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，说明了扭转在算术群同调中的重要性。在与同余算术群相关的局部对称空间的同调上，Hecke算子的特征值系统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）结果证明了算术群同调中的扭转（至少有时）是丰富的，那么Z上的朗兰兹规划将是空洞的。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性证据。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，如果我们期望在显式算术群的同调中存在大量的扭转，那么我们当然应该能够“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，而且在可能的情况下，用计算来证实它们似乎是非常值得的。然而，计算算术群的同调性的最先进的方法是特别的，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的同调性（以及Hecke在其上的动作）是非常值得的。今后五年，我的研究计划将围绕以上两个主题展开。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是：（A*）（从一年前开始）与Mark Stern合作，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，我们设计了一种计算同余拓扑、算术局部对称空间及其Hecke作用的通用、高效算法。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，这种进展将具有很大的效用。（A*）和（B*）中提到的对我与Stern和Page合作的思考，启发了我在本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.