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Self-conjugate core partitions and modular forms

A recent paper by Hanusa and Nath states many conjectures in the study of self-conjugate core partitions. We prove all but two of these conjectures asymptotically by number-theoretic means. We also obtain exact formulas for the number of self-conjugate t-core partitions for "small" t via explicit computations with modular forms. For instance, self-conjugate 9-core partitions are related to counting points on elliptic curves over \Q with conductor dividing 108, and self-conjugate 6-core partitions are related to the representations of integers congruent to 11 mod 24 by 3X^2 + 32Y^2 + 96Z^2, a form with finitely many (conjecturally five) exceptional integers in this arithmetic progression, by an ineffective result of Duke--Schulze-Pillot.