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\textbf{On the arithmetic of operations and cooperations in elliptic
cohomology.}

by Imma G\'{a}lvez i Carrillo (London Metropolitan University)

(joint work in progress with Sarah Whitehouse (University of
Sheffield)\medskip \medskip 

\strut In this talk, we want to report on our current project with Sarah
Whitehouse \cite{gw} on (co)operations in elliptic cohomology. In recent
work \cite{ccw},\cite{ccw2}, she and her coworkers have succeeded in giving
explicit bases for cooperations and operations in $p$-local $K$-theory.
(Co)operations in elliptic cohomology have been studied (see e.g. \cite
{baker95}a.o.,\cite{ando},\cite{cj},\cite{tan} for many important results).

\strut 

In both theories, cooperations have a very nice description as elements
satisfying certain integrality conditions inside bigger rational rings. In
the case of $K$-theory, (stably)numerical polynomials appear, and in \cite
{ccw} the authors were able to use Gaussian polynomials to obtain the
desired basis out of the integrality conditions. In the elliptic case, the
works above quoted have determined that the ring of cooperations is some
kind of divided congruences ring in the spirit of \cite{katz}.  We will
discuss the situation in the elliptic cohomology case and see how far the
analogy with the $K$-theory case can go.

\strut 

For doing so, we will take advantage of what general theory for
(co)operations in generalized cohomology theories as described e.g. in \cite
{bmn71} and \cite{bn71} has to say about it. We focus in level 1 elliptic
cohomology theory (eventually made periodic) for the sake of simplicity, but
we hope to hint towards more general settings, as topological modular forms (%
\cite{h}) when possible. 

\strut 

In the case of $p$-local $K$-theory, Adams operations give us everything (%
\cite{ccw2}). In the elliptic case, we have not only Adams operations, but
at least also Hecke operations (\cite{baker90}a.o.). Both Adams and Hecke
operations correspond to well known families of operators on modular forms,
that is, the coefficients of our theories. Explicit relations between the $p$%
-local elliptic version of Adams operations and classical polynomials from
the theory of modular forms appear hence naturally, as do Dirichlet series
associated to families of Adams and Hecke operations, and to the logarithm
of the formal group law involved, in the line of \cite{ls87}. 

\strut 

We may then go ahead on this way and take advantage of the fact that our
orientation classes are meromorphic Jacobi forms (\cite{ez85}). Roughly
speaking, Jacobi forms are functions of two complex variables who transform
like modular forms in the first one and like theta functions in the second.
We will see how the definitions of Adams and Hecke operations on the
orientation classes give us back essentially the classical operators in the
theory of Jacobi forms as they appear in (\cite{ez85},\cite{b92}). We will
see for instance how some of Borcherds product formulae show up in this
context. This takes us to consider the r\^{o}le of differential operators in
this picture, as and the Halphen-Fricke and Ramanujan \cite{g2003} and the
ones in \cite{gt}.

\begin{thebibliography}{99}
\bibitem{ando}  M.Ando, \emph{Power operations in elliptic cohomology and
representations of loop groups. }

\bibitem{b92}  R.Borcherds, \emph{Monstruous moonshine and mounstrous Lie
superalgebras}, Invent.Math. \textbf{109} (1992) 405-444.

\bibitem{baker90}  A.Baker, \emph{Hecke operators as operations in elliptic
cohomology}, Journal of Pure and Applied Algebra, \textbf{63}(1990), 1--11.

\bibitem{baker95}  A. Baker, \emph{Operations and cooperations in elliptic
cohomology, Part 1: generalized modular forms and the cooperation algebra},
New York J. Math. \textbf{1} (1995), 39--74.

\bibitem{bmn71}  Bukhshtaber, V.M., Mishchenko, A.S., Novikov, S.P. \emph{%
Formal groups and their role in the apparatus of algebraic topology},
Uspekhi Mat. Nauk \textbf{26}-2(1971), 131-154

\bibitem{bn71}  Bukhshtaber, V.M., Novikov, S.P. Formal groups, power
systems and Adams operators. Mat. Sb. \textbf{84} (1971) 116-153.

\bibitem{ccw}  F. Clarke, M. D. Crossley and S. Whitehouse, \emph{Bases for
cooperations in $K$-theory}, K-theory \textbf{23} (2001), 237-250.

\bibitem{ccw2}  F. Clarke, M. D. Crossley and S. Whitehouse, \emph{%
Operations and cooperations in $p$-local $K$-theory}, in preparation.

\bibitem{cj}  F. Clarke and K. Johnson, \emph{Cooperations in elliptic
homology}, Adams Memorial Symposium on Algebraic Topology, vol. 2, (N. Ray
and G. Walker, eds.), London Mathematical Society Lecture Notes, no. 175,
London Mathematical Society, London, 1992, 131--143.

\bibitem{ez85}  M.Eichler, D.Zagier, \emph{The theory of Jacobi forms},
Progress in Mathematics, \textbf{43}(1985) Birkh\"{a}user.

\bibitem{g2003}  I.G\'{a}lvez, \emph{Some differential equations related to
level 1 elliptic genera}, Poster, XI Encuentro de Topolog\'{i}a, Bilbao,
2003.

\bibitem{gt}  I.G\'{a}lvez,A.Tonks, \emph{Differential operators and the
Witten genus for projective spaces and Milnor manifolds.}
Math.Proc.Camb.Soc. 2003,\textbf{135,1}.

\bibitem{gw}  I.G\'{a}lvez,S.Whitehouse, \emph{Divided congruences and
elliptic cohomology}, in preparation.

\bibitem{h}  M.J.Hopkins, \emph{Topological modular forms}, Proceedings of
the 2002 ICM, Beijing.

\bibitem{katz}  N. M. Katz, \emph{Higher congruences between modular forms},
Annals of Math., \textbf{101} (1975), 332--367.

\bibitem{laures}  G. Laures, \emph{The topological $q$-expansion principle},
Topology \textbf{38} (1999), 387--425.

\bibitem{ls87}  L.Smith and R.E.Stong, \emph{Dirichlet series and Homology
theories}, pp.134-149, in P.S.Landweber (Ed.), \emph{Elliptic curves and
modular forms in algebraic topology}, Lecture Notes in Mathematics, \textbf{%
1326}(1987), Springer Verlag,

\bibitem{tan}  M.Tanabe \emph{A remark on Baker operations on the elliptic
cohomology of finite groups, }J.Math.Kyoto Univ. \textbf{40}-1(2000) 165-176
\end{thebibliography}

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