closes #112

Author: ecpeterson

sigma.tex

@@ -1217,7 +1217,7 @@ \section{Modular forms and \texorpdfstring{\(MU[6, \infty)\)}{MU[6, oo)}--manifo
 \end{example}
 
 \begin{example}\label{OrdinaryHomologyInUpperHalfPlaneEx}
-This leads us to consider all curves \(E_\Lambda\) simultaneously---or, equivalently, to consider modular forms.  The lattice \(\Lambda\) can be put into a standard form, by picking a basis and scaling it so that one vector lies at \(1\) and the other vector lies in the upper half-plane, \(\h\).  This gives a cover \[\h \to \moduli{ell} \times \Spec \C\] which is well-behaved (i.e., unramified) away from the special points \(\mathrm i\) and \(\mathrm e^{2 \pi i / 6}\).  A \index{modular form}\textit{complex modular form of weight \(n\)} is an analytic function \(\h \to \C\) which satisfies a certain decay condition and which is quasi-periodic for the action of \(SL_2(\Z)\), i.e.,\footnote{That is, for the action of change of basis vectors.} \[f\left(M; \frac{a \tau + b}{c \tau + d} \right) = (c \tau + d)^n f(M; \tau).\]
+This leads us to consider all curves \(E_\Lambda\) simultaneously---or, equivalently, to consider modular forms.  The lattice \(\Lambda\) can be put into a standard form, by picking a basis and scaling it so that one vector lies at \(1\) and the other vector lies in the upper half-plane, \(\h\).  This gives a cover \[\h \to \moduli{ell} \times \Spec \C\] which is well-behaved (i.e., unramified) away from the special points \(i\) and \(\mathrm e^{2 \pi i / 6}\).  A \index{modular form}\textit{complex modular form of weight \(n\)} is an analytic function \(\h \to \C\) which satisfies a certain decay condition and which is quasi-periodic for the action of \(SL_2(\Z)\), i.e.,\footnote{That is, for the action of change of basis vectors.} \[f\left(M; \frac{a \tau + b}{c \tau + d} \right) = (c \tau + d)^n f(M; \tau).\]
 
 Using these ideas, we construct a cohomology theory \(H \sheaf O_{\h} P\), where \(\sheaf O_{\h}\) is the ring of complex-analytic functions on the upper half-plane.  The \(\h\)--parametrized family of elliptic curves \[\h \times \C / (1, \tau) \to \h,\] together with the logarithm, present \(H \sheaf O_{\h} P\) as an elliptic spectrum \(H\h P\).  The canonical map \(\Phi\co MU[6, \infty) \to H\h P\) specializes at a point to give the functions \(\Phi(-; \Lambda)\) considered above, and hence \(\Phi(M) \in u^k \cdot \sheaf O_{\h}\) is itself a complex modular form of weight \(k\).
 \end{example}