Transactions of the American Mathematical Society

Author(s): Jeremy Rouse
Journal: Trans. Amer. Math. Soc. 358 (2006), 4637-4651.
MSC (2000): Primary 11F25; Secondary 11F72
Posted: May 9, 2006
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Abstract: Using a reformulation of the Eichler-Selberg trace formula, due to Frechette, Ono and Papanikolas, we consider the problem of the vanishing (resp. non-vanishing) of traces of Hecke operators on spaces of even weight cusp forms with trivial Nebentypus character. For example, we show that for a fixed operator and weight, the set of levels for which the trace vanishes is effectively computable. Also, for a fixed operator the set of weights for which the trace vanishes (for any level) is finite. These results motivate the ``generalized Lehmer conjecture'', that the trace does not vanish for even weights $2k \geq 16$ or

1.: G. Everest, A. van der Poorten, I. Shparlinski, and T. Ward, Recurrence sequences, Mathematical Surveys and Monographs, vol. 104, American Mathematical Society, Providence, RI, 2003. MR 1990179 (2004c:11015)
2.: S. Frechette, K. Ono, and M. Papanikolas, The combinatorics of traces of Hecke operators, Proc. Natl. Acad. Sci. 101 (2004), 17016-17020. MR 2114776 (2005h:11090)
3.: G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, The Clarendon Press, Oxford University Press, New York, 1979. MR 0568909 (81i:10002)
4.: H. Hijikata, A. Pizer, and T. Shemanske, The basis problem for modular forms on $\Gamma \sb{0}(N)$ , Proc. Japan Acad. Ser. A Math. Sci. 56 (1980), no. 6, 280-284. MR 0581471 (81k:10045)
5.: N. Koblitz, Introduction to elliptic curves and modular forms, Graduate Texts in Mathematics, vol. 97, Springer-Verlag, New York, 1993. MR 1216136 (94a:11078)
6.: D. H. Lehmer, The vanishing of Ramanujan's function $\tau(n)$ , Duke Math. J. 14 (1947), 429-433. MR 0021027 (9:12b)
7.: K. Ono, The web of modularity: arithmetic of the coefficients of modular forms and -series, CBMS Regional Conference Series in Mathematics, vol. 102, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2004. MR 2020489 (2005c:11053)
8.: The PARI Group, Bordeaux, Pari/gp, version 2.0.20, 1999, available from http://pari.math.u-bordeaux.fr/.
9.: B. M. Phong, On generalized Lehmer sequences, Acta Math. Hungar. 57 (1991), no. 3-4, 201-211. MR 1139314 (92m:11021)
10.: H. P. Schlickewei and W. M. Schmidt, The number of solutions of polynomial-exponential equations, Compositio Math. 120 (2000), no. 2, 193-225. MR 1739179 (2001b:11022)
11.: J.-P. Serre, A course in arithmetic, Springer-Verlag, New York, 1973. MR 0344216 (49:8956)
12.: -, Divisibilité de certaines fonctions arithmétiques, Enseignement Math. (2) 22 (1976), no. 3-4, 227-260. MR 0434996 (55:7958)
13.: -, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. (1981), no. 54, 323-401. MR 0644559 (83k:12011)

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Jeremy Rouse
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email: rouse@math.wisc.edu

DOI: 10.1090/S0002-9947-06-03896-7
PII: S 0002-9947(06)03896-7
Received by editor(s): July 15, 2004
Received by editor(s) in revised form: November 8, 2004
Posted: May 9, 2006
Additional Notes: This research was supported by the NDSEG Fellowship Program, which is sponsored by the Department of Defense and the Office of Naval Research.
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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