---
title: "Self-conjugate core partitions and modular forms"
authors:
  - "Levent Alpoge"
year: 2014
venue: "Journal of Number Theory"
arxiv: "1307.0158"
doi: "10.1016/j.jnt.2014.01.010"
source: "https://arxiv.org/abs/1307.0158"
topics:
  - "number-theory"
  - "modular-forms"
seed_rank: 1214
seed_batch: "alpoge-math-2026-08-24"
reviewed: "2026-08-24"
pool: "maths-foundations"
relevance_score: 8
lineage: arithmetic-geometry
cites:
  []
see:
  - "1213-proof-of-a-conjecture-of-stanley-zanello"
---

# Self-conjugate core partitions and modular forms

## One-sentence takeaway

Prove all but two of the Hanusa-Nath conjectures on self-conjugate t-cores asymptotically, and give exact formulas for small t via modular forms (9-cores related to elliptic curves of conductor dividing 108).

## Why it's lovely

Undergraduate modular forms meeting partition combinatorics.

## Problem

Count self-conjugate t-core partitions; Hanusa-Nath listed conjectures.

## Design

Asymptotics plus explicit modular-form identities for small t.

## Evidence

J. Number Theory 140 (2014), 60-92. DOI 10.1016/j.jnt.2014.01.010. arXiv:1307.0158.

## Limitations

Two conjectures left. Duke-Schulze-Pillot ineffectivity for 6-cores.

## Implications for Broadside

Early JNT paper.

## Bottom line

Mint the journal version.

## Links

- arXiv: [1307.0158](https://arxiv.org/abs/1307.0158)
- DOI: [10.1016/j.jnt.2014.01.010](https://doi.org/10.1016/j.jnt.2014.01.010)