# Initiative and Materiality: Exploring Mixed-Initiative Calculators
with the Tangible Human-A.I. Interaction Framework

Lunshi Zhou  Cornell UniversityIthaca, NYUSA<EMAIL_ADDRESS>,  Alexandra
Bremers  Cornell TechNew York, NYUSA<EMAIL_ADDRESS>and  Wendy Ju
0000-0002-3119-611X Cornell TechNew York, NYUSA<EMAIL_ADDRESS>
(Date: 2024)

###### Abstract.

How can interactions with A.I. systems be designed? This paper explores the
design space for A.I. interaction to develop tools for designers to think
about tangible and physical A.I. interactions. Our proposed framework consists
of two dimensions: initiative (human, mixed, or machine) and materiality
(physical, combined, or digital form). A particularly interesting area of
interactions we identify is the quadrant of physical, machine-initiated
interactions. With our framework, we examine calculator interactions and
attempt to expand these to the tangible, mixed-initiative space. We illustrate
each area in our proposed framework with one representative example of a
calculator – a common and well-known example of a computing device. We discuss
existing examples of calculators and speculative future interactions with
mixed-initiative and physical calculator systems. We reflect on the
implications of our framework for the larger task of designing human-A.I.
collaborative systems. Designers can also apply this framework as a guideline
for analogous solutions to problems in the same domain.

human-A.I. collaboration; design space

##  1\. Introduction

Figure 1. Schematic diagram of the Tangible Human-A.I. Interaction framework

Artificial Intelligence (A.I.) is a manufactured system that uses computations
to perceive information and take actions to achieve goals (Russell and Norvig,
2003; Poole et al., 1998). A.I. systems are usually good at specific tasks,
yet they can struggle with complex environments – designing smooth
interactions with A.I. is still a challenge (Amershi et al., 2019; Blackwell,
2015). A.I. applications can be either wholly digital or involve a physical
interaction component, such as when computation is embodied in a robot. From
years of study in tangible and embodied interaction, it is known that
materiality influences interaction on many levels (Hornecker, 2011). For
instance, when robots are collaborative, influence, order, and space must be
negotiated. Some recent works have investigated the potential for non-
anthropomorphic robots to communicate initiative through the movement of their
physical body (Ariccia et al., 2022; Grinberg et al., 2023).

In this late-breaking report, we approach how A.I. systems could be designed
by developing a framework that can serve as a design tool (Figure 1). Our
proposed framework spans the two dimensions of materiality and initiative.
Initiative and materiality highlight this topic’s relevance to human-robot
interaction. We explore and illustrate the framework by reflecting on the
design of a specific, extensively developed type of computational application,
namely calculators. Calculators are the foundations or one of the most well-
studied manifestations of computing systems, and they provide a concrete
example to explore the more abstract design space of future A.I. systems. The
examples of calculators mentioned in the framework are not intended to provide
a systematic literature review but to illustrate the design space’s utility.
This paper contributes a framework to examine and design human-A.I.
interactions that integrate initiative and materiality.

##  2\. Related Work

###  2.1. What are design spaces?

The theoretical concept of a ”design space” was first introduced to the realm
of design research by Herbert Simon’s Sciences of the Artificial, when he
mapped the computer science notion of a problem or search space, for example,
of all the moves that can be made in a chess game. This notion was adopted
rather literally by researchers incorporating A.I. to perform design space
exploration for computer-aided design in the 1990s (Esbensen and Kuh, 1996,
1996; Moya et al., 1999). In HCI and HRI, the notion of design space was
interpreted more abstractly, for example, as the space of possibility and
rationale embodied in early design prototypes that go beyond the mere
specification or artifact or the realm of questions, options, and criteria
being considered to approach a design problem (MacLean et al., 1991; Bellotti
et al., 1995). Nowadays, the conceptualization of design space is more
metaphorical. Gaver, for example, describes ”the notion of a ’design space’”
as a metaphor for the designer’s perception of possibility. He argues, ”Design
creates the spaces in which it operates. They do not pre-exist their
manifestation, whether as undiscovered design ideas or as the abstract and
rationalized parameter spaces that some of those who follow Simon might
suggest” (Gaver, 2011).

###  2.2. Frameworks for the design of tangible interactions with A.I.

Various relevant frameworks have been proposed to consider the design of
interactions. For instance, Ju and Leifer (2008) presented the Implicit
Interaction Framework which considers the dimensions of initiative and
attentional demand; Hornecker and Buur (2006) presented a framework that
integrates four different perspectives on designing tangible interactions,
including a focus on social aspects of such interactions; Mazalek and Van Den
Hoven (2009) provides a meta-analysis of the frameworks that have been
designed around the topic of tangible interactions and conclude that
frameworks tend to focus on design and have less focus on knowledge
abstraction or the technical development of interactive applications. Some
prior work has looked explicitly into the promise of physicality for making
A.I. more ”graspable”, using the form as a modality for explainability
(Ghajargar et al., 2021, 2022).

###  2.3. A design space for A.I. interactions

We believe there are numerous aspects of human-A.I. interaction where
differences in the initiative, embodiment, and other interaction parameters
create different genres of interaction, which generalize across applications.
For example, in the space of human-autonomous vehicle interaction, the sensing
and prediction capabilities of an ”intelligent vehicle” could be used to act
as a chauffeur for people, or it could be used to augment the driver’s
capability as a guardian, as illustrated in (Hodson, 2016). The
Chauffeur/Guardian dichotomy can be useful for people working far outside
vehicle interaction. To this end, our exploration of interactions with
calculators is intended to be a foundation for potential directions of the
development of human A.I. interactive systems and a way to make the abstract
design space of A.I. systems more concrete.

##  3\. The Tangible Human-A.I. Interaction Framework

###  3.1. Framework dimensions

Figure 1 shows two dimensions that we consider for human-A.I. interaction:

  * •

Initiative (human-initiated/machine-initiated): Does initiative lie with the
human, the machine, or with both? If a human uses a typical calculator to get
an answer to their mathematical question, the interaction is human-initiated.
However, if we consider an intelligent program that can test users’ progress
in math, then the interaction is machine-initiated. Moreover, a third category
exists of mixed-initiative interactions, where the users and machine
collaborate (Horvitz, 1999). In our framework, we thus divide the vertical
dimension into machine-initiated, mixed-initiative, and human-initiated
interactions.

  * •

Materiality (physical/digital): Some calculators are totally programmed to run
on computers, like WolframAlpha, which we would classify as digital. On the
other hand, ancient calculators that do math computations purely on mechanical
parts would be classified as physical. Here again, the dimension covers a
spectrum between the two extremes. Many calculators have both mechanical parts
and digital parts. Therefore, we also divide the horizontal into three areas:
from left to right, purely digital, combined digital and physical, and purely
physical interactions.

Figure 2. Schematic diagram of the Tangible Human-A.I. Interaction framework
when applied to the case of calculators, with an example given in each grid
space. Red fonts (top row) indicate three speculative examples.

###  3.2. Example calculator interactions

To better understand the range of the framework, we pick up existing examples
and discuss people’s interactions with these calculators. Figure 2 shows
examples we choose in each domain:

  * •

Abacus (physical & human-initiated): The abacus is an ancient calculator in
human history that emerged before the adoption of the Hindu-Arabic numerical
system (Boyer, 1991). Figure 3 shows that it usually looks like a frame with
beans and sticks connecting beams. This falls into the domain of physical,
human-initiated interaction. People have developed many tricks for using the
abacus as a supplementary tool to increase calculation speed.

1,2 Image source: Wikimedia

Figure 3. Calculating using abacus1, left, Texas Instruments Calculator2,
right

  * •

Texas Instruments standard calculator (combined materiality & human-
initiated):

Texas Instruments calculators are representative of classic mechanical
calculators with physical keys to input math symbols and a screen to
demonstrate results (see Figure 3). Hence, its materiality has a combination
of physical and digital parts. The calculator heavily depends on the user’s
input and does not take initiative independently. People mainly use such a
category of calculators as an essential tool to support complex calculations,
like multi-digit multiplications, that are time-consuming or require great
precision. It appears in a wide range of engineering scenarios to provide help
with calculations.

  * •

WolframAlpha web calculator (digital & human-initiated):

WolframAlpha is a web-based calculating engine that provides solutions to a
wide range of Math topics (see Figure 4)(Wol, [n. d.]). Users mainly use it
through web browsers. The users can input math symbols through the mouse by
clicking the desired functionalities listed on the WolframAlpha page. It can
demonstrate answers and steps to reach answers on the computer. Therefore,
such a calculator is categorized as digital and mixed-initiative. People
mainly use WolframAlpha as a tool to do complicated calculations beyond the
scope of mechanical calculators and default calculators in computers.

3Image source: www.wolframalpha.com

Figure 4. WolframAlpha Main Page3

4Image reproduction permission granted by Wellner

Figure 5. Digital Desk by Pierre Wellner4(Wellner, 1991)

  * •

Siri (digital & mixed-initiative): Siri is a virtual assistant developed by
Apple that can perform calculations based on spoken inputs, with the
possibility to ask clarifying follow-up questions (Sir, [n. d.]). For example,
a user can ask, ”What is 1 plus 1?” and Siri will answer, ”The answer is 2”.
Siri can also perform complicated calculations by using internet resources.
People can use Siri when they have a hard time using their hands to access
other calculators, such as when a mechanic is busy with their hands but needs
to do quick math for a price quote. Since the user needs to give verbal
instructions for the calculations and Siri can perform them independently, we
classify it as an example of a mixed-initiative interaction. As Siri is purely
digital, as software is installed in Apple products, we say it is a digital
and mixed-initiative calculator.

  * •

Pierre Wellner’s digital desk (combined materiality & mixed-initiative):
Pierre Wellner developed a prototype of a DigitalDesk that allows users to
interact with paper and electronic objects by touching them with a bare finger
on a physical desk equipped with a computer-controlled camera and projector
above it (see Figure 5). (Wellner, 1991). Since this prototype allows users to
interact directly with physical and digital materials, we classify it as a
combined format. And because of its ability to recognize physical
manipulations and produce corresponding digital changes under bare-finger
manipulations, we classify it as a mixed initiative.

  * •

QAMA (physical & mixed-initiative):

QAMA is a physical calculator that shows the answer only if a user provides a
precise enough estimate (QAM, [n. d.]). Its target is students who are
studying math concepts, aiming to help students better understand math
concepts through mental estimates. QAMA uses algorithms to enable the
calculator to decide which estimations are precise enough and hence, evaluate
whether the student masters the concept. Based on this, QAMA is a mixed-
initiative calculator with a physical body since it still requires users to
input equations actively.

##  4\. Discussion

###  4.1. Evaluating the framework

Frameworks are like models–as the saying goes, all models are wrong, and some
models are useful. The evaluation of our framework, then, should be based on
its utility. The exploration of calculators in the framework serves as an
example of the application of other forms of computing systems. Since the
dimensions considered in this framework are pertinent to all forms of A.I., we
believe the framework could inspire designers to find analogous solutions
across different A.I. systems. For example, there may be some common ground in
interactions between an A.I. program that teaches people to drive in virtual
reality and an A.I. app that teaches students math. There can be shared
features across such A.I. systems, and designers can look for inspirations or
solutions across different systems in the same framework domain. In the next
stage of this project, we are looking to refine and evaluate our framework by
including interview data from speaking to users and practitioners in the space
of A.I. system design, using our framework and certain specific instances of
its application as design probes.

###  4.2. Future Work: Trying out three modes of mixed-initiative and machine-
initiated calculators with Wizard-of-Oz

Through our framework development and exploration of the design space of
calculator interactions, we included six examples of interactions that are
already fully developed products or prototypes. The domains of digital,
physical, and human-initiated interactions seem to offer plenty of examples of
interactions that have been investigated and fine-tuned for decades.

Importantly, we identified key areas of the framework that are largely gaps in
human-A.I. interactions, namely machine-initiated and mixed-initiative
interactions, particularly when physical materiality is involved. Since there
are no existing examples yet in some spaces, we plan to explore these areas
through a case study in which we will design mixed-initiative interactions
with a calculator. We will consider three designs that cover the three types
of materiality in our framework:

  1. (1)

Digital: Such a calculator can have the form of an app installed on computers
or smartphones. It can have an A.I.-controlled program that reminds students
to do math based on their schedules and sets the difficulty of the questions
based on previous answers of students.

  2. (2)

Combined materiality: This kind of calculator can have a combined format of
physical and digital content. We propose a physical calculator equipped with
screens and cameras, which enables it to read students’ facial expressions to
determine whether they are confused about questions it asks students to
compute. If so, it will demonstrate detailed calculation steps and guide
students in following them. If not, it will simply show answers.

  3. (3)

Physical: This type particularly emphasizes the physical component. Based on
this principle, we propose a self-moving abacus with a screen to demonstrate
some calculation techniques and can self-move beans to simulate student
situations.

We will evaluate the three calculator interaction concepts through a user
study with 18 participants, where users are video-recorded. At the same time,
they perform calculation tasks with the calculators in counter-balanced order.
The participants consist of designers as well as users of computer systems.
After the task, the users will be asked to complete quantitative
questionnaires to rank the various calculator designs. They will participate
in a semi-structured qualitative interview where they are asked about their
experience and how it makes them think about using or designing for A.I.
systems. For setting the interaction, we plan to rely on interaction
prototyping and Wizard-of-Oz systems, such as described in (Dahlbäck et al.,
1993; Hoffman and Ju, 2014; Riek, 2012).

##  5\. Conclusion

In this late-breaking report, we explore human-A.I. interaction through a
proposed framework of two dimensions: initiative and materiality. These are
pertinent elements across the design of different A.I. systems. This paper
further provides illustrative examples of calculators for six areas that have
already been well-developed with existing calculators and identifies areas of
the framework with no existing calculator examples, notably machine-initiated
or mixed-initiative interactions. Future work will investigate the application
of the framework to concrete systems tested with users and practitioners to
understand the value such a framework could offer designers. With our
illustrated framework, we hope designers can look for analogous solutions or
gain inspiration across different A.I. systems in the same domain.

##  6\. Acknowledgements

We would like to thank members of the Cornell Tech research community for
their suggestions and ideas regarding this project.

## References

  * (1)
  * Wol ([n. d.]) Wolfram Alpha LLC [n. d.].  _Frequently Asked Questions_.  Wolfram Alpha LLC.  https://www.wolframalpha.com/faqs/
  * QAM ([n. d.]) QAMA LLC [n. d.].  _QAMA Calculator_.  QAMA LLC.  http://qamacalculator.com/
  * Sir ([n. d.]) Apple Inc [n. d.].  _Siri_.  Apple Inc.  https://www.apple.com/siri/
  * Amershi et al. (2019) Saleema Amershi, Dan Weld, Mihaela Vorvoreanu, Adam Fourney, Besmira Nushi, Penny Collisson, Jina Suh, Shamsi Iqbal, Paul N Bennett, Kori Inkpen, et al. 2019\.  Guidelines for human-AI interaction. In _Proceedings of the 2019 chi conference on human factors in computing systems_. 1–13.
  * Ariccia et al. (2022) Avital Dell’ Ariccia, Alexandra Bremers, Johan Michalove, and Wendy Ju. 2022.  How to Make People Think You’re Thinking if You’re a Drawing Robot: Expressing Emotions Through the Motions of Writing. In _2022 17th ACM/IEEE International Conference on Human-Robot Interaction (HRI)_. IEEE, 1190–1191.
  * Bellotti et al. (1995) Victoria Bellotti, Simon Buckingham Shum, Allan MacLean, and Nick Hammond. 1995.  Multidisciplinary Modelling in HCI Design… in theory and in practice. In _Proceedings of the SIGCHI conference on Human factors in computing systems_. 146–153.
  * Blackwell (2015) Alan F Blackwell. 2015\.  Interacting with an inferred world: The challenge of machine learning for humane computer interaction. In _Proceedings of The Fifth Decennial Aarhus Conference on Critical Alternatives_. 169–180.
  * Boyer (1991) Uta C Boyer, Carl B.; Merzbach. 1991\.  _A History of Mathematics (2nd ed)_.  252 – 253.
  * Dahlbäck et al. (1993) Nils Dahlbäck, Arne Jönsson, and Lars Ahrenberg. 1993\.  Wizard of Oz studies—why and how.  _Knowledge-based systems_ 6, 4 (1993), 258–266.
  * Esbensen and Kuh (1996) Henrik Esbensen and Ernest S Kuh. 1996.  Design space exploration using the genetic algorithm. In _1996 IEEE International Symposium on Circuits and Systems (ISCAS)_ , Vol. 4. IEEE, 500–503.
  * Gaver (2011) William Gaver. 2011\.  Making spaces: how design workbooks work. In _Proceedings of the SIGCHI conference on human factors in computing systems_. 1551–1560.
  * Ghajargar et al. (2021) Maliheh Ghajargar, Jeffrey Bardzell, Alison Smith Renner, Peter Gall Krogh, Kristina Höök, David Cuartielles, Laurens Boer, and Mikael Wiberg. 2021.  From” explainable ai” to” graspable ai”. In _Proceedings of the Fifteenth International Conference on Tangible, Embedded, and Embodied Interaction_. 1–4.
  * Ghajargar et al. (2022) Maliheh Ghajargar, Jeffrey Bardzell, Alison Marie Smith-Renner, Kristina Höök, and Peter Gall Krogh. 2022.  Graspable AI: Physical forms as explanation modality for explainable AI. In _Sixteenth International Conference on Tangible, Embedded, and Embodied Interaction_. 1–4.
  * Grinberg et al. (2023) Itay Grinberg, Alexandra Bremers, Louisa Pancoast, and Wendy Ju. 2023.  Implicit collaboration with a drawing machine through dance movements.  _arXiv preprint arXiv:2310.00215_ (2023).
  * Hodson (2016) Hal Hodson. 2016\.  I am your guardian angel.  _New Scientist_ 230 (May 2016), 22–23.  Issue 3073.
  * Hoffman and Ju (2014) Guy Hoffman and Wendy Ju. 2014.  Designing Robots with Movement in Mind.  _J. Hum.-Robot Interact._ 3, 1 (feb 2014), 91–122.  https://doi.org/10.5898/JHRI.3.1.Hoffman
  * Hornecker (2011) Eva Hornecker. 2011\.  The Role of Physicality in Tangible and Embodied Interactions. In _Interaction, Volume 18, Issue 2_. 19 – 23.
  * Hornecker and Buur (2006) Eva Hornecker and Jacob Buur. 2006.  Getting a grip on tangible interaction: a framework on physical space and social interaction. In _Proceedings of the SIGCHI conference on Human Factors in computing systems_. 437–446.
  * Horvitz (1999) Eric Horvitz. 1999\.  Principles of Mixed-Initiative User Interfaces. In _CHI ’99: Proceedings of the SIGCHI conference on Human Factors in Computing Systems_. 159 – 166.
  * Ju and Leifer (2008) Wendy Ju and Larry Leifer. 2008.  The design of implicit interactions: Making interactive systems less obnoxious.  _Design Issues_ 24, 3 (2008), 72–84.
  * MacLean et al. (1991) Allan MacLean, Richard M Young, Victoria ME Bellotti, and Thomas P Moran. 1991.  Questions, options, and criteria: Elements of design space analysis.  6 (1991), 201–250.  Issue 3&4\.
  * Mazalek and Van Den Hoven (2009) Ali Mazalek and Elise Van Den Hoven. 2009.  Framing tangible interaction frameworks.  _AI EDAM_ 23, 3 (2009), 225–235.
  * Moya et al. (1999) Francisco Moya, José M Moya, and Juan C López. 1999\.  Evaluation of design space exploration strategies. In _Proceedings 25th EUROMICRO Conference. Informatics: Theory and Practice for the New Millennium_ , Vol. 1. IEEE, 472–476.
  * Poole et al. (1998) David Poole, Alan Mackworth, and Randy Goebel. 1998\.  _Computational Intelligence and Knowledge_.  Oxford University Press, 1.
  * Riek (2012) Laurel D Riek. 2012\.  Wizard of oz studies in hri: a systematic review and new reporting guidelines.  _Journal of Human-Robot Interaction_ 1, 1 (2012), 119–136.
  * Russell and Norvig (2003) Stuart J. Russell and Peter Norvig. 2003.  _Artificial Intelligence: A Modern Approach (2nd ed)_.  Prentice Hall, 55.
  * Wellner (1991) Pierre Wellner. 1991\.  The DigitalDesk Calculator: Tangible Manipulation on a Desk Top Display. In _Proceedings of the 4th annual ACM symposium on User interface software and technology_. 27 – 33.


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

Hongxiang Zhao

###### Abstract

Ando established an algebraic criterion for when a complex orientation for a
Morava E-theory is an $H_{\infty}$-map. The criterion relates such an
orientation to a specific property of the formal group associated to the
E-theory, namely, a norm coherence condition on its coordinate. On the other
hand, Coleman constructed a norm operator for interpolating division values in
local fields, which depends on a Lubin–Tate formal group law. These formal
group laws are important tools in explicit local class field theory.

In this article, we give a conceptual proof for Ando’s theorem using the
Coleman norm operator via the bridge of formal group laws between topology and
arithmetic.

###### Contents

  1. 1 Introduction
  2. 2 Coleman norm operators from explicit local class field theory after Lubin and Tate
  3. 3 Proof of Ando’s theorem in a special case
  4. 4 Norm coherence condition
  5. 5 Generalization of the norm operators
  6. References

##  1 Introduction

The purpose of this article is to supply a number-theoretic, conceptual proof
for an old, topological theorem of Matthew Ando’s from his study of structured
ring spectra and their multiplicative operations. We follow a suggestion by
Charles Rezk and Yifei Zhu (cf. [Zhu20, Remark 1.3]).

  * •

We identify Ando’s algebraic criterion for $H_{\infty}$ complex orientations
of Morava E-theories as a coherence condition for certain division values in
local fields with respect to a Coleman norm operator.

  * •

We then coherently construct these division values by an infinite iteration of
the operator.

In a concrete way, the ingredients going into the proof hint towards a higher-
algebraic theory for class fields (c.f., e.g., [BSY22]).

Throughout, $k$ denotes a perfect field of characteristic $p>0$, $R$ denotes a
complete local ring with maximal ideal $\mathfrak{m}$ and whose residue field
$R/\mathfrak{m}$ contains $k$. Given a formal group law $F\in R\llbracket
X,Y\rrbracket$, we will write $X+_{F}Y:=F(X,Y)$.

Morava E-theories $E_{n}$ are complex oriented cohomology theories in stable
homotopy theory, whose coefficient ring $(E_{n})_{*}$ in degree $0$ classifies
deformations of a $1$-dimensional formal group law $G$ of height $n$ over $k$
to $R$. Each E-theory is represented by an $E_{\infty}$-spectrum [GH04,
Corollary 7.6] and carries power operations associated to this multiplicative
structure. Let ${\rm MU}$ be the complex cobordism theory. It is well-known
that ${\rm MU}$ also admits power operations ([tD68] and [May77, §IV.2]). A
complex orientation on $E_{n}$ is a map ${\rm MU}\to E_{n}$ of homotopy
commutative ring spectra. Upon taking the connective cover ${\rm MU}\langle
0\rangle$, this is equivalent to a coordinate (i.e., local uniformizer) of the
universal deformation formal group associated with $E_{n}$, which in turn
determines a formal group law [Zhu20, Section 2]. In the case that $G$ is a
Honda formal group law over $\mathbb{F}_{p}$, Ando gave a criterion for when
power operations on ${\rm MU}$ and $E_{n}$ are compatible along the map ${\rm
MU}\to E_{n}$, i.e., when this map is $H_{\infty}$ [And95, Theorem 5]. The
criterion is formulated in terms of the formal group law $F$ of the universal
deformation associated to the coordinate.

######  Theorem 1.1 ([And95, Theorem 4]).

Suppose $k=\mathbb{F}_{p}$ and $G$ is the Honda formal group law of height $n$
over $k$, so that $[p]_{G}(T)=T^{p^{n}}$, where $[p]_{G}$ is the $p$-series of
$G$. Then in each $\star$-isomorphism class of lifts of $G$ to
$\pi_{0}(E_{n})\cong W(k)\llbracket u_{1},\cdots,u_{n-1}\rrbracket$, there is
a unique lift $F$ such that

$[p]_{F}(T)=\prod_{\lambda}(T+_{F}\lambda)$

where the product runs over all roots $\lambda$ of $[p]_{F}$.

Such a coordinate is said to be norm-coherent, in that the right-hand side has
the form of a norm map. The condition says that the canonical lift of
Frobenius, which is just the map of multiplication by $p$ corresponding to the
$p$-series in this case, coincides with the norm map. For a detailed
discussion about the condition, see Section 4 and [Zhu20, Section 6].

On the number-theoretic side, an important tool invented in Lubin and Tate’s
explicit construction of the local Artin map in local class field theory is
the family of Lubin–Tate formal group laws. Suppose a prime number $p$ is a
uniformizer of a local field $K$, i.e., $K$ is an unramified extension of
$\mathbb{Q}_{p}$. Then a Lubin–Tate formal group law reduces to a Honda formal
group law over the residue field. In 1979, Coleman proved an interpolation
theorem on division values in local fields, with applications to $p$-adic
$L$-functions and modular units [Col79, Theorem A]. For that, he constructed a
norm operator $\mathscr{N}_{F}$ depending on a Lubin–Tate formal group law $F$
such that

$\mathscr{N}_{F}(g)\circ[p]_{F}(T)=\prod_{\\{\lambda\colon[p]_{F}(\lambda)=0\\}}g(T+_{F}\lambda)$

Rezk conjectured that the Coleman norm and Ando’s algebraic criteria were
closely related. Here, we prove Theorem 1.1 via the norm operators
$\mathscr{N}_{F}$.

In more detail, the original definition of a norm operator restricts to the
special case when $R$ is a complete discrete valuation ring with uniformizer
$p$. We shall first give a description for Coleman’s norm operators and prove
Ando’s theorem in this special case. Our proof will only depend on several
properties of the norm operators and do not require $G$ to be a Honda formal
group law. In view of this, we will generalize the definition of a norm
operator to complete local domains in which $p\neq 0$, and show that the
generalized norm operator satisfies the desired properties. In particular,
$\pi_{0}(E_{n})$ is a complete local domain with $p\neq 0$. The main result of
this article is the following.

######  Theorem 1.2.

Suppose $G$ is a formal group law of finite height over $k$ and $R$ is a
complete local domain with $p\neq 0$ whose residue field contains $k$. Then in
each $\star$-isomorphism class of lifts of $G$ to $R$, there is a unique lift
whose corresponding coordinate is norm-coherent.

A more precise and explicit formulation of Theorem 1.2 will be given as
Theorem 4.8.

######  Remark 1.3.

Zhu generalized the above theorem to apply to arbitrary complete local rings
$R$ following the original proof by Ando [Zhu20, Theorem 1.2]. It would be
interesting to have an alternative approach from number-theoretic
constructions in this generality.

##  2 Coleman norm operators from explicit local class field theory after
Lubin and Tate

In this section, suppose that $k=\mathbb{F}_{q}$ with $q=p^{n}$. Suppose $K$
is a local field with integer ring $\mathcal{O}_{K}$, maximal ideal
$\mathfrak{m}$, and residue field $k$. Pick a uniformizer $\pi$ of
$\mathcal{O}_{K}$ and let

$\mathscr{F}_{\pi}:=\\{\alpha(T)\in\mathcal{O}_{K}\llbracket
T\rrbracket\colon\alpha(T)\equiv\pi T\mod{T^{2}},\ \ \alpha(T)\equiv
T^{q}\mod{\pi}\\}$ (1)

In explicit local class field theory, we have the following.

######  Proposition 2.1.

For any $\alpha\in\mathscr{F}_{\pi}$, there is a unique formal group law
$F_{\alpha}$ over $\mathcal{O}_{K}$ such that $\alpha\in{\rm
End}(F_{\alpha})$.

###### Proof.

See [Mil20, I, 2.12]. ∎

######  Proposition 2.2.

  * (a)

For any $\alpha,\beta\in\mathscr{F}_{\pi}$ and $a\in\mathcal{O}_{K}$, there is
a unique element $[a]_{\beta,\alpha}(T)$ in $T\mathcal{O}_{K}\llbracket
T\rrbracket$ such that $[a]_{\beta,\alpha}(T)\equiv aT\mod{T^{2}}$ and
$[a]_{\beta,\alpha}\in{\rm Hom}(F_{\alpha},F_{\beta})$.

  * (b)

Moreover, the map $a\mapsto[a]_{\alpha,\alpha}(T)$ gives an isomorphism
$\mathcal{O}_{K}\to{\rm End}(F_{\alpha})$. In particular,
$\alpha(T)=[\pi]_{\alpha,\alpha}(T)$.

###### Proof.

See [Mil20, I, 2.14 and 2.17]. ∎

The formal group laws $F_{\alpha}$ characterized by Proposition 2.1 are the
Lubin–Tate formal group laws. Let $\alpha\in\mathscr{F}_{\pi}$ and
$\alpha=\tilde{\alpha}v$ for some unit $v\in\mathcal{O}_{K}\llbracket
T\rrbracket$ and some polynomial $\tilde{\alpha}\in\mathcal{O}_{K}[T]$ by the
Weierstrass preparation theorem. We then define a finite set

$\Lambda_{\alpha,1}:=\\{\mbox{roots of }\tilde{\alpha}\mbox{ in a fixed
algebraic closure of K}\\}$ (2)

Suppose $\mathcal{O}_{K}(\\!(T)\\!)$ is the ring of formal Laurent series with
coefficients in $\mathcal{O}_{K}$. We assign the “compact–open” topology to
$\mathcal{O}_{K}(\\!(T)\\!)$, i.e., a sequence $\\{g_{n}\\}$ converges to $g$
if and only if for any compact subset $A$ not containing $0$ in
$\mathfrak{m}$, and for each $\epsilon>0$, there exists a positive integer
$N=N(A,\epsilon)$ such that $|g_{n}(a)-g(a)|<\epsilon$ for all $a\in A$ and
$n\geqslant N$. Given a Lubin–Tate formal group law $F_{\alpha}$, the Coleman
norm operator is characterized by the following.

######  Theorem 2.3 ([Col79, Theorem 11 and Corollary 12]).

As notations above, there exists a unique function
$\mathscr{N}_{F_{\alpha}}\colon\mathcal{O}_{K}(\\!(T)\\!)\to\mathcal{O}_{K}(\\!(T)\\!)$
satisfying

$\mathscr{N}_{F_{\alpha}}(g)\circ[p]_{F_{\alpha}}(T)=\prod_{\lambda\in\Lambda_{\alpha,1}}g(T+_{F_{\alpha}}\lambda)$

for every $g\in\mathcal{O}_{K}(\\!(T)\\!)$. Moreover,
$\mathscr{N}_{F_{\alpha}}$ is continuous and multiplicative.

The norm operator has the following properties.

######  Lemma 2.4.

Let $i\geqslant 1$, $g\in 1+\mathfrak{m}^{i}\llbracket T\rrbracket$ and $h$ be
a unit in $\mathcal{O}_{K}(\\!(T)\\!)$. Then

  1. (a)

$\mathscr{N}_{F_{\alpha}}(g)\in 1+\mathfrak{m}^{i+1}\llbracket T\rrbracket$
and

  2. (b)

$\mathscr{N}_{F_{\alpha}}^{i}(h)/\mathscr{N}_{F_{\alpha}}^{i-1}(h)\in
1+\mathfrak{m}^{i}\llbracket T\rrbracket$, where
$\mathscr{N}_{F_{\alpha}}^{i}$ denotes $i$ iterations of applying the norm
operator $\mathscr{N}_{F_{\alpha}}$.

###### Proof.

See [Col79, Lemma 13]. Here, part (b) looks different from [Col79, Lemma
13(b)],
$\mathscr{N}_{F_{\alpha}}^{i}(h)/\phi\mathscr{N}_{F_{\alpha}}^{i-1}(h)\in
1+\pi^{i}\mathcal{O}_{K}\llbracket T\rrbracket$, where $\phi\in{\rm Gal}(H/K)$
is the Frobenius map for a complete unramified extension $H/K$ and $\pi$ is
the corresponding uniformizer. For our applications, we need only consider $K$
itself with $\phi$ the identity. ∎

As a consequence of Lemma 2.4(b), the sequence
$\\{\mathscr{N}_{F_{\alpha}}^{i}(h)\\}$ converges in
$\mathcal{O}_{K}(\\!(T)\\!)$. Let
$\mathscr{N}_{F_{\alpha}}^{\infty}(h):=\displaystyle\lim_{i\to\infty}\mathscr{N}_{F_{\alpha}}^{i}(h)$.
In particular, by Lemma 2.4(a), we have that

$\mathscr{N}_{F_{\alpha}}^{\infty}\bigl{(}1+\mathfrak{m}\llbracket
T\rrbracket\bigr{)}=1$ (3)

Since $\mathscr{N}_{F_{\alpha}}$ is continuous,

$\mathscr{N}_{F_{\alpha}}\bigl{(}\mathscr{N}_{F_{\alpha}}^{\infty}(h)\bigr{)}=\mathscr{N}_{F_{\alpha}}\bigl{(}\displaystyle\lim_{i\to\infty}\mathscr{N}_{F_{\alpha}}^{i}(h)\bigr{)}=\displaystyle\lim_{i\to\infty}\mathscr{N}_{F_{\alpha}}\bigl{(}\mathscr{N}_{F_{\alpha}}^{i}(h)\bigr{)}=\mathscr{N}_{F_{\alpha}}^{\infty}(h)$

Moreover, the operator$\mathscr{N}_{F_{\alpha}}^{\infty}$ is multiplicative
since $\mathscr{N}_{F_{\alpha}}$ is.

##  3 Proof of Ando’s theorem in a special case

We will first prove Theorem 1.1 (and 1.2) in a special case using Coleman norm
operators.

In this section, suppose that $K$ is an unramified extension of
$\mathbb{Q}_{p}$ of degree $n$ and $G$ is the Honda formal group law over
$k\cong\mathbb{F}_{p^{n}}$ of height $n$. Here, $p$ is a uniformizer of $K$.

In (1), choose the uniformizer $\pi=p$. Given any
$\alpha\in\mathscr{F}_{\pi}$, let $F_{\alpha}$ be the associated Lubin–Tate
formal group law so that
$[p]_{F_{\alpha}}(T)=[\pi]_{\alpha,\alpha}(T)=\alpha(T)$ by Proposition
2.2(b). Thus, $F_{\alpha}$ is a lift of $G$ to $\mathcal{O}_{K}$. Conversely,
every lift of $G$ to $\mathcal{O}_{K}$ has $p$-series in $\mathscr{F}_{\pi}$,
so by the uniqueness in Proposition 2.1 it is a Lubin–Tate formal group law.

######  Definition 3.1 ($\star$-isomorphisms).

Two formal group laws $F,F^{\prime}$ over $\mathcal{O}_{K}$ are said to be
$\star$-isomorphic if there is an isomorphism $u\colon F\to F^{\prime}$ such
that $u$ restricts to the identity series to $k$. A more general definition of
$\star$-isomorphism will be given in Section 4.

######  Theorem 3.2.

With notations as above and in (2), in each $\star$-isomorphism class of lifts
of $G$ to $\mathcal{O}_{K}$, there is a unique formal group law
$F=F_{\alpha}$, necessarily a Lubin–Tate formal group law for some
$\alpha\in\mathscr{F}_{\pi}$, satisfying

$[p]_{F_{\alpha}}(T)=\prod_{\lambda\in\Lambda_{\alpha,1}}(T+_{F_{\alpha}}\lambda)$
(4)

To prove this special case of Ando’s theorem, we proceed as follows. In terms
of the norm operator, we see that a Lubin–Tate formal group law satisfies (4)
if and only if

$[p]_{F_{\alpha}}(T)=\prod_{\lambda\in\Lambda_{\alpha,1}}(T+_{F_{\alpha}}\lambda)=\bigl{(}\mathscr{N}_{F_{\alpha}}(T)\circ[p]_{F_{\alpha}}\bigr{)}(T)$
(5)

Since $p$ is invertible in $K$, $[p]_{F_{\alpha}}$ has a composition inverse
in $K\llbracket T\rrbracket$. We can thus cancel the term from both sides
above, so that condition (4) is equivalent to

$\mathscr{N}_{F_{\alpha}}(T)=T$ (6)

Begin with any lift $F_{\alpha}$ of $G$ to $\mathcal{O}_{K}$. Let $u\in T+\pi
T\mathcal{O}_{K}\llbracket T\rrbracket=T+T\mathfrak{m}\llbracket T\rrbracket$.
Then there is an element $\alpha_{u}\in\mathscr{F}_{\pi}$ such that $u\circ
F_{\alpha}\circ u^{-1}=F_{\alpha_{u}}$. Indeed, since
$\alpha=[p]_{F_{\alpha}}$ and $\alpha_{u}=[p]_{F_{\alpha_{u}}}$, we have
$\alpha_{u}=u\circ\alpha\circ u^{-1}$. Clearly $F_{\alpha}$ and
$F_{\alpha_{u}}$ are $\star$-isomorphic. In order to show that there is a
unique $u\in T+T\mathfrak{m}\llbracket T\rrbracket$ such that $F_{\alpha_{u}}$
satisfies (4), we are reduced to show that there is a unique $u\in
T+T\mathfrak{m}\llbracket T\rrbracket$ such that

$\mathscr{N}_{F_{\alpha_{u}}}(T)=T$ (7)

Note that $u$ induces a bijection from $\Lambda_{\alpha,1}$ to
$\Lambda_{\alpha_{u},1}$. By definition,

$\bigl{(}\mathscr{N}_{F_{\alpha_{u}}}(T)\circ[p]_{F_{\alpha_{u}}}\bigr{)}(T)=\prod_{\lambda\in\Lambda_{\alpha_{u},1}}(T+_{F_{\alpha_{u}}}\lambda)$

We rewrite this identity as follows:

$\displaystyle\bigl{(}\mathscr{N}_{F_{\alpha_{u}}}(T)\circ
u\circ[p]_{F_{\alpha}}\circ u^{-1}\bigr{)}(T)$
$\displaystyle=\prod_{\lambda\in\Lambda_{\alpha,1}}\bigl{(}T+_{F_{\alpha_{u}}}u(\lambda)\bigr{)}$
$\displaystyle=\prod_{\lambda\in\Lambda_{\alpha,1}}F_{\alpha_{u}}\biggl{(}u\bigl{(}u^{-1}(T)\bigr{)},u(\lambda)\biggr{)}$
$\displaystyle=\prod_{\lambda\in\Lambda_{\alpha,1}}u\circ
F_{\alpha}\bigl{(}u^{-1}(T),\lambda\bigr{)}$
$\displaystyle=\prod_{\lambda\in\Lambda_{\alpha,1}}u\circ\bigl{(}u^{-1}(T)+_{F_{\alpha}}\lambda\bigr{)}$
$\displaystyle=\bigl{(}\mathscr{N}_{F_{\alpha}}(u)\circ[p]_{F_{\alpha}}\bigr{)}\bigl{(}u^{-1}(T)\bigr{)}$

By canceling $[p]_{F_{\alpha}}\circ u^{-1}$ from both sides, we obtain
$\bigl{(}\mathscr{N}_{F_{\alpha_{u}}}(T)\circ
u\bigr{)}(T)=\mathscr{N}_{F_{\alpha}}(u)(T)$. Therefore, (7) is equivalent to

$\mathscr{N}_{F_{\alpha}}(u)=u$

Consequently, it remains to show the following.

######  Proposition 3.3.

Given any $\alpha\in\mathcal{F}_{\pi}$, there is a unique $u\in
T+T\mathfrak{m}\llbracket T\rrbracket$, such that
$\mathscr{N}_{F_{\alpha}}(u)=u$.

###### Proof.

Existence: By Lemma 2.4(b), let
$h_{i}:=\mathscr{N}_{F_{\alpha}}^{i}(T)/\mathscr{N}_{F_{\alpha}}^{i-1}(T)\in
1+\mathfrak{m}^{i}\llbracket T\rrbracket$. Then we have
$\mathscr{N}_{F_{\alpha}}^{\infty}(T)=Th_{1}h_{2}\cdots$. It is easy to see
that $h_{1}h_{2}\cdots\in 1+\mathfrak{m}\llbracket T\rrbracket$, so
$\mathscr{N}_{F_{\alpha}}^{\infty}(T)\in T+T\mathfrak{m}\llbracket
T\rrbracket$. Therefore, $u=\mathscr{N}_{F_{\alpha}}^{\infty}(T)$ satisfies
the condition.

Uniqueness: If $\mathscr{N}_{F_{\alpha}}(u)=u$, then
$\mathscr{N}_{F_{\alpha}}^{i}(u)=u$ for each $i$. Thus,
$\mathscr{N}_{F_{\alpha}}^{\infty}(u)=u$ after taking the limit. Since $u\in
T+T\mathfrak{m}\llbracket T\rrbracket$, there is $\tilde{u}\in
1+\mathfrak{m}\llbracket T\rrbracket$ such that $u=T\tilde{u}$. Then, in view
of (3),

$u=\mathscr{N}_{F_{\alpha}}^{\infty}(u)=\mathscr{N}_{F_{\alpha}}^{\infty}(T)\mathscr{N}_{F_{\alpha}}^{\infty}(\tilde{u})=\mathscr{N}_{F_{\alpha}}^{\infty}(T)$

∎

######  Remark 3.4.

We can interpret the equality $\mathscr{N}_{F_{\alpha}}(u)=u$ as a condition
of norm coherence in the context of [Col79]. To be precise, suppose
$\Lambda_{\alpha,n}$ is the set of roots of $[p^{n}]_{F_{\alpha}}$ in the
fixed algebraic closure of $K$. Let $K_{\pi,n}:=K(\Lambda_{\alpha,n})$ and
$N_{n+1,n}:=N_{K_{\pi,n+1}/K_{\pi,n}}$ be the norm map. It can be shown that
$\Lambda_{\alpha,n}\cong\mathcal{O}_{K}/\mathfrak{m}^{n}$ [Mil20, I, 3.4].
Suppose $v_{n}$ is a generator of $\Lambda_{\alpha,n}$ as an
$\mathcal{O}_{K}$-module for each $n$ such that
$[p]_{F_{\alpha}}(v_{n+1})=v_{n}$. We then have

$\mathscr{N}_{F_{\alpha}}(u)(v_{n})=N_{n+1,n}\bigl{(}u(v_{n+1})\bigr{)}$

by [Col79, Corollary 12(ii)]. Thus, $\mathscr{N}_{F_{\alpha}}(u)=u$ is
equivalent to saying that

$u(v_{n})=N_{n+1,n}\bigl{(}u(v_{n+1})\bigr{)}$

i.e., $u$ transforms the sequence $\\{v_{n}\\}$ to a sequence compatible with
the norm maps.

######  Remark 3.5.

Let
$\mathscr{M}_{\infty,\alpha}=\\{g\in\mathcal{O}_{K}(\\!(T)\\!)^{*}\colon\mathscr{N}_{F_{\alpha}}(g)=g\\}$
be the subset in $\mathcal{O}_{K}(\\!(T)\\!)^{*}$ consisting of norm-coherent
series in the sense above. Then the uniqueness of $u$ follows from an exact
sequence of groups.

$1\to 1+\mathfrak{m}\llbracket
T\rrbracket\to\mathcal{O}_{K}(\\!(T)\\!)^{\times}\stackrel{{\scriptstyle\mathscr{N}_{F_{\alpha}}^{\infty}}}{{\longrightarrow}}\mathscr{M}_{\infty,\alpha}\to
1$

as in [Col79, Proposition 14].

##  4 Norm coherence condition

In a more general case of Theorem 3.2, the left-hand side of (4) should not
simply be $[p]_{F_{\alpha}}(T)$. Indeed, it should be a canonical lift of the
relative Frobenius map. To define this lift, we have to first recall some
notions in formal groups and deformations of formal group laws.

Suppose $\mathcal{F}$ is a formal group over $R$ and $\mathcal{G}$ is a formal
group over $k$ of finite height $n$. Let $F$ and $G$ be the respective formal
group laws associated to chosen coordinates of these formal groups. Let $X$ be
a chosen coordinate on $\mathcal{F}$.

######  Definition 4.1 (Quotients of formal groups).

Suppose $\mathcal{D}$ is a subgroup of $\mathcal{F}$ of degree $p^{r}$ defined
over a complete local ring $S$ containing $R$. Then we define the quotient
group $\mathcal{F}/\mathcal{D}$ over $S$ as follows. Let
$m\colon\mathcal{F}\times\mathcal{D}\to\mathcal{F}$ be the multiplication map
and $pr_{1}\colon\mathcal{F}\times\mathcal{D}\to\mathcal{F}$ be the projection
onto the first factor. The coordinate ring of $\mathcal{F}/\mathcal{D}$ is
defined by the equalizer

${{\mathcal{O}_{\mathcal{F}/\mathcal{D}}}}$${{\mathcal{O}_{\mathcal{F}}}}$${{\mathcal{O}_{\mathcal{F}\times\mathcal{D}}}}$$\scriptstyle{pr_{1}^{*}}$$\scriptstyle{m^{*}}$$\scriptstyle{f_{\mathcal{D}}^{*}}$

It can be shown that $\mathcal{F}/\mathcal{D}$ is a formal group over $S$. In
addition, viewing $\mathcal{O}_{\mathcal{F}}$ as an
$\mathcal{O}_{\mathcal{F}/\mathcal{D}}$-module through $f_{\mathcal{D}}^{*}$,
we obtain $X_{\mathcal{D}}=\mbox{Norm}_{f_{\mathcal{D}}^{*}}(X)$ as a
coordinate of $\mathcal{F}/\mathcal{D}$ [Str97, Theorem 19].

If $\mathcal{D}(S)$ contains exactly $p^{r}$ elements, then

$f_{\mathcal{D}}^{*}(X_{\mathcal{D}})=\prod_{P\in\mathcal{D}(S)}\bigl{(}X+_{F}X(P)\bigr{)}$
(8)

by direct calculation. Thus, reducing (8) to the residue field of $R$ we have

$f_{\mathcal{D}}^{*}(X_{\mathcal{D}})\equiv X^{p^{r}}\mod{\mathfrak{m}}$ (9)

Notation. For simplicity, from now on, we will not distinguish an isogeny
between formal groups and the power series to which it corresponds as a map
between coordinate rings. For instance, we will simply write

$f_{\mathcal{D}}(T)=\prod_{P\in\mathcal{D}(S)}\bigl{(}T+_{F}X(P)\bigr{)}$

as an isogeny between formal group laws.

The following definition generalizes Definition 3.1.

######  Definition 4.2 (Deformations of formal group laws and
$\star$-isomorphisms).

Let $\pi\colon R\to R/\mathfrak{m}$ be the natural projection. A deformation
of $G$ to $R$ is a triple $(F,i,\eta)$, where $F$ is a formal group law over
$R$, $i\colon k\to R/\mathfrak{m}$ is an inclusion and
$\eta\colon\pi^{*}(F)\to i^{*}(G)$ is an isomorphism. Here $\pi$ and $i$ act
on each coefficient.

Suppose $(F,i,\eta)$ and $(F^{\prime},i^{\prime},\eta^{\prime})$ are two
deformations of $G$ to $R$ such that $i=i^{\prime}$. Then we say $(F,i,\eta)$
and $(F^{\prime},i^{\prime},\eta^{\prime})$ are $\star$-isomorphic if there is
an isomorphism $\psi\colon F\to F^{\prime}$ of formal group laws such that
$\eta^{\prime}\circ\pi^{*}(\psi)=\eta$.

Furthermore, if $\eta=\eta^{\prime}$ and $\pi^{*}(\psi)=\rm{id}$, we call
$\psi\colon F\to F^{\prime}$ a $\star$-isomorphism as well.

######  Definition 4.3 (Deformations of Frobenius).

Suppose $\Phi$ is the relative Frobenius map on $G$ over $k$ and $\sigma$ is
the absolute Frobenius map over $k$. Suppose $(F,i,\eta)$ and
$(F^{\prime},i^{\prime},\eta^{\prime})$ are two deformations of $G$ to $R$. An
isogeny $\psi\colon F\to F^{\prime}$ of degree $p^{r}$ is a deformation of
Frobenius if $i^{\prime}=i\circ\sigma^{r}$, and
$\eta^{\prime}\circ\pi^{*}(\psi)=i^{*}(\Phi^{r})\circ\eta$, i.e., the
following diagram commutes.

${{\mathcal{F}}}$${{\pi^{*}(\mathcal{F})}}$${{i^{*}(\mathcal{G})}}$${{i^{*}(\mathcal{G}^{(p^{r})})}}$${{\mathcal{F^{\prime}}}}$${{\pi^{*}(\mathcal{F^{\prime}})}}$${{{i^{\prime}}^{*}(\mathcal{G})}}$$\scriptstyle{\psi}$$\scriptstyle{\pi^{*}(\psi)}$$\scriptstyle{i^{*}(\Phi^{r})}$$\scriptstyle{\eta}$$\scriptstyle{\eta^{\prime}}$

Two deformations of Frobenius
$(F_{1},i_{1},\eta_{1})\to(F_{1}^{\prime},i_{1}^{\prime},\eta_{1}^{\prime})$
and
$(F_{2},i_{2},\eta_{2})\to(F_{2}^{\prime},i_{2}^{\prime},\eta_{2}^{\prime})$
are said to be isomorphic if $(F_{1},i_{1},\eta_{1}),(F_{2},i_{2},\eta_{2})$
are $\star$-isomorphic and
$(F_{1}^{\prime},i_{1}^{\prime},\eta_{1}^{\prime}),(F_{2}^{\prime},i_{2}^{\prime},\eta_{2}^{\prime})$
are $\star$-isomorphic.

The following theorem classifies deformations of Frobenius.

######  Theorem 4.4 (cf. [Str97, Theorem 42]).

There is a universal deformation $(F_{\rm univ},{\rm id},{\rm id})$ of $G$ to
the Lubin–Tate ring $W(k)\llbracket u_{1},\cdots,u_{n-1}\rrbracket$, in the
following sense. For each $r\geqslant 0$, there is a complete local ring
$A^{r}$ such that

$\\{\mbox{deformations
}(F,i,\eta)\to(F^{\prime},i^{\prime},\eta^{\prime})\mbox{ of }\Phi^{r}\mbox{
to }T\\}/\mbox{isomorphisms}\cong{\rm Hom}(A^{r},T)$

Moreover, we have $A^{0}=W(k)\llbracket u_{1},\cdots,u_{n-1}\rrbracket$ and
$A^{r}$ is a bimodule over $A^{0}$ with structure maps $s^{r},t^{r}\colon
A^{0}\to A^{r}$, which are local homomorphisms.

The isomorphism is given as follows: for any deformation
$(F,i,\eta)\to(F^{\prime},i^{\prime},\eta^{\prime})$ of $\Phi^{r}$ to $T$,
there is a unique local homomorphism $\rho^{r}\colon A^{r}\to T$ such that
$\rho^{r}\circ s^{r}$ and $\rho^{r}\circ t^{r}$ restrict to $i$ and
$i^{\prime}$ on the residue fields respectively and that there are unique
$\star$-isomorphisms $(F,i,\eta)\to({\rho^{r}}^{*}{s^{r}}^{*}F_{\rm
univ},i,\rm{id})$ and
$(F^{\prime},i^{\prime},\eta^{\prime})\to({\rho^{r}}^{*}{t^{r}}^{*}F_{\rm
univ},i^{\prime},{\rm id})$.

######  Remark 4.5.

According to [Zhu20, Remark 6.2], given any deformation $(F,i,\eta)$ of $G$ to
$R$, there exists a unique deformation $(\tilde{F},i,\rm{id})$ given by [LT66,
Theorem 3.1] such that the two deformations are $\star$-isomorphic. Thus, we
will assume $\eta=\rm{id}$ in the following.

Suppose from now on that $(F,i,\rm{id})$ is a deformation of $G$ to $R$, where
$R$ is a complete local domain with $p\neq 0$.

Since $[p]_{F}\equiv[p]_{G}\mod{\mathfrak{m}}$ and $G$ has height $n<\infty$,
not all coefficients of $[p]_{F}$ are in $\mathfrak{m}$. By the Weierstrass
preparation theorem, there is a unit $v$ in $R\llbracket T\rrbracket$ and a
monic polynomial $\beta$ of degree $p^{n}$, such that $[p]_{F}=v\cdot\beta$.
Note that roots of $[p]_{F}$ are the same as the roots of $\beta$. Let

$\Lambda_{[p]_{F},1}:=\\{\mbox{roots of $[p]_{F}$ in a larger ring $S$
obtained from $R$ by adjoining roots of $\beta$}\\}$ (10)

Since $p\neq 0$ in $R$, $0$ is a simple root of $[p]_{F}$. For any
$\lambda\in\Lambda_{[p]_{F},1}$,
$[p]_{F}\bigl{(}T-_{F}\lambda\bigr{)}=[p]_{F}(T)$. Therefore, $\lambda$ is
also a simple root of $[p]_{F}$. We conclude that the set
$\Lambda_{[p]_{F},1}$ has exactly $p^{n}$ elements.

Let $\mathcal{D}:=\mathcal{F}[p]$ be the subgroup of $\mathcal{F}$ of
$p$-torsions defined over $S$. Then the quotient isogeny
$f_{p}\colon\mathcal{F}\to\mathcal{F}/\mathcal{F}[p]$ is given by

$f_{p}(T)=\prod_{\lambda\in\Lambda_{[p]_{F},1}}\bigl{(}T+_{F}\lambda\bigr{)}$

Note that $f_{p}(T)$ is invariant under the action of $\mbox{Aut}(S/R)$, so
$f_{p}(T)\in R\llbracket T\rrbracket$. As a consequence,
$\mathcal{F}/\mathcal{F}[p]$ can be defined over $R$. Thus, $f_{p}$ is a
deformation of Frobenius between $(F,i,\rm{id})$ and
$(F/F[p],i\circ\sigma^{n},\rm{id})$ in view of (9). By Theorem 4.4, there
exists a unique local homomorphism $\rho^{n}\colon A^{n}\to R$ with a unique
$\star$-isomorphism $g_{p}\colon F/F[p]\to{\rho^{n}}^{*}{t^{n}}^{*}F_{\rm
univ}$.

######  Definition 4.6.

Define

$l_{p}:=g_{p}\circ f_{p}\colon F\to{\rho^{n}}^{*}{t^{n}}^{*}F_{\rm univ}$

######  Remark 4.7.

According to [Zhu20, Remark 6.7], the isogeny $l_{p}$ is an isogeny of formal
group laws over $R$ characterized by the following properties.

  * (a)

It is an isogeny from $F$ to ${\rho^{n}}^{*}{t^{n}}^{*}F_{\rm univ}$.

  * (b)

The kernel of $l_{p}$ is the same as that of $[p]_{F}$.

  * (c)

The reduction of $l_{p}$ to the residue field is the $p^{n}$-power relative
Frobenius $T^{p^{n}}$, i.e., $l_{p}(T)\equiv T^{p^{n}}\mod{\mathfrak{m}}$.

To proceed to the general case, we formulate the norm coherence condition of
Theorem 1.2 precisely as follows.

######  Theorem 4.8.

Suppose $R$ is a complete local domain with $p\neq 0$ and residue field
containing $k$. In each $\star$-isomorphism class of deformations of $G$ to
$R$, there is a unique element $(F,i,{\rm id})$ such that

$l_{p}(T)=f_{p}(T)=\prod_{\lambda\in\Lambda_{[p]_{F},1}}\bigl{(}T+_{F}\lambda\bigr{)}$
(11)

i.e., $g_{p}(T)=T$.

######  Remark 4.9.

According to the proof for [Zhu20, Proposition 7.1], the coordinate given by
Theorem 4.8 is norm-coherent with respect to any finite subgroup
$D\subset\mathcal{F}$ in the sense of [Zhu20, Definition 6.21].

######  Remark 4.10.

Back to the topological side, an orientation on the Morava E-theory $E_{n}$
corresponds to a coordinate on the formal group ${\rm
Spf}\bigl{(}\pi_{0}\bigl{(}E_{n}^{\mathbb{CP}_{+}^{\infty}}\bigr{)}\bigr{)}$
[AHS01, Example 2.53], so it induces a deformation $(F,i,\eta)$ of $G$ to
$\pi_{0}\bigl{(}E_{n}^{\mathbb{CP}_{+}^{\infty}}\bigr{)}$. A change of
orientation on $E_{n}$ will induce a $\star$-isomorphism between the induced
deformations [Zhu20, Example 4.9]. One should be aware that the
$\star$-isomorphism here may change the $\eta$-component. Recall that we set
$\eta=\rm{id}$ in Remark 4.5. Thus, Theorem 4.8 implies that there is a unique
$H_{\infty}$-orientation ${\rm MU}\langle 0\rangle\to E_{n}$ such that the
induced deformation has $\eta=\rm{id}$ and satisfies $\eqref{Generalized
Ando's criterion}$.

In general, we say a deformation $(F,i,\eta)$ (also the orientation inducing
this deformation) is norm-coherent if the component $\tilde{F}$ in
$(\tilde{F},i,{\rm id})$ is norm-coherent in the sense of (11), where the
latter deformation is given by Remark 4.5 (cf. [Zhu20, Definition 6.21]).
Therefore, there is only a family of norm-coherent orientations on $E_{n}$,
which are $\star$-isomorphic to the deformation $(F,i,{\rm id})$ given by
Theorem 4.8.

##  5 Generalization of the norm operators

In this section, we aim to prove Theorem 4.8 (and hence Theorem 1.2) following
the proof of the special case in Section 3. Observe that our earlier proof
actually only requires $\mathcal{O}_{K}$ to be a complete local domain such
that $[p]_{F_{\alpha}}$ is right-cancellative and the definition and
properties of the Coleman norm operator over $\mathcal{O}_{K}$ in Section 2.
Therefore, we need only show that $l_{p}$ is right-cancellative and generalize
Theorem 2.3 and Lemma 2.4 to the case of a complete local domain $R$ such that
$p\neq 0$ and residue field contains $k$ substituting $[p]_{F}$ by $l_{p}$.
The proof then applies mutatis mutandis.

######  Remark 5.1.

In the general case, we do not require $G$ to be a Honda formal group law and
$F$ to be a Lubin-Tate formal group law as in Section 3, since the property
that $[p]_{F}\equiv T^{p^{n}}\mod{\mathfrak{m}}$ has been replaced by Remark
4.7(c).

######  Lemma 5.2.

The power series $l_{p}$ is right-cancellative, i.e., if there are $g,h\in
R\llbracket T\rrbracket$ such that $g\circ l_{p}=h\circ l_{p}$, we have $g=h$.

###### Proof.

We may assume $h=0$, so that $g\bigl{(}l_{p}(T)\bigr{)}=0$. Since $R$ is
complete with respect to $\mathfrak{m}$, we need only prove by induction on
$i$ that $g\equiv 0\mod{\mathfrak{m}^{i}}$ for each $i\geqslant 0$. The
statement is vacuous for $i=0$. Suppose we have proven that $g\equiv
0\mod{\mathfrak{m}^{i}}$. Since $l_{p}(T)\equiv T^{p^{n}}\mod{\mathfrak{m}}$
by Remark 4.7(c), $g(T^{p^{n}})\equiv 0\mod{\mathfrak{m}^{i+1}}$, and hence
$g\equiv 0\mod{\mathfrak{m}^{i+1}}$. ∎

The following proofs essentially follow those in [Col79].

######  Lemma 5.3 (cf. [Col79, Lemma 3]).

If $g\in R\llbracket T\rrbracket$ and $g(T+_{F}\lambda)=g(T)$ for all
$\lambda\in\Lambda_{[p]_{F},1}$, then there is a unique $h\in R\llbracket
T\rrbracket$ such that $h\circ l_{p}=g$.

###### Proof.

The uniqueness follows from Lemma 5.2.

For the existence, we inductively construct formal power series $g_{m}$ for
each $m\geqslant 0$. Let $g_{0}=g$. Suppose that we have constructed $a_{i}\in
R$ for $0\leqslant i\leqslant m-1$ such that

$g-\sum_{i=0}^{m-1}a_{i}l_{p}^{i}=l_{p}^{m}\cdot g_{m}$

for some $g_{m}\in R\llbracket T\rrbracket$. Note that by assumption
$g(T+_{F}\lambda)=g(T)$ and by Remark 4.7(b) $l_{p}(T+_{F}\lambda)=l_{p}(T)$.
We thus have $g_{m}(T+_{F}\lambda)=g_{m}(T)$. In particular,
$\bigl{(}g_{m}-g_{m}(0)\bigr{)}(\lambda)=0$ for all
$\lambda\in\Lambda_{[p]_{F},1}$. By the Euclidean algorithm for power series,
there exists elements $g_{m+1}\in R\llbracket T\rrbracket$ with $r_{m}\in
R[T]$ such that $g_{m}-g_{m}(0)=l_{p}\cdot g_{m+1}+r_{m}$ and
$\deg(r_{m})<p^{n}$. Then $r_{m}$ vanishes on $\Lambda_{[p]_{F},1}$. Since we
have deduced in the paragraph after (10) that $\Lambda_{[p]_{F},1}$ has
$p^{n}$ elements, we have $r_{m}=0$. Let $a_{m}=g_{m}(0)$. This finishes the
inductive step. Thus, we obtain

$g-\sum_{i=0}^{\infty}a_{i}l_{p}^{i}\in\bigcap_{i=0}^{\infty}l_{p}^{i}R\llbracket
T\rrbracket=0$

Then $h(T):=\sum_{i=0}^{\infty}a_{i}T^{i}$ is the desired element. ∎

Now we also endow $R\llbracket T\rrbracket$ with the compact–open topology
similar to that on $\mathcal{O}_{K}\llbracket T\rrbracket$ as in Section 2.
Here $R$ has the $\mathfrak{m}$-adic topology.

######  Theorem 5.4 (cf. Theorem 2.3).

There is a unique operator $\mathscr{N}_{F}\colon R\llbracket T\rrbracket\to
R\llbracket T\rrbracket$ such that for any $g\in R\llbracket T\rrbracket$,

$\mathscr{N}_{F}(g)\circ
l_{p}(T)=\prod_{\lambda\in\Lambda_{[p]_{F},1}}g(T+_{F}\lambda)$

Moreover, $\mathscr{N}_{F}$ is multiplicative and continuous.

###### Proof.

Note that the right-hand side of the above identity satisfies the condition of
$\Lambda_{[p]_{F},1}$-invariance from Lemma 5.3. Thus, there is a unique
$\mathscr{N}_{F}$ satisfying the formula.

Given any $g,h\in R\llbracket T\rrbracket$,

$\displaystyle\begin{split}\mathscr{N}_{F}(gh)\circ
l_{p}(T)&=\prod_{\lambda\in\Lambda_{[p]_{F},1}}gh(T+_{F}\lambda)\\\
&=\bigl{(}\mathscr{N}_{F}(g)\circ
l_{p}(T)\bigr{)}\cdot\bigl{(}\mathscr{N}_{F}(h)\circ l_{p}(T)\bigr{)}\\\
&=\bigl{(}\mathscr{N}_{F}(g)\cdot\mathscr{N}_{F}(h)\bigr{)}\circ
l_{p}(T)\end{split}$

By Lemma 5.2, we then obtain
$\mathscr{N}_{F}(gh)=\mathscr{N}_{F}(g)\cdot\mathscr{N}_{F}(h)$.

Suppose $\\{g_{n}\\}$ converges to $g$. Then

$\displaystyle\begin{split}\bigl{(}\lim\mathscr{N}_{F}(g_{n})\bigr{)}\circ
l_{p}(T)&=\lim\bigl{(}\mathscr{N}_{F}(g_{n})\circ
l_{p}\bigr{)}(T)=\lim\prod_{\lambda\in\Lambda_{[p]_{F},1}}g_{n}(T+_{F}\lambda)\\\
&=\prod_{\lambda\in\Lambda_{[p]_{F},1}}g(T+_{F}\lambda)=\mathscr{N}_{F}(g)\circ
l_{p}(T)\end{split}$

Again, it follows from Lemma 5.2 that
$\lim\mathscr{N}_{F}(g_{n})=\mathscr{N}_{F}(g)$. ∎

Note that in the proof in Section 3 we only took limits of
$\mathscr{N}_{F_{\alpha}}$ applied to elements in $1+\mathfrak{m}\llbracket
T\rrbracket$ and to $T$. Thus, it remains to show the following for
$\mathscr{N}_{F}$.

######  Lemma 5.5 (cf. Lemma 2.4).

Let $g\in 1+\mathfrak{m}^{i}\llbracket T\rrbracket$ and $i\geqslant 1$. Then

  1. (a)

$\mathscr{N}_{F}(g)\in 1+\mathfrak{m}^{i+1}\llbracket T\rrbracket$ and

  2. (b)

$\mathscr{N}_{F}^{i}(T)/\mathscr{N}_{F}^{i-1}(T)\in
1+\mathfrak{m}^{i}\llbracket T\rrbracket$.

###### Proof.

  1. (a)

By definition, $\mathscr{N}_{F}(g)\circ
l_{p}(T)=\prod_{\lambda\in\Lambda_{[p]_{F},1}}g(T+_{F}\lambda)$. Suppose that
$g(T)=1+\sum_{j=0}^{\infty}b_{j}T^{j}$, where $b_{j}\in\mathfrak{m}^{i}$.
Since $i\geqslant 1$, terms divided by $b_{j_{1}}b_{j_{2}}$ for some
$j_{1},j_{2}$ must lie in $\mathfrak{m}^{i+1}$. Therefore, modulo
$\mathfrak{m}^{i+1}$,

$\displaystyle\begin{split}\mathscr{N}_{F}(g)\circ l_{p}(T)&\equiv
1+\sum_{\lambda\in\Lambda_{[p]_{F},1}}\sum_{j=0}^{\infty}b_{j}(T+_{F}\lambda)^{j}\\\
&=1+\sum_{j=0}^{\infty}\sum_{\lambda\in\Lambda_{[p]_{F},1}}b_{j}(T+_{F}\lambda)^{j}\\\
&=1+\sum_{j=0}^{\infty}b_{j}\bigl{(}p^{n}T^{j}+\sum_{k=0}^{\infty}p_{k}(\Lambda_{[p]_{F},1})T^{k}\bigr{)}\end{split}$

where each $p_{k}(\Lambda_{[p]_{F},1})$ is a symmetric function on
$\lambda\in\Lambda_{[p]_{F},1}$. Recall from Section 4 that $\beta$ is a
polynomial of degree $p^{n}$ dividing $[p]_{F}$ from the Weierstrass
preparation theorem, and $\Lambda_{[p]_{F},1}$ is the set of roots of $\beta$.
Thus, $\beta(T)\equiv T^{p^{n}}\mod{\mathfrak{m}}$. It follows that
$p_{k}(\Lambda_{[p]_{F},1})\equiv 0\mod{\mathfrak{m}}$, since
$p_{k}(\Lambda_{[p]_{F},1})$ is a polynomial in the non-leading coefficients
of $\beta$ (i.e., those elementary symmetric functions). Therefore,

$\mathscr{N}_{F}(g)\circ l_{p}(T)\equiv 1\mod{\mathfrak{m}^{i+1}}$

Next we prove by induction on $j$ that if $h\in R\llbracket T\rrbracket$ and
$h\circ l_{p}\in\mathfrak{m}^{j}\llbracket T\rrbracket$, then
$h\in\mathfrak{m}^{j}\llbracket T\rrbracket$ (here $j\geqslant 0$). Setting
$h=\mathscr{N}_{F}(g)-1$ completes the proof of part (a). The case is vacuous
when $j=0$. Suppose $j\geqslant 1$ and the statement holds for $j-1$. By the
induction hypothesis, $h\in\mathfrak{m}^{j-1}\llbracket T\rrbracket$. Since
$l_{p}(T)\equiv T^{p^{n}}\mod{\mathfrak{m}}$, we have

$h\circ l_{p}(T)\equiv h(T^{p^{n}})\mod{\mathfrak{m}^{j}}$

Since $h\circ l_{p}\in\mathfrak{m}^{j}\llbracket T\rrbracket$, we must have
$h\in\mathfrak{m}^{j}\llbracket T\rrbracket$.

  2. (b)

By part (a) and the multiplicativity of $\mathscr{N}_{F}$ from Theorem 5.4, we
need only show the case when $i=1$. Since $l_{p}(T)\equiv
T^{p^{n}}\mod{\mathfrak{m}}$,

$\mathscr{N}_{F}(T)(T^{p^{n}})\equiv\mathscr{N}_{F}(T)\circ
l_{p}(T)=\prod_{\lambda\in\Lambda_{[p]_{F},1}}(T+_{F}\lambda)\mod{\mathfrak{m}}$

By an argument with symmetric functions similar to that for part (a), we
obtain that $\prod_{\lambda\in\Lambda_{[p]_{F},1}}(T+_{F}\lambda)\equiv
T^{p^{n}}\mod{\mathfrak{m}}$. Hence, $\mathscr{N}_{F}(T)\equiv
T\mod{\mathfrak{m}}$. Therefore, we have that $\mathscr{N}_{F}(T)/T\equiv
1\mod{T^{-1}\mathfrak{m}\llbracket T\rrbracket}$. It remains to show that $T$
divides $\mathscr{N}_{F}(T)$ in $R\llbracket T\rrbracket$, or
$\mathscr{N}_{F}(T)(0)=0$. Since $0\in\Lambda_{[p]_{F},1}$,

$\mathscr{N}_{F}(T)(0)=\mathscr{N}_{F}(T)\circ
l_{p}(0)=\prod_{\lambda\in\Lambda_{[p]_{F},1}}\lambda=0$

∎

######  Remark 5.6.

Comparing (5) from the proof in Section 3, we see that $f_{p}$ as defined in
Section 4 now factors as $\mathscr{N}_{F}(T)\circ l_{p}$. Recall that
$l_{p}=g_{p}\circ f_{p}$ by definition. By Lemma 5.2, $l_{p}$ is right-
cancellative with respect to composition, and hence so does $f_{p}$.
Therefore, $\mathscr{N}_{F}(T)$ turns out to be the inverse to $g_{p}$, i.e.,
it is a (necessarily unique) $\star$-isomorphism
${\rho^{n}}^{*}{t^{n}}^{*}F_{\rm univ}\to F/F[p]$. This agrees with the
construction of the coordinate associated with $F/F[p]$ in Definition 4.1.

In view of this, the proof in Section 3 is actually similar to the one for
[And95, Theorem 2.6.4]. In [And95, Theorem 2.6.4], Ando constructed a series
of $\star$-isomorphisms to modify the coordinates such that $g_{p}(T)$ can
approximate $T$ inductively modulo $\mathfrak{m}^{r}$. On the other hand, our
proof constructed a $\star$-isomorphism $u=\mathscr{N}_{F}^{\infty}(T)$
directly so that $g_{p}^{-1}(T)=\mathscr{N}_{F}(T)(T)=T$ after a change of
coordinate by $u$. Indeed, properties of this limit of operators (i.e., Lemma
2.4 and Lemma 5.5) are deduced inductively on $r$ modulo $\mathfrak{m}^{r}$
similar to the one in Ando’s construction. However, our proof cannot proceed
by constructing a series of $\star$-isomorphisms, since
$\mathscr{N}_{F}^{\infty}(T)$ is not the infinite composite of
$\mathscr{N}_{F}(T)$.

######  Remark 5.7.

Walker has also observed the relationship between the Coleman norm operator
and Ando’s criterion [Wal08, Chapter 5]. In particular, he has deduced that
Ando’s criterion is equivalent to (6) in [Wal08, Lemma 5.0.5 and (5.0.10)].
However, he did not prove Theorem 1.1 via the Coleman norm operator.

######  Remark 5.8.

As mentioned in Remark 1.3, Zhu generalized Theorem 4.8 to apply to arbitrary
complete local rings. To apply our proof to arbitrary complete local rings, we
need to generalize the Coleman norm operator to such cases. However, recall in
the argument around (10), we need $R$ to be a domain so that we can count the
number of roots of $\beta$, and we need $p\neq 0$ in $R$ so that $\beta$ is
separable. Thus, there is a question whether the Coleman norm operator can be
defined over arbitrary complete local rings.

## Acknowledgements

The author is grateful to Charles Rezk for his insight of connecting Coleman’s
norm operator and Ando’s theorem. Furthermore, the author would like to give
sincere gratitude to Yifei Zhu for introducing him this idea. The author would
also like to thank Eric Peterson and Robert Burklund for useful discussions
and suggestions. The author thank Tongtong Liang for teaching me basic notions
on power operations. Thanks also to the organizers and speakers of IWoAT
(International Workshop on Algebraic Topology) 2022 for the wonderful lectures
on chromatic homotopy theory.

This work was partly supported by the National Natural Science Foundation of
China grant 11701263.

## References

  * [AHS01] M. Ando, M. J. Hopkins, and N. P. Strickland, _Elliptic spectra, the Witten genus and the theorem of the cube_ , Invent. Math. 146 (2001), no. 3, 595–687.
  * [And95] Matthew Ando, _Isogenies of formal group laws and power operations in the cohomology theories $E_{n}$_, Duke Math. J. 79 (1995), no. 2, 423–485.
  * [BSY22] Robert Burkland, Tomer M. Schlank, and Allen Yuan, _The chromatic nullstellensatz_ , 2022.
  * [Col79] Robert F. Coleman, _Division values in local fields_ , Invent. Math. 53 (1979), no. 2, 91–116.
  * [GH04] Paul G. Goerss and Michael J. Hopkins, _Moduli spaces of commutative ring spectra_ , London Math. Soc. Lecture Noe Ser., no. 315, Cambridge Univ. Press, Cambridge, 2004.
  * [LT66] Jonathan Lubin and John Tate, _Formal moduli for one-parameter formal Lie groups_ , Bull. Soc. Math. France 94 (1966), 49–59.
  * [May77] J. Peter May, _$E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra_, Lecture Notes in Mathematics, vol. 577, Springer-Verlag, Berlin-New York, 1977\.
  * [Mil20] James Milne, _Class field theory_ , https://www.jmilne.org/math/CourseNotes/CFT.pdf, 2020.
  * [Rez14] Charles Rezk, _Isogenies, power operations, and homotopy theory_ , Proceedings of the International Congress of Mathematicians—Seoul 2014 (Seoul), vol. II, Kyung Moon Sa, 2014, pp. 1125–1145.
  * [Str97] Neil P. Strickland, _Finite subgroups of formal groups_ , Journal of Pure and Applied Algebra 121 (1997), no. 2, 161–208.
  * [tD68] Tammo tom Dieck, _Steenrod-Operationen in Kobordismen-Theorien_ , Math. Z. 107 (1968), 380–401.
  * [Wal08] Barry John Walker, _Multiplicative orientations of K-Theory and p-adic analysis_ , ProQuest LLC, Ann Arbor, MI, 2008, Thesis (Ph.D.)–University of Illinois at Urbana-Champaign.
  * [Zhu20] Yifei Zhu, _Norm coherence for descent of level structures on formal deformations_ , J. Pure Appl. Algebra (2020), no. 10, 35.

Email<EMAIL_ADDRESS>
Webpage: hongxiang-zhao.github.io