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，说明了扭转在算术群同调中的重要性。作用在与同余*算术群有关的局部对称空间同调上的Hecke算子的特征值系*现在被期望成为其Galois群*是有限类型李群的数域的通用来源。这一期望被称为“Z上的朗兰兹程序”；它*包括了Serre关于奇数二维mod p Galois表示的模性的猜想作为*的一个特例。如果不是定量的(渐近的)结果证明*算术群的同调中的挠率(至少有时)是充分的，那么Z上的朗兰兹程序将是空的。Bergeron-Venkatesh的工作中提出了一个令人信服的*模型，它准确地预测了哪些算术群的同调中应该包含大量的挠率*；重要的支持证据在同上以及马歇尔-穆勒、穆勒-普法夫和其他人的*工作中得到了证明。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”方面是非常有限的。*此外，*-如果我们期望显式算术群的同调中有丰富的挠率，那么我们肯定*我们应该能够“在计算机上看到它”。*-Z上的互惠图景是“完全开放的”，充满了猜想。其中最重要的*猜想是可证伪的，而且在可能的情况下，似乎非常值得在计算上确认它们。*计算算术群的同调的最新方法是特别的，并且*由于算法效率问题*范围有限。因此，设计算法来有效地计算算术群的同调(以及在其上的Hecke作用)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。也就是说，它将研究：*(A)有限体积局部对称空间族中的拓扑不变量的增长性，特别是同调中的*挠性。*(B)如何有效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*)(一年前)与Mark Stern合作，我证明了双曲三维流形上微小的1-形式Laplace特征值，一个已知的阻碍第一同调群中挠率增长的障碍，与短环不是`有效有界的有关。*(B*)(正在进行)与Aurel Page合作，我们设计了一个通用、高效的算法来计算同余的拓扑、算术局部对称空间和在其上的Hecke作用。*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，这样的进展将具有很大的实用价值。我在(A*)和(B*)中提到的对我与斯特恩*和佩奇的作品的沉思启发了我在*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.