Combinatorics of traces of Hecke operators
- *Department of Mathematics and Computer Science, College of the Holy Cross, 1 College Street, Worcester, MA 01610; †Department of Mathematics, Van Vleck Hall, University of Wisconsin, Madison, WI 53706; and §Department of Mathematics, Texas A&M University, College Station, TX 77843-3368
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Communicated by George E. Andrews, Pennsylvania State University, University Park, PA, September 30, 2004 (received for review March 17, 2004)
Abstract
We investigate the combinatorial properties of the traces of the nth Hecke operators on the spaces of weight 2k cusp forms of level N. We establish examples in which these traces are expressed in terms of classical objects in enumerative combinatorics (e.g., tilings and Motzkin paths). We establish in general that Hecke traces are explicit rational linear combinations of values of Gegenbauer (also known as ultraspherical) polynomials. These results arise from “packaging” the Hecke traces into power series in weight aspect. These generating functions are easily computed by using the Eichler–Selberg trace formula.
Footnotes
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↵ ‡ To whom correspondence should be addressed. E-mail: ono{at}math.wisc.edu.
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Author contributions: K.O., S.F., and M.P. performed research and wrote the paper.
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Abbreviation: resp., respectively.
- Copyright © 2004, The National Academy of Sciences
