\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{SO orientation of elliptic cohomology}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{elliptic_cohomology}{}\paragraph*{{Elliptic cohomology}}\label{elliptic_cohomology}

[[!include elliptic cohomology -- contents]]

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{induced_relation_between_cobordism_and_homology}{Induced relation between cobordism and homology}\dotfill \pageref*{induced_relation_between_cobordism_and_homology} \linebreak
\noindent\hyperlink{relation_to_the_atiyahbottshapiro_spin_orientation_of_ko}{Relation to the Atiyah-Bott-Shapiro Spin orientation of KO}\dotfill \pageref*{relation_to_the_atiyahbottshapiro_spin_orientation_of_ko} \linebreak
\noindent\hyperlink{homotopy_type_of_the_spectrum_}{Homotopy type of the spectrum $Ell$}\dotfill \pageref*{homotopy_type_of_the_spectrum_} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The [[Ochanine elliptic genus]] $\Omega^{SO}_\bullet \to M_\bullet = \mathbb{Z}[\tfrac{1}{2}][\delta, \epsilon]$ lifts to a map of [[ring spectra]]

\begin{displaymath}
M SO \longrightarrow Ell
\end{displaymath}
(\hyperlink{LandweberRavenelStong93}{Landweber-Ravenel-Stong 93}). Here $Ell[\tfrac{1}{6}]$ is equivalently [[tmf0(2)]] (\hyperlink{Behrens05}{Behrens 05}) and as such this lift is analogous to the [[string orientation of tmf]] $M String \to tmf$.

If this map of [[ring spectra]] could be shown to be ``highly structured'' in that it preserves [[E-∞ ring]] structure, then it would equivalently be a universal [[orientation in generalized cohomology|orientation]] (see at \href{orientation+in+generalized+cohomology#RelationToGenera}{relation between orientations and genera}).

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

After inversion of the [[prime number]] 2, the oriented [[cobordism ring]] is a [[polynomial ring]] over $\mathbb{Z}[\tfrac{1}{2}]$ on generators in degrees $4k$

\begin{displaymath}
\Omega^{SO}_\bullet[\tfrac{1}{2}]
  \simeq
  \mathbb{Z}[\tfrac{1}{2}] [x_4, x_8, x_{12}, \cdots  ]
\end{displaymath}
where $x_4$ is the class of the complex [[projective space]] $\mathbb{C}P^2$ and $x_8$ that of $\mathbb{H}P^2$ and where all [[elliptic genera]] vanish on all the other generators (\hyperlink{LandweberRavenelStong93}{Landweber-Ravenel-Stong 93, prop. 3.2}).

From this one gets that the [[quotient]] by the [[ideal]] generated by these higher elements is

\begin{displaymath}
\Omega^{SO}_\bullet[\tfrac{1}{2}]/(x_{4(k \geq 3)})
  \simeq
  MF_0(2)_\bullet \coloneqq \mathbb{Z}[\tfrac{1}{2}][\delta, \epsilon]
\end{displaymath}
where the right hand side here is naturally identified as the ring of those [[modular forms]] for the [[congruence subgroup]] $\Gamma_0(2)$ which have half-integral [[coefficients]] in their $q$-expansion at the [[nodal curve]] (\hyperlink{LandweberRavenelStong93}{Landweber-Ravenel-Stong 93, theorem 1.5}).

Now by a general construction due to (\hyperlink{Baas73}{Baas 73}) this induces a [[generalized homology theory]]

\begin{displaymath}
\Omega^{SO}_\bullet[\tfrac{1}{2}](-)
\end{displaymath}
[[Brown representability theorem|represented]] by some [[spectrum]] $Ell$,whose coefficient ring is as above

\begin{displaymath}
Ell_\bullet \simeq MF_0(2)_\bullet
  \,.
\end{displaymath}
By construction, this comes with a quotient map

\begin{displaymath}
M SO[\tfrac{1}{2}] \longrightarrow Ell
\end{displaymath}
which is a map of [[ring spectra]] by (\hyperlink{Mironov78}{Mironov 78}). This maplifts the universal [[elliptic genus]] (in that it reproduces it on [[homotopy groups]]) (\hyperlink{LandweberRavenelStong93}{Landweber-Ravenel-Stong 93, section 4.6, 4.7})

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{induced_relation_between_cobordism_and_homology}{}\subsubsection*{{Induced relation between cobordism and homology}}\label{induced_relation_between_cobordism_and_homology}

The SO orientation of elliptic cohomology makes it expressible in terms of the [[cobordism cohomology theory]], see at \emph{[[cobordism theory determining homology theory]]} (\hyperlink{LandweberRavenelStong93}{Landweber-Ravenel-Stong 93, theorem 1.2}).

\hypertarget{relation_to_the_atiyahbottshapiro_spin_orientation_of_ko}{}\subsubsection*{{Relation to the Atiyah-Bott-Shapiro Spin orientation of KO}}\label{relation_to_the_atiyahbottshapiro_spin_orientation_of_ko}

There are maps of spectra

\begin{displaymath}
Ell [\epsilon^{-1}] \longrightarrow KO[\tfrac{1}{2}]
\end{displaymath}
and

\begin{displaymath}
Ell [(\delta^2- \epsilon)^{-1}] \longrightarrow KO[\tfrac{1}{2}]
\end{displaymath}
such that postcomposition of the above SO-orientation with reproduces the [[signature genus]] and the [[Atiyah-Bott-Shapiro orientation]] of [[KO]], respective, hence the [[A-hat genus]] (\hyperlink{LandweberRavenelStong93}{Landweber-Ravenel-Stong 93, prop. 4.9}).

Notice that here the second localization correponds again to including the [[nodal curve]]:

[[!include moduli stack of curves -- table]]

\hypertarget{homotopy_type_of_the_spectrum_}{}\subsubsection*{{Homotopy type of the spectrum $Ell$}}\label{homotopy_type_of_the_spectrum_}

After suitable localization the spectrum Ell is a wedge sum of suspensions of the [[Morava E-theory]] $E(2)$ (\hyperlink{Baker97}{Baker 97}).

Specifically after [[K(n)-local spectrum|K(2)-localization]] and inversion of 6 it coincides with [[tmf0(2)|TMF0(2)]]

\begin{displaymath}
L_{K(2)} TMF_0(2) \simeq L_{K(2)}(E(2) \vee \Sigma^8 E(2))
  \,.
\end{displaymath}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

[[!include genera and partition functions - table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

The construction is due to

\begin{itemize}%
\item [[Peter Landweber]], [[Douglas Ravenel]], [[Robert Stong]], \emph{Periodic cohomology theories defined by elliptic curves}, in [[Haynes Miller]] et. al. (eds.), \emph{The Cech centennial: A conference on homotopy theory}, June 1993, AMS (1995) (\href{http://www.math.sciences.univ-nantes.fr/~hossein/GdT-Elliptique/Landweber-Ravenel-Stong.pdf}{pdf})

\end{itemize}
based on general constructions of multiplicative homology theories from cobordism theories due to

\begin{itemize}%
\item [[Nils Baas]], \emph{On bordism theory of manifolds with singularities}, Math. Scand. 33 (1973), 279--302.


\item O. K. Mironov, \emph{Multiplications in cobordism theories with singularities}, and Steenrod-tom Dieck operations\_, Izv. Akad. Nauk SSSR, Ser. Mat. 42 (1978), 789--806; English transl. in Math. USSR Izvestiya 13 (1979), 89--106.



\end{itemize}
Analysis of $Ell$ is in

\begin{itemize}%
\item [[Andrew Baker]], The homotopy type of the spectrum representing elliptic cohomology, Proceedings of the American Mathematical Society 107.2 (1989): 537-548. (\href{http://www.maths.gla.ac.uk/~ajb/dvi-ps/homell.pdf}{pdf})


\item [[Andrew Baker]], \emph{On the Adams $E_2$-term for elliptic cohomology}, 1997 (\href{http://www.maths.gla.ac.uk/~ajb/dvi-ps/ell-ext.pdf}{pdf})



\end{itemize}
The interpretation of $Ell$ in terms of [[tmf0(2)|TMF0(2)]] is discussed in

\begin{itemize}%
\item [[Mark Behrens]], \emph{A modular description of the K(2)-local sphere at the prime 3} (\href{http://arxiv.org/abs/math/0507184}{arXiv:math/0507184})

\end{itemize}
More is in

\begin{itemize}%
\item [[Dylan Wilson]], \emph{Orientations and Topological Modular Forms with Level Structure} (\href{http://arxiv.org/abs/1507.05116}{arXiv:1507.05116})

\end{itemize}


\end{document}