2-Elongated Plane Partitions and Powers of 7: The Localization Method Applied to a Genus 1 Congruence Family

TL;DR

This paper extends the localization method to analyze a genus 1 congruence family related to 2-elongated plane partitions and powers of 7, demonstrating its effectiveness beyond genus 0 cases.

Contribution

It adapts the localization technique to a genus 1 modular curve and applies it to a new congruence family involving plane partitions and powers of 7.

Findings

Successfully applied the method to a genus 1 congruence family

Compared the technique with existing methods for genus 1 cases

Conjectured a new congruence family potentially analyzable with similar methods

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.