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\begin{document}

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\section*{level structure on an elliptic curve}

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

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

[[!include elliptic cohomology -- contents]]

\hypertarget{geometry}{}\paragraph*{{Geometry}}\label{geometry}

[[!include higher geometry - 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{congruence_subgroups}{Congruence subgroups}\dotfill \pageref*{congruence_subgroups} \linebreak
\noindent\hyperlink{over_general_rings}{Over general rings}\dotfill \pageref*{over_general_rings} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RelationToSpinStructures}{Relation to spin structures on elliptic curves}\dotfill \pageref*{RelationToSpinStructures} \linebreak
\noindent\hyperlink{properties_2}{Properties}\dotfill \pageref*{properties_2} \linebreak
\noindent\hyperlink{topological_modular_forms_with_level_structure}{Topological modular forms with level structure}\dotfill \pageref*{topological_modular_forms_with_level_structure} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{over_the_complex_numbers}{Over the complex numbers}\dotfill \pageref*{over_the_complex_numbers} \linebreak
\noindent\hyperlink{over_general_base_rings}{Over general base rings}\dotfill \pageref*{over_general_base_rings} \linebreak


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

Over the [[complex numbers]] an [[elliptic curve]] (hence a [[complex manifold|complex]] [[torus]]) may be, and often is, presented as a [[quotient]] of the [[complex plane]] by a framed [[lattice]], determined by a point $\tau$ in the [[upper half plane]] $\mathfrak{h}$. But indeed this data determines a [[framed elliptic curve]], namely the underlying [[curve]] $\Sigma$ together with the ``edges along which it is glued'' by this construction. These are equivalently a choice of [[ordering|ordered]] [[basis]]

\begin{displaymath}
(a_1,a_2)\in H_1(\Sigma,\mathbb{Z})
\end{displaymath}
of the first [[ordinary homology]] of $\Sigma$ with [[integer]] [[coefficients]] (and vanishing [[intersection number]]). The [[special linear group]] $SL_2(\mathbb{Z})$ naturally acts on this data (by [[Möbius transformations]] on $\tau$) and the [[quotient]] [[projection]] exhibits an infinite [[covering]] ([[atlas]]) of the [[moduli stack of elliptic curves]] over the complex numbers

\begin{displaymath}
\mathfrak{h} \longrightarrow \mathfrak{h}//SL_2(\mathbb{C}) \simeq \mathcal{M}_{ell}(\mathbb{C})
  \,.
\end{displaymath}
The concept of \emph{level structure} on an elliptic curve is a structure weaker than that of a [[framed elliptic curve|framing]] which analogously gives a [[finite number|finite]] [[covering]]. Instead of considering cycles in integral homology, a \emph{level $n$-structure} for [[natural number]] $B$ is given by cycles in homology with [[coefficients]] just in the [[cyclic group]] $\mathbb{Z}/n\mathbb{Z}$ (e.g. \hyperlink{Hain08}{Hain 08, def. 4.6}).

On such level-$n$ data now acts instead just the group $SL_2(\mathbb{Z}/n\mathbb{Z})$. The [[kernel]] of the [[projection]] maps is called the \emph{level $n$-subgroup} (an example of a \emph{[[congruence subgroup]]})

\begin{displaymath}
\Gamma(n) \to SL_2(\mathbb{Z}) \to SL_2(\mathbb{Z}/n\mathbb{Z})
  \,.
\end{displaymath}
There is then a [[moduli space]] of complex elliptic curves equipped with level $n$-structure

\begin{displaymath}
\mathcal{M}_{ell}(\mathbb{C})[n] \simeq \mathfrak{h}//\Gamma_n
\end{displaymath}
called the \emph{[[modular curve]]}, and this is now a finite cover (of rank the [[order of a group|order]] of the [[finite group]] $SL_2(\mathbb{Z}/n\mathbb{Z})$) of the actual [[moduli stack of elliptic curves|moduli stack of complex elliptic curves]]

\begin{displaymath}
\mathcal{M}_{ell}[n](\mathbb{C})
  \longrightarrow
  \mathcal{M}_{ell}[n](\mathbb{C})//SL_2(\mathbb{Z}/n\mathbb{Z})
  \simeq
  \mathcal{M}_{ell}(\mathbb{C})
  \,.
\end{displaymath}
\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{congruence_subgroups}{}\subsubsection*{{Congruence subgroups}}\label{congruence_subgroups}

Let $n \in \mathbb{N}$ be a [[natural number]]. Write

\begin{displaymath}
p_n \;\colon\; SL_2(\mathbb{Z}) \to SL_2(\mathbb{Z}/\mathbb{Z}_n)
\end{displaymath}
for the projection from the [[special linear group]] induced by the [[quotient]] [[projection]] $\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$ to the [[cyclic group]].

The [[congruence subgroups]] of the [[special linear group]] $SL_2(\mathbb{Z})$ (essentially the [[modular group]]) are defined as follows.

The \textbf{principal congruence subgroup} is

\begin{displaymath}
\Gamma(n) \coloneqq ker(p_n) = p_n^{-1}\left(\left\{\itexarray{ 1 & 0 \\ 0 & 1}\right\}\right)
\end{displaymath}
The other two are

\begin{displaymath}
\Gamma_0(n) \coloneqq p_n^{-1}\left(\left\{\itexarray{ \ast & \ast \\ 0 & \ast}\right\}\right)
\end{displaymath}
\begin{displaymath}
\Gamma_1(n) \coloneqq p_n^{-1}\left(\left\{\itexarray{ 1 & \ast \\ 0 & \ast}\right\}\right)
\end{displaymath}
\hypertarget{over_general_rings}{}\subsubsection*{{Over general rings}}\label{over_general_rings}

(e.g \hyperlink{Voloch}{Voloch, def. 1.1} \hyperlink{Ando00}{Ando 00, section 1.4}, \hyperlink{AndoHopkinsStrickland02}{Ando-Hopkins-Strickland 02, section 15.2}, \hyperlink{HillLawson13}{Hill-Lawson 13, section 3.6})

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

\hypertarget{RelationToSpinStructures}{}\subsubsection*{{Relation to spin structures on elliptic curves}}\label{RelationToSpinStructures}

For [[elliptic curves]] over the [[complex numbers]] ([[complex manifold|complex]] oriented pointed [[tori]]) the congruence subgroup $\Gamma_0(2)$ has the interpretation as being precisely the subgroup of the [[modular group]] which preserves one of the ``NS-R'' [[spin structures]].

In detail, the elliptic curve $\Sigma$, being [[framed manifold|framed]] has a canonical [[spin structure]] given by the trivial [[double cover]]. The space of all spin structures is a [[torsor]] over $H^1(\Sigma, \mathbb{Z}/2\mathbb{Z}) \simeq [\pi_1(\sigma), \mathbb{Z}/2\mathbb{Z}] \simeq [\mathbb{Z} \times \mathbb{Z}, \mathbb{Z}/2\mathbb{Z}] \simeq (\mathbb{Z}/2\mathbb{Z})^2$. In terms of this action the canonical one is labeled $(0,0)$ and then there are three more, labeled $(1,0)$, $(0,1)$ and $(1,1)$. The [[modular group]] acts on these via the quotient map $p_2 \;\colon\;  SL_2(\mathbb{Z}) \to SL_{2}(\mathbb{Z}/2\mathbb{Z})$. Hence it preserves $(0,0)$ and mixes the other three spin structures. Precisely $\Gamma_0(2)$ preserves $(1,0)$ (and an isomorphic subgroup of course preserves $(0,1)$). The principal congruence subgroup $\Gamma(2)$ is the one which preserves all four spin structures jointly.

In terms of [[type II string theory]] the spin structure $(1,0)$ is called the ``NS-R boundary condition'' for the spinors. The [[partition function]] of the [[type II superstring]] ``in the NS-R sector'' is therefore (at best, indeed it is, being the universal [[Ochanine elliptic genus]]) a [[modular form]] not for the full [[modular group]], but for $\Gamma_0(2)$ (\href{Witten87a}{Witten 87a, below (13)}). For more on this see at \emph{\href{Witten+genus#ModularityForTypeIISuperstring}{Witten genus -- Modularity -- For the type II string}}. The homotopy-theoretic refinement of this involves [[tmf0(2)]], see at \emph{[[spin orientation of Ochanine elliptic cohomology]]}.

\hypertarget{properties_2}{}\subsection*{{Properties}}\label{properties_2}

\hypertarget{topological_modular_forms_with_level_structure}{}\subsubsection*{{Topological modular forms with level structure}}\label{topological_modular_forms_with_level_structure}

The construction of [[topological modular forms]] ([[tmf]]) may be generalized to curves with level structure (\hyperlink{MahowaldRezk09}{Mahowald-Rezk 09}). A systematic kind of ``[[modular equivariant elliptic cohomology]]'' in this sense is discussed in (\hyperlink{HillLawson13}{Hill-Lawson 13}).

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

\hypertarget{over_the_complex_numbers}{}\subsubsection*{{Over the complex numbers}}\label{over_the_complex_numbers}

The principal congruence subgroups are discussed for instance in

\begin{itemize}%
\item Richard Hain, section 4.2 of \emph{Lectures on Moduli Spaces of Elliptic Curves} (\href{http://arxiv.org/abs/0812.1803}{arXiv:0812.1803})

\end{itemize}
The relation of $\Gamma_1(2)$ to [[spin structures]] is discussed for instance in

\begin{itemize}%
\item [[Daniel Freed]], pages 24-25 of \emph{On determinant line bundles}, 1987 (\href{http://www.ma.utexas.edu/users/dafr/detsur.pdf}{pdf})


\item [[Edward Witten]], \emph{Elliptic Genera And Quantum Field Theory} , Commun.Math.Phys. 109 525 (1987) (\href{http://projecteuclid.org/euclid.cmp/1104117076}{Euclid})



\end{itemize}
\hypertarget{over_general_base_rings}{}\subsubsection*{{Over general base rings}}\label{over_general_base_rings}

The concept of level structure on an elliptic curve is due to

\begin{itemize}%
\item Nicholas M. Katz, [[Barry Mazur]], \emph{Arithmetic moduli of elliptic curves}, Princeton University Press, Princeton, NJ, 1985

\end{itemize}
Review of the definition includes

\begin{itemize}%
\item [[Felipe Voloch]], \emph{Modular curves -- 1. Level structures} (\href{https://www.ma.utexas.edu/users/voloch/Ellnotes/feb23.pdf
}{pdf})

\end{itemize}
General discussion is in

\begin{itemize}%
\item [[Aaron Greicius]], \emph{Elliptic curves with surjective adelic Galois representations} (\href{http://arxiv.org/abs/0901.2513}{arXiv:0901.2513})


\item David Zywina, \emph{Elliptic curves with maximal Galois action on their torsion points} (\href{http://arxiv.org/abs/0809.3482}{arXiv:0809.3482})



\end{itemize}
Discussion of the corresponding [[moduli stack]] and its [[tmf]]$(n)$-spectrum is in

\begin{itemize}%
\item [[Matthew Ando]], section 1.4 of \emph{Power operations in elliptic cohomology and representations of loop groups} Transactions of the American Mathematical Society 352, 2000, pp. 5619-5666. (\href{http://www.jstor.org/stable/221905}{JSTOR}, \href{http://www.math.uiuc.edu/~mando/papers/POECLG/poeclg.pdf}{pdf})


\item [[Matthew Ando]], [[Michael Hopkins]], [[Neil Strickland]], part 3 of \emph{The sigma orientation is an H-infinity map}. American Journal of Mathematics Vol. 126, No. 2 (Apr., 2004), pp. 247-334 (\href{http://arxiv.org/abs/math/0204053}{arXiv:math/0204053})


\item [[Mark Mahowald]] [[Charles Rezk]], \emph{Topological modular forms of level 3}, Pure Appl. Math. Quar. 5 (2009) 853-872 (\href{http://www.math.uiuc.edu/~rezk/tmf3-paper-final.pdf}{pdf})


\item [[Michael Hill]], [[Tyler Lawson]], \emph{Topological modular forms with level structure} (\href{http://arxiv.org/abs/1312.7394}{arXiv:1312.7394})



\end{itemize}
Specifically Level-2 structure in this context is discussed in

\begin{itemize}%
\item [[Vesna Stojanoska]], \emph{Duality for Topological Modular Forms} (\href{http://arxiv.org/abs/1105.3968}{arXiv:1105.3968})


\item [[Mark Behrens]], section 1.3 of \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}
[[!redirects level structures on an elliptic curve]]

[[!redirects level structures on elliptic curves]]

[[!redirects level N-structure]] [[!redirects level N structure]] [[!redirects level N-structures]] [[!redirects level N structures]]

[[!redirects level-n structure]] [[!redirects level-n structures]]

[[!redirects level-n structure]] [[!redirects level-n structures]]

[[!redirects level n-structure]] [[!redirects level n structure]] [[!redirects level n-structures]] [[!redirects level n structures]]

[[!redirects level n subgroup]] [[!redirects level n subgroups]]

[[!redirects level n-structure on an elliptic curve]] [[!redirects level n-structure on elliptic curves]]

[[!redirects elliptic curve with level-n structure]] [[!redirects elliptic curves with level-n structure]]

[[!redirects elliptic curve with level structure]] [[!redirects elliptic curve with level structures]]

[[!redirects elliptic curves with level structure]] [[!redirects elliptic curves with level structures]]



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