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.

工作塞尔，灰，Calegari，Venkatesh，贝杰龙，Venkatesh，和Scholze描绘了一个非常引人注目的图片的意义扭转的同源性算术群。Hecke算子作用于与同余算术群相关联的局部对称空间的同调上的特征值系统现在被期望成为其伽罗瓦群是有限型李群的数域的普遍来源。这种期望被称为“Z上的朗兰兹计划”;它包括塞尔猜想的模块化奇数，2维模p伽罗瓦表示作为一种特殊情况。如果没有定量（渐进）结果证明算术群同调中的挠率（至少有时）是丰富的，Z上的朗兰兹纲领将是空洞的。一个令人信服的模型准确地预测了算术群应该包含丰富的挠在他们的同源性是在工作的Bergeron文卡特什;重要的支持证据被证明同上和工作的马歇尔穆勒，穆勒普法夫，和其他人。然而，目前对挠增长的理解在挠和有理上同调共存的“权重”上非常有限。此外，-如果我们期望扭转是丰富的同调的一个明确的算术群，那么我们当然应该能够“看到它在计算机上”。- Z上的互惠景观是“开放的”，充满了猜测。其中最密切相关的代数是可证伪的，似乎非常值得在计算上确认它们，只要它是可能的。然而，用于计算算术群的同调的现有方法是特别的，并且由于算法效率问题而具有有限的范围。因此，设计算法来有效地计算算术群（及其上的Hecke作用）的同调是非常值得的。我的研究计划，在未来五年，将围绕上述两个主题。即研究：（A）有限体积局部对称空间族中拓扑不变量的增长性，特别是同调中的挠。(B)如何有效地和高效地计算这些不变量。我在这些问题上最重要的进展：（A*）（从一年前）与Mark Stern联合，我证明了双曲3-流形上微小的1-形式Laplacian特征值，这是第一同调群中挠率增长的已知障碍，与短循环的失败有关。“（B*）（进行中）与Aurel Page合作，设计了一个通用的、有效的算法来计算同余、算术局部对称空间及其上的Hecke作用的拓扑。算术群的同调中的挠是一个“热门话题”;在问题（A）和（B）上取得进展的时机已经成熟，这样的进展将具有很大的效用。 在（A*）和（B*）中提到的对我与斯特恩和佩奇的作品的思考，激发了我在本提案中提出的许多具体问题。