Systoles of Arithmetic Hyperbolic Surfaces and 3-manifolds
Systoles of Arithmetic Hyperbolic Surfaces and 3-manifolds
复制标题
算术双曲曲面和 3 流形的收缩
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
Lola Thompson
中科院分区：
文献类型：
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作者：
Benjamin Linowitz;D. McReynolds;P. Pollack;Lola Thompson
我们的主要结果是，对于足够大的$x0>；0$，具有固定不变迹域$k$的算术双曲2或3-orbolold的可公度类的集合和具有$x_0$下界的收缩的可公度类的集合在具有不变迹场$k$的算术双曲2或3-orbolold的所有可公度类的集合中具有密度1。证明依赖于Silverman，Brindza和Hajdu给出的代数整数的绝对对数Weil高度的界，以及对不允许嵌入任何具有小判别式的二次域的有理四元数代数数目的精确估计。当迹为$mathbf{q}$时，利用Granville和Soundararajan的工作，我们建立了一个更强的结果，它允许我们的常数下界$x_0$随着面积的增加而增长。作为应用，我们建立了算术双曲曲面的一个收缩上界，这与Buser-Sarnak和Katz-Schaps-Vishne的工作有关。最后，我们建立了具有小面积全测地$2的算术双曲3-orbilloles的可公度性类的相似密度结果。
Our main result is that for all sufficiently large $x_0>0$, the set of commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with fixed invariant trace field $k$ and systole bounded below by $x_0$ has density one within the set of all commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with invariant trace field $k$. The proof relies upon bounds for the absolute logarithmic Weil height of algebraic integers due to Silverman, Brindza and Hajdu, as well as precise estimates for the number of rational quaternion algebras not admitting embeddings of any quadratic field having small discriminant. When the trace field is $mathbf{Q}$, using work of Granville and Soundararajan, we establish a stronger result that allows our constant lower bound $x_0$ to grow with the area. As an application, we establish a systolic bound for arithmetic hyperbolic surfaces that is related to prior work of Buser-Sarnak and Katz-Schaps-Vishne. Finally, we establish an analogous density result for commensurability classes of arithmetic hyperbolic 3-orbifolds with small area totally geodesic $2$-orbifolds.