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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=657706
• 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的作品绘制了非常引人注目的***图片，描述了扭转在算术组同源性中的重要性。针对与一致性相关的本地对称空间同源的Hecke运算符的特征值的系统*** ***算术群现在预计将是数字字段的通用来源。这种期望被称为“ Z上的Langlands计划”； IT ***包括Serre对奇数，二维mod p galois表示的模块化的猜想，为***一种特殊情况。 ******如果不是定量（渐近）结果，Z上的Langlands计划将是空置的，证明***算术组的同源性扭转（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该在其同源性中包含丰富的扭转***在Bergeron-Venkatesh的工作中布置；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。 ****** *** - *** - 如果我们期望在显式算术组的同源性中扭曲丰富，那么***肯定应该能够“在计算机上看到它”。 *** - Z上互惠的景观是“宽开”的，充满了猜想。其中最隐秘的猜想是可伪造的，而且似乎非常值得在计算上确认它们，如果可能的可能。因此，设计算法有效地计算***算术群的同源性（和Hecke Actions）非常值得。 ************我的研究计划在接下来的五年中，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的拓扑不变性的增长，尤其是同源性的***扭转。 ***（b）如何有效，有效地计算这些不变的。 “ ***（b*）（持续）与Aurel页面的联合，我们设计了一种一般，有效的算法来计算一致性，算术在本地对称的空间和Hecke的拓扑结构。我与Stern ***和Page的合作，在（A*）和（B*）中提到，启发了我在***目前的建议中提出的许多具体问题。