Automorphic forms and rational homology 3--spheres
Automorphic forms and rational homology 3--spheres
复制标题
DOI：
10.2140/gt.2006.10.295
复制
发表时间：
2005-08
影响因子：
2
通讯作者：
Frank Calegari;N. Dunfield
中科院分区：
文献类型：
--
作者：
Frank Calegari;N. Dunfield
研究了与虚哈肯猜想相邻的一个库珀问题.在数论中假设一定的条件，证明了存在具有任意大内射半径的双曲有理同调3-球面。这些例子来自显式算术3-流形的阿贝尔覆盖塔。我们必须假定的假设是广义黎曼假设和泰勒等人关于虚二次域GL 2的朗兰兹纲领的一部分的结果的一个温和的加强。这个定理的证明涉及排除Gal（Qbar/Qsqrt-2）的不可约二维Galois表示rho的存在性，该表示rho满足某些指定的分歧条件。与此相反，类似的问题，这种形式，ρ是允许有任意分歧，在一些总理π Z [sqrt-2]。在本卷的下一篇论文中，Boston和Ellenberg将pro-p技术应用于我们的例子，并证明我们的结果是无条件正确的。在这里，我们给出了他们的技术适用的其他示例，包括一些非算术示例。最后，我们研究了扭结orbifolds的同余覆盖。我们的实验证据表明，这些拓扑相似的orbifold有相当不同的行为取决于他们是否算术。特别地，非算术orbifold的同余覆盖具有缺乏的同源性。
We investigate a question of Cooper adjacent to the Virtual Haken Conjecture. Assuming certain conjectures in number theory, we show that there exist hyperbolic rational homology 3–spheres with arbitrarily large injectivity radius. These examples come from a tower of abelian covers of an explicit arithmetic 3–manifold. The conjectures we must assume are the Generalized Riemann Hypothesis and a mild strengthening of results of Taylor et al on part of the Langlands Program for GL2 of an imaginary quadratic field. The proof of this theorem involves ruling out the existence of an irreducible two dimensional Galois representation rho of Gal(Qbar/Qsqrt-2) satisfying certain prescribed ramification conditions. In contrast to similar questions of this form, rho is allowed to have arbitrary ramification at some prime pi of Z[sqrt -2]. In the next paper in this volume, Boston and Ellenberg apply pro–p techniques to our examples and show that our result is true unconditionally. Here, we give additional examples where their techniques apply, including some non-arithmetic examples. Finally, we investigate the congruence covers of twist-knot orbifolds. Our experimental evidence suggests that these topologically similar orbifolds have rather different behavior depending on whether or not they are arithmetic. In particular, the congruence covers of the non-arithmetic orbifolds have a paucity of homology.