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SUMMARY:Modular forms of negative weight - John Duncan (Cambridge)
DTSTART:20091124T143000Z
DTEND:20091124T153000Z
UID:TALK20559@talks.cam.ac.uk
CONTACT:Tom Fisher
DESCRIPTION:In 1939 Rademacher derived a conditionally convergent series\n
 expression for the elliptic modular invariant. This motivated\ninvestigati
 ons by various authors in to the problem of constructing\nmodular function
 s\, and even modular forms of negative weight\, for\ndiscrete groups of is
 ometries of the hyperbolic plane.\n\nWe will describe a generalization of 
 Rademacher's construction that\nfurnishes spanning sets for a certain subs
 pace of the space of\nmeromorphic modular forms of even integral weight\, 
 for any group\ncommensurable with the modular group. In the course of this
  we are\nled to an analytic continuation of the elliptic modular invariant
 \,\nand an association of Dirichlet series to groups commensurable with\nt
 he modular group.\n\nThe expressions we obtain behave well under the actio
 ns of Hecke\noperators and have simple branching rules. These facts lead t
 o\napplications in monstrous moonshine and three dimensional quantum\ngrav
 ity.\n\n
LOCATION:MR13
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