Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms

By Jan H. Bruinier, Ken Ichi Ono. 2011

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Abstract We prove that the coefficients of certain weight − 1 / 2 harmonic Maass forms are “traces” of singular moduli for weak Maass forms. To prove this theorem, we construct a theta lift from spaces of weight − 2 harmonic weak Maass forms to spaces of weight − 1 / 2 vector-valued harmonic weak Maass forms on Mp 2 ( Z ) , a result which is of independent interest. We then prove a general theorem which guarantees (with bounded denominator) when such Maass singular moduli are algebraic. As an example of these results, we derive a formula for the partition function p ( n ) as a finite sum of algebraic numbers which lie in the usual discriminant − 24 n + 1 ring class field.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b36e6e1247c8ab9b8e1ceab621162bf7dd7468a1

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In Algebra and Number Theory

In Mathematical Physics

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Maximal Multiplicative Properties of Partitions (2014, background): “By evaluating Ta(1) and Sa(1) directly, one easily finds that (2.”