Orientations and p-Adic Analysis

The Ando-Hopkins-Rezk orientation is surjective

We show that the map $π_\ast \mathrm{MString}\to π_\ast \mathrm{tmf}$ induced by the Ando-Hopkins-Rezk orientation is surjective. This proves an unpublished claim of Hopkins and Mahowald. We do so by constructing an $\mathbf{E}_1$-ring $B$ and a map $B \to \mathrm{MString}$ such that the composite $B \to \mathrm{MString} \to \mathrm{tmf}$ is surjective on homotopy. Applications to differential topology, and in particular to Hirzebruch's prize question, are discussed.

Strictly Commutative Complex Orientations of $Tmf_1(N)$

We construct by adapting methods and results of Ando, Hopkins, Rezkand Wilson combined with results of Hopkins and Lawson strictly commu-tative complex orientations for the spectra of topological modular forms with level $Γ_1(N)$.

通过 Coleman 范数算子证明 Ando 关于范数相干坐标的定理

A proof for Ando's theorem on norm-coherent coordinates via the Coleman norm operator

本文利用 Coleman 范数算子，通过形式群律连接拓扑与算术，为 Ando 关于 Morava E-理论复定向的范数相干坐标定理提供了概念性证明。

Weierstrass椭圆函数迭代导数中的p进同余关系

p-adic congruences in iterated derivatives of the Weierstrass elliptic function

本研究反向运用Ando–Hopkins–Rezk的$\mathbb E_\infty$定向机制，结合Tate不动点对象的定向存在性结果，证明Weierstrass椭圆函数迭代导数间的p进Kümmer同余。

Orientations and Topological Modular Forms with Level Structure

Using the methods of Ando-Hopkins-Rezk, we describe the characteristic series arising from E-infinity genera valued in topological modular forms with level structure. We give examples of such series for tmf_0(N) and show that the Ochanine genus comes from an E-infinity ring map. We also show that, away from 6, certain tmf orientations of String cobordism descend to orientations of Spin cobordism.

Simultaneous Kummer congruences and $\mathbb{E}_\infty$-orientations of KO and tmf

Building on results of M. Ando, M.J. Hopkins and C. Rezk, we show the existence of uncountably many $\mathbb{E}_\infty$-String orientations of real K-theory KO and of topological modular forms tmf, generalizing the $\hat{A}$- (resp. the Witten) genus. Furthermore, the obstruction to lifting an $\mathbb{E}_\infty$-String orientations from KO to tmf is identified with a classical Iwasawa-theoretic condition. The common key to all these results is a precise understanding of the classical Kummer congruences, imposed for all primes simultaneously. This result is of independent arithmetic interest.