Arithmetic properties of non-harmonic weak Maass forms
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- by Kathrin Bringmann and David Penniston PDF
- Proc. Amer. Math. Soc. 137 (2009), 825-833 Request permission
Abstract:
We prove the existence of an infinite family of non-harmonic weak Maass forms of varying weights and Laplace eigenvalues having algebraic coefficients, and show that the coefficients of these forms satisfy congruences of Ramanujan type.References
- M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, Dover Publications, 1965.
- K. Bringmann, Asymptotics for rank partition functions, Trans. Amer. Math. Soc., accepted for publication.
- K. Bringmann, On certain congruences for Dyson’s ranks, Int. J. Number Theory, accepted for publication.
- K. Bringmann, J. Bruinier, A. Folsom and K. Ono, Harmonic Maass forms and Borcherds products, in preparation.
- K. Bringmann, A. Folsom and K. Ono, q-series and weight $3/2$ Maass forms, submitted for publication.
- Kathrin Bringmann and Ken Ono, The $f(q)$ mock theta function conjecture and partition ranks, Invent. Math. 165 (2006), no. 2, 243–266. MR 2231957, DOI 10.1007/s00222-005-0493-5
- K. Bringmann and K. Ono, Dyson’s ranks and Maass forms, Ann. of Math., accepted for publication.
- Kathrin Bringmann and Ken Ono, Arithmetic properties of coefficients of half-integral weight Maass-Poincaré series, Math. Ann. 337 (2007), no. 3, 591–612. MR 2274544, DOI 10.1007/s00208-006-0048-0
- Kathrin Bringmann and Ken Ono, Lifting cusp forms to Maass forms with an application to partitions, Proc. Natl. Acad. Sci. USA 104 (2007), no. 10, 3725–3731. MR 2301875, DOI 10.1073/pnas.0611414104
- Jan Hendrik Bruinier and Jens Funke, On two geometric theta lifts, Duke Math. J. 125 (2004), no. 1, 45–90. MR 2097357, DOI 10.1215/S0012-7094-04-12513-8
- Jan Hendrik Bruinier, Paul Jenkins, and Ken Ono, Hilbert class polynomials and traces of singular moduli, Math. Ann. 334 (2006), no. 2, 373–393. MR 2207703, DOI 10.1007/s00208-005-0723-6
- S. Garthwaite, The coefficients of the $\omega (q)$ mock theta function, Int. J. Number Theory, accepted for publication.
- S. Garthwaite and D. Penniston, p-adic properties of Maass forms arising from theta series, Math. Res. Lett., accepted for publication.
- F. Hirzebruch and D. Zagier, Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus, Invent. Math. 36 (1976), 57–113. MR 453649, DOI 10.1007/BF01390005
- Ken Ono, Distribution of the partition function modulo $m$, Ann. of Math. (2) 151 (2000), no. 1, 293–307. MR 1745012, DOI 10.2307/121118
- Srinivasa Ramanujan, The lost notebook and other unpublished papers, Springer-Verlag, Berlin; Narosa Publishing House, New Delhi, 1988. With an introduction by George E. Andrews. MR 947735
- Goro Shimura, On modular forms of half integral weight, Ann. of Math. (2) 97 (1973), 440–481. MR 332663, DOI 10.2307/1970831
Additional Information
- Kathrin Bringmann
- Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
- Address at time of publication: Mathematisches Institut, Universität Köln, Weyertal 86-90, 50931 Köln, Germany
- MR Author ID: 774752
- Email: kbringma@math.uni-koeln.de
- David Penniston
- Affiliation: Department of Mathematics, Furman University, Greenville, South Carolina 29613
- Email: david.penniston@furman.edu
- Received by editor(s): July 23, 2007
- Received by editor(s) in revised form: February 26, 2008
- Published electronically: September 12, 2008
- Communicated by: Ken Ono
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 825-833
- MSC (2000): Primary 11F33
- DOI: https://doi.org/10.1090/S0002-9939-08-09541-5
- MathSciNet review: 2457420