Heegner点在水平方向上的次凸性和等分布性
Subconvexity and equidistribution of Heegner points in the level aspect
摘要：
设q是素数，-D < -4是奇基本判别式，使得q在Q（\sqrt{-D}）中分裂。对于f是q层的权为零的Hecke-Maass新形式，h是D层的权为1 θ的级数，对应于Q（\sqrt{-D}）的一个理想类群特征标，当q = D^η且0 < η< 1时，我们建立了L（f \times h，s）在中心点的混合次凸界.通过这个循环，我们证明了当q，D变大且q < D^{1/20- \varepsilon }时，水平q和判别式D的Heegner点在自然意义上变得等分布。我们解决这些问题的方法与估计限制于Heegner点集的大水平Maass形式的L^2-限制范数有关。进一步建立了同时具有大水平和大二次扭曲的Hecke-Maass L-函数的二次扭曲的界，以及二次Dirichlet L-函数在一定范围内的混合界.
英文摘要：
Let q be a prime and -D < -4 be an odd fundamental discriminant such that q splits in Q(\sqrt{-D}). For f a weight zero Hecke-Maass newform of level q and h the weight one theta series of level D corresponding to an ideal class group character of Q(\sqrt{-D}), we establish a hybrid subconvexity bound for L(f \times h,s) at the central point when q = D^η for 0 < η< 1. With this circle of ideas, we show that the Heegner points of level q and discriminant D become equidistributed, in a natural sense, as q, D become large with q < D^{1/20-\varepsilon}. Our approach to these problems is connected to estimating the L^2-restriction norm of a Maass form of large level when restricted to the collection of Heegner points. We furthermore establish bounds for quadratic twists of Hecke-Maass L-functions with simultaneously large level and large quadratic twist, and hybrid bounds for quadratic Dirichlet L-functions in certain ranges.
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