最高-最低零点及正性的其他应用
The Highest-Lowest Zero and other Applications of Positivity
摘要：
黎曼$ζ$函数的第一个非平凡零点是$\approx 1/2+\pm14.13472i$。我们研究的问题是否有任何其他L-函数有一个更高的最低零点。为了做到这一点，我们试图量化这样一个概念，即一个“小”自守表示（即一个小级别和阿基米德类型）的L函数没有小零，反之亦然。我们证明了许多类型的自守L-函数的第一个零点比$ζ$的第一个零点低。这是使用Weil的显式公式和精心选择的检验函数完成的。当这个方法不能立即显示某个类型的L-函数有低零点时，我们就试图扭转局面，显示不存在这种类型的L-函数。因此，论证是证明低零点存在和某些尖点形式不存在的组合。因此，我们能够证明上同调消灭定理和改进现有的上界的拉普拉斯谱$L^2(\quo n)$。
英文摘要：
The first nontrivial zeroes of the Riemann $ζ$ function are $\approx 1/2+\pm14.13472i$. We investigate the question of whether or not any other L-function has a higher lowest zero. To do so we try to quantify the notion that the L-function of a ``small'' automorphic representation (i.e. one with small level and archimedean type) does not have small zeroes, and vice-versa. We prove that many types of automorphic L-functions have a lower first zero than $ζ$'s. This is done using Weil's explicit formula with carefully-chosen test functions. When this method does not immediately show L-functions of a certain type have low zeroes, we then attempt to turn the tables and show no L-functions of that type exist. Thus the argument is a combination of proving low zeroes exist and that certain cusp forms do not. Consequently we are able to prove cohomological vanishing theorems and improve upon existing bounds on the Laplace spectrum on $L^2(\quo n)$.
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