Dualizing modules in algebra and geometry

PI建议研究数论和代数几何中同伦对偶模的存在性及其含义。具体地说，将有两个这样的对偶化模块的有价值的实例。其中一个将与Tomer Schlank共同研究；它将给出Tate-Poitou的算术对偶定理的同伦推广，该定理指出某些绝对Galois群的上同调具有自对偶的扭曲形式。PI和Tomer Schlank还将研究这种对偶结果在代数簇上有理点存在问题上的应用。一个不同的例子涉及具有或不具有能级结构的拓扑模形式的谱。PI以前的工作表明，广义椭圆曲线的谱导出模堆叠具有简单可描述的对偶化交换环谱；所提出的项目的一个目标是证明这一结果。拓扑模形式对于理解球谱中的v2周期同伦至关重要；尽管PI与对偶主题略有偏离，但它将与Mark Behrens、Kyle Ormsby和Nathaniel Stapleton合作计算基于连通拓扑模形式的同调中的合作。二元性在数学中是一个普遍存在的概念；拟议的项目将在统一的框架内研究不同类型的二元性，从而得出新的和以其他方式无法获得的信息。特别是，该项目的动机是引入同伦观点可以为我们对几何和算术的理解提供新的启示。其中一个具有奇特的自对偶性质的对象，即拓扑模形式，适合于广泛的应用，一些已经被探索过，另一些还没有被探索过，因为它在同伦、代数几何、数论，甚至是量子场论中反映了自己作为Witten亏格的容器。

The PI proposes to investigate the existence of homotopical dualizing modules in number theory and algebraic geometry and the implications thereof. Specifically, there will be two valuable instances of such dualizing modules. One will be studied in joint work with Tomer Schlank; it will give a homotopical extension of the arithmetic duality theorems of Tate-Poitou, which state that the cohomology of certain absolute Galois groups has a twisted form of self-duality. The PI and Tomer Schlank will also investigate applications of such duality results to problems of existence of rational points on algebraic varieties. A different example is related to the spectra of topological modular forms with or without level structures. Previous work of the PI suggests that the spectrally derived moduli stack of generalized elliptic curves has a simply describable dualizing sheaf of commutative ring spectra; an objective of the proposed project is to prove that result. Topological modular forms are crucial for understanding v2 periodic homotopy in the sphere spectrum; though somewhat removed from the theme of duality, the PI will collaborate with Mark Behrens, Kyle Ormsby, and Nathaniel Stapleton to compute the cooperations in the homology based on connective topological modular forms. Duality is a pervasive concept in mathematics; the proposed project will study different types of duality in a unified framework, thereby arriving at novel and otherwise inaccessible information. In particular, the project is motivated by the idea that introducing a homotopical viewpoint can shed new light on our understanding of geometry and arithmetic. One of the objects with such curious self-duality properties, namely topological modular forms, lends itself to vast applications, some already explored and others not, as it mirrors itself in homotopy, algebraic geometry, number theory, and even quantum field theory as the receptacle of the Witten genus.