmetry. The proof requires us to bound the contribution of oldforms from level $p$.
