Homology growth in families of locally symmetric spaces

局部对称空间族中的同源增长

• 批准号: RGPIN-2018-04784
• 负责人: Lipnowski, Michael
• 金额: $ 1.17万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759044
• 关键词: Homology; growth; families; locally; symmetric

Work of Serre, Ash, Calegari-Venkatesh, Bergeron-Venkatesh, and Scholze paints a very compelling picture of the significance of torsion in the homology of arithmetic groups. Systems of eigenvalues for Hecke operators acting on the homology of locally symmetric spaces associated with congruence arithmetic groups are now expected to be the universal source of number fields whose Galois groups are Lie groups of finite type. This expectation has been dubbed the "Langlands program over Z"; it includes Serre's conjecture on the modularity of odd, 2-dimensional mod p Galois representations as a special case. The Langlands Program over Z would be vacuous if not for quantitative (asymptotic) results proving that torsion in the homology of arithmetic groups is (at least sometimes) abundant. A convincing model predicting exactly which arithmetic groups should contain abundant torsion in their homology was laid out in work of Bergeron-Venkatesh; significant supporting evidence was proven ibid and in works of Marshall-Muller, Muller-Pfaff, and others. Current understanding of torsion growth, however, is very limited in "weights" for which both torsion and rational cohomology coexist. Furthermore, - If we expect torsion to be abundant in the homology of an explicit arithmetic group, then surely we should be able to "see it on a computer". - The landscape of reciprocity over Z is "wide open" and full of conjecture. The most germane conjectures therein are falsifiable and it seems very worthwhile to confirm them computationally, insofar as it is possible.State of the art approaches for computing the homology of arithmetic groups, however, are ad hoc and have limited scope due to algorithmic efficiency issues. Devising algorithms to efficiently compute the homology of arithmetic groups (and Hecke actions thereon) is therefore very worthwhile. My research program, over the next five years, will center around the above two themes. Namely, it will study: (A) growth of topological invariants in families of finite volume locally symmetric spaces, especially torsion in homology. (B) how to effectively and efficiently compute these invariants.My most significant progress on these problems:(A*) (from one year ago) Joint with Mark Stern, I show that tiny 1-form Laplacian eigenvalues on hyperbolic 3-manifolds, a known obstruction to growth of torsion in the first homology group, are related to the failure of short loops to be ``efficiently bounded." (B*) (ongoing) Joint with Aurel Page, we devise a general, efficient algorithm to computing the topology of congruence, arithmetic locally symmetric spaces and Hecke actions thereon. Torsion in the homology of arithmetic groups is a "hot topic"; the time is ripe for progress on problems (A) and (B) and such progress would have great utility. Meditation on my works with Stern and Page, alluded in (A*) and (B*), inspired many of specific problems I suggest in the present proposal.

Serre，Ash，Calegari-Venkatesh，Bergeron-Venkatesh和Scholze的作品描绘了扭转在算术组同源性中的重要性。 Hecke操作员的特征值系统作用于与一致性算术组相关的本地对称空间的同源性系统，现在预计将是数字字段的通用来源，其galois组是有限类型的谎言组。这种期望被称为“ Z上的Langlands计划”；它包括Serre对奇数二维mod p galois表示的模块化的猜想。如果不是定量（渐近）结果，则Z上的Langlands计划将是空置的，证明算术组同源性的扭转（至少有时）丰富。令人信服的模型准确地预测了哪些算术群应包含其同源性大量扭转，这是在Bergeron-Venkatesh的工作中列出的。在Marshall-Muller，Muller-Pfaff等人的作品中证明了大量的支持证据。然而，当前对扭转生长的理解在“权重”中非常有限，扭转和理性的共同体共存。此外，如果我们期望扭转在明确的算术组的同源性中很丰富，那么我们肯定应该能够“在计算机上看到它”。 - Z上互惠的景观是“大开”的，充满了猜想。其中最敏锐的猜想是可伪造的，并且在可能的情况下，在计算上确认它们似乎非常值得。因此，设计算法以有效计算算术群（和Hecke Actions）的同源性非常值得。在接下来的五年中，我的研究计划将围绕上述两个主题。也就是说，它将研究：（a）有限体积局部对称空间（尤其是同源性扭转）中拓扑不变的生长。 （b）如何有效，有效地计算这些不变的问题。在这些问题上，我最重大的进展：（a*）（从一年前开始）与马克·斯特恩（Mark Stern）的关节，我表明，在双曲线3个manifolds上，很小的1形laplacian特征值，这是一个已知的障碍，已知与第一个同级小组的扭曲相关的障碍，与失败的效果相关。 （正在进行的）与AUREL页面的关节，我们将一般的，有效的算法计算出一致性的拓扑，算术在局部对称的空间和静脉弧线的扭转。 （b*），我在本提案中提出了许多具体问题。