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 的工作描绘了一幅关于算术群同调中挠率重要性的非常引人注目的图画。作用于与同余算术群相关的局部对称空间的同源性的 Hecke 算子的特征值系统现在预计将成为其伽罗瓦群是有限类型李群的数域的通用源。这一期望被称为“Z 上的朗兰兹计划”；它包括 Serre 关于奇数二维 mod p Galois 表示的模块化的猜想，作为一个特例。如果没有定量（渐近）结果证明算术群同调性中的挠率（至少有时）丰富，那么 Z 上的朗兰兹纲领将是空洞的。 Bergeron-Venkatesh 的工作中提出了一个令人信服的模型，可以准确预测哪些算术群在其同源性中应包含丰富的挠率。重要的支持证据已在同上以及 Marshall-Muller、Muller-Pfaff 等人的著作中得到证实。然而，目前对挠率增长的理解对于挠率和有理上同调共存的“权重”非常有限。此外，如果我们期望显式算术群的同调性中挠率丰富，那么我们当然应该能够“在计算机上看到它”。 - Z 上的互惠景观是“敞开的”并且充满了猜想。其中最密切的猜想是可证伪的，并且只要可能，似乎非常值得通过计算来确认它们。然而，计算算术群同源性的最先进的方法是临时的，并且由于算法效率问题而具有有限的范围。因此，设计算法来有效计算算术群的同源性（以及其上的赫克动作）是非常值得的。未来五年我的研究计划将围绕上述两个主题。即，它将研究：（A）有限体积局部对称空间族中拓扑不变量的增长，特别是同调中的挠率。 (B) 如何有效且高效地计算这些不变量。我在这些问题上最重要的进展：(A*)（一年前）与 Mark Stern 联合，我证明了双曲 3 流形上微小的 1 型拉普拉斯特征值，这是第一同源群中挠率增长的已知障碍，与短环“有效限制”的失败有关。(B*) （正在进行）与 Aurel Page 合作，我们设计了一种通用、高效的算法来计算同余拓扑、算术局部对称空间及其 Hecke 动作。算术群同调中的扭转是一个“热门话题”；在问题（A）和（B）上取得进展的时机已经成熟，并且这种进展将具有巨大的效用。 对我与斯特恩和佩奇的作品的思考，在（A*）中提到 (B*)，激发了我在本提案中提出的许多具体问题。