No CrossRef data available.
Article contents
$23$-REGULAR PARTITIONS AND MODULAR FORMS WITH COMPLEX MULTIPLICATION
Published online by Cambridge University Press: 23 December 2022
Abstract
A partition of a positive integer n is called $\ell $-regular if none of its parts is divisible by
$\ell $. Denote by
$b_{\ell }(n)$ the number of
$\ell $-regular partitions of n. We give a complete characterisation of the arithmetic of
$b_{23}(n)$ modulo
$11$ for all n not divisible by
$11$ in terms of binary quadratic forms. Our result is obtained by establishing a relation between the generating function for these values of
$b_{23}(n)$ and certain modular forms having complex multiplication by
${\mathbb Q}(\sqrt {-69})$.
Keywords
MSC classification
- Type
- Research Article
- Information
- Bulletin of the Australian Mathematical Society , Volume 108 , Issue 2 , October 2023 , pp. 254 - 263
- Copyright
- © The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
References
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230910093347184-0049:S0004972722001393:S0004972722001393_inline231.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230910093347184-0049:S0004972722001393:S0004972722001393_inline232.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230910093347184-0049:S0004972722001393:S0004972722001393_inline233.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230910093347184-0049:S0004972722001393:S0004972722001393_inline234.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230910093347184-0049:S0004972722001393:S0004972722001393_inline235.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230910093347184-0049:S0004972722001393:S0004972722001393_inline236.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230910093347184-0049:S0004972722001393:S0004972722001393_inline237.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230910093347184-0049:S0004972722001393:S0004972722001393_inline238.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20230910093347184-0049:S0004972722001393:S0004972722001393_inline239.png?pub-status=live)