Overconvergent cohomology of higher rank groups

在过去的五年里，模形式的p-adic理论取得了巨大的进展。 斯金纳和厄本对主要猜想的证明，哈雷和温特伯格对塞尔猜想的证明，埃默顿和基辛对丰泰因-马祖猜想的独立工作，只是最近发生的令人难以置信的进展的一部分。 此外，有一个丰富的新兴的p-adic理论的自守形式推广相应的p-adic理论的模形式。 然而，大量的计算方法可用于研究经典的模形式，还不存在于更高的秩的情况下。 目前，在这种更一般的环境中出现的许多基本物体的形状和结构的理解方面存在差距。 Pollack提出要沿着他和Glenn Stevens在经典模形式的情况下所做的路线，对高秩群的过收敛上同调进行沿着具体而明确的研究。 以前的研究有理论和计算两个方面，特别是，导致更深入地了解两个变量的p-adic L-函数和p-adic方法计算椭圆曲线上的全局点。 他希望，在更高等级的情况下进行类似的研究，将有助于更好地理解这种更普遍的情况下的类似现象。一百多年来，数学家们一直在研究模形式，模形式是某些具有许多对称性的复解析函数。这些对象的定义是通过分析（微积分的严格研究）;尽管如此，这些对象已经进入了数论的中心，因为它们拥有丰富的算术信息，这些信息会使欧几里得，高斯，费马等感兴趣。 此外，自守形式理论--模形式的一个意义深远的推广--也被证明是极其丰富的算术。 然而，与可以相对容易地计算的经典模形式不同，一般自守形式可能非常难以计算和数值实验。 Pollack建议对某些自守形式的计算进行深入研究，希望通过猜想和数值探索对它们有更好的算术理解。

In the last five years, a tremendous amount of progress has been made in the p-adic theory of modular forms. Skinner and Urban's proof of the main conjecture, Khare and Winterberger's proof of Serre's conjecture, Emerton's and Kisin's independent work on the Fontaine-Mazur conjecture represent only a partial list of the incredible recent progress that has occurred. Moreover, there is a rich emerging p-adic theory of automorphic forms generalizing the corresponding p-adic theory of modular forms. However, the vast computational methods available in the study of classical modular forms, do not yet exist in the higher rank case. Currently, there is a gap in the understanding of the shape and structures of many of the fundamental objects that appear in this more general setting. Pollack proposes to make a concrete and explicit study of overconvergent cohomology of higher rank groups along the lines that he and Glenn Stevens did in the case of classical modular forms. This previous study had both theoretical and computational aspects that, in particular, led to a deeper understanding of two variable p-adic L-functions and to a p-adic method of computing (conjecturally) global points on elliptic curves. He hopes that a similar study in the higher rank case will lead to a better understanding of analogous phenomena in this more general setting. For over a hundred years, mathematicians have studied modular forms which are certain complex analytic functions possessing many symmetries. The definition of these objects is through analysis (the rigorous study of calculus); nonetheless, these objects have made their way into the center of number theory as they possess a wealth of arithmetic information of the type that would have interested Euclid, Gauss, Fermat, etc. A deep understanding of these arithmetic properties of modular forms was one of the key steps in Andrew Wiles' proof of Fermat's Last Theorem. Moreover, the theory of automorphic forms -- a far-reaching generalization of modular forms -- has proven to also be extremely rich arithmetically. However, unlike classical modular forms where one can compute with relative ease, general automorphic forms can be extremely difficult to compute and experiment with numerically. Pollack proposes to make an in depth study of the computation of certain kinds of automorphic forms with the hope of gaining a better arithmetic understanding of them through conjecture and numerical exploration.