Homology growth in families of locally symmetric spaces

Serre, Ash, Calegari-Venkatesh， Bergeron-Venkatesh和Scholze的工作描绘了一幅非常引人注目的图画，描绘了扭转在算术群同调中的意义。与同余算术群相关的局部对称空间的同调上的Hecke算子的特征值系统***现在被期望成为伽罗瓦群***为有限型李群的数域的普遍来源。这种期望被称为“朗兰兹Z计划”；它***包含了Serre关于奇数，二维模p伽罗瓦表示的模性的猜想，作为***一个特例。******如果不是定量的（渐近的）结果证明***算术群同调中的扭转（至少有时）是丰富的，Z上的朗兰兹程序将是空洞的。在Bergeron-Venkatesh的工作中，提出了一个令人信服的***模型，准确地预测了哪些算法群在其同调***中应该包含丰富的扭转；在Marshall-Muller， Muller-Pfaff和其他人的***作品中证明了重要的支持证据。然而，目前对扭转增长的理解，***在扭转和有理上同调共存的“权重”上是非常有限的。******此外，***-如果我们期望扭转在显式算术群的同调中是丰富的，那么当然***我们应该能够“在计算机上看到它”。***- Z的互惠景观是“开放的”，充满了猜测。其中最密切相关的猜想是可证伪的，并且似乎非常值得在可能的情况下通过计算来证实它们。******计算算术群的同调性的最先进的方法，然而，是特别的和***有有限的范围由于算法效率的问题。因此，设计算法来有效地计算算法群的同调性（以及Hecke在其上的动作）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***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***扭转。(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*)（从一年前开始）与Mark Stern联合，我证明了双曲3流形上的微小1-形式拉普拉斯特征值，这是已知的第一个同调群中扭转增长的障碍，与短环的失败有关。***(B*)（正在进行中）结合Aurel Page，我们设计了一个通用的、高效的算法来计算同余拓扑、算法局部对称空间及其上的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，而且这种进展将具有巨大的效用。（A*）和（B*）中提到的我与Stern **和Page的作品，启发了我在***提案中提出的许多具体问题。

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.