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CONGRUENCES MODULO 4 FOR WEIGHT $\textbf{3/2}$ ETA-PRODUCTS

Part of: Forms and linear algebraic groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  05 October 2020

RONG CHEN Open the ORCID record for RONG CHEN [Opens in a new window]  and
F. G. GARVAN Open the ORCID record for F. G. GARVAN [Opens in a new window]
RONG CHEN*
Affiliation:
School of Mathematical Sciences, East China Normal University, Shanghai, People’s Republic of China
F. G. GARVAN
Affiliation:
Department of Mathematics, University of Florida, Gainesville, FL32611-8105, USA e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

We find and prove a class of congruences modulo 4 for eta-products associated with certain ternary quadratic forms. This study was inspired by similar conjectured congruences modulo 4 for certain mock theta functions.

Keywords

ternary quadratic formsRamanujan-type congruences

MSC classification

Primary: 11E20: General ternary and quaternary quadratic forms; forms of more than two variables
Secondary: 11F33: Congruences for modular and $p$-adic modular forms
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 103 , Issue 3 , June 2021 , pp. 405 - 417
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

The second author was supported in part by a grant from the Simons Foundation (#318714).

References

Berndt, B. C., Ramanujan’s Notebooks, Part III (Springer-Verlag, New York, 1991).CrossRefGoogle Scholar
Borevich, Z. I. and Shafarevich, I. R., Number Theory (Academic Press, New York, 1966).Google Scholar
Bryson, J., Ono, K., Pitman, S. and Rhoades, R. C., ‘Unimodal sequences and quantum and mock modular forms’, Proc. Natl. Acad. Sci. USA 109(40) (2012), 16063–16067.CrossRefGoogle Scholar
Cooper, S. and Lam, H. Y., ‘On the Diophantine equation ${n}^2={x}^2+b{y}^2+c{z}^2$ ’, J. Number Theory 133 (2013), 719–737.CrossRefGoogle Scholar
Gauss, C. F., Disquisitiones Arithmeticae (Fleischer, Leipzig, 1801).CrossRefGoogle Scholar
Grosswald, E., Representations of Integers as Sums of Squares (Springer-Verlag, Berlin, 1984).Google Scholar
Guo, X. J., Peng, Y. Z. and Qin, H. R., ‘On the representation numbers of ternary quadratic forms and modular forms of weight $3/ 2$ ’, J. Number Theory 140 (2014), 235–266.10.1016/j.jnt.2014.01.024CrossRefGoogle Scholar
Hirschhorn, M. D. and Sellers, J., ‘On representations of a number as a sum of three squares’, Discrete Math. 199 (1999), 85–101.CrossRefGoogle Scholar
Kim, B., Lim, S. and Lovejoy, J., ‘Odd-balanced unimodal sequences and related functions: Parity, mock modularity and quantum modularity’, Proc. Amer. Math. Soc. 144 (2016), 3687–3700.CrossRefGoogle Scholar
Shemanske, T. R., ‘Representations of ternary quadratic forms and the class number of imaginary quadratic fields’, Pacific J. Math. 122(1) (1986), 223–250.CrossRefGoogle Scholar