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2-Elongated plane partitions and powers of 7: the localization method applied to a genus 1 congruence family

Part of: Additive number theory; partitions Riemann surfaces

Published online by Cambridge University Press:  22 August 2025

Koustav Banerjee  and
Nicolas Smoot Open the ORCID record for Nicolas Smoot [Opens in a new window]
Koustav Banerjee*
Affiliation:
Research Institute of Symbolic Computation (RISC), Johannes Kepler University , Linz 4040, Austria
Nicolas Smoot
Affiliation:
Faculty of Mathematics, Universität Wien , Wien 1010, Austria e-mail: nicolas.allen.smoot@univie.ac.at
*
e-mail: koustav.banerjee@risc.uni-linz.ac.at
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Abstract

Over the last century, a large variety of infinite congruence families have been discovered and studied, exhibiting a great variety with respect to their difficulty. Major complicating factors arise from the topology of the associated modular curve: classical techniques are sufficient when the associated curve has cusp count 2 and genus 0. Recent work has led to new techniques that have proven useful when the associated curve has cusp count greater than 2 and genus 0. We show here that these techniques may be adapted in the case of positive genus. In particular, we examine a congruence family over the 2-elongated plane partition diamond counting function $d_2(n)$ by powers of 7, for which the associated modular curve has cusp count 4 and genus 1. We compare our method with other techniques for proving genus 1 congruence families, and conjecture a second congruence family by powers of 7, which may be amenable to similar techniques.

Keywords

Partition congruencesmodular functionsplane partitionsmodular curveRiemann surface

MSC classification

Primary: 11P83: Partitions; congruences and congruential restrictions 30F35: Fuchsian groups and automorphic functions

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Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 43
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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