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

作者：Bruinier Jan Hendrik; Ono Ken^*

来源：Advances in Mathematics, 2013, 246: 198-219.

DOI：10.1016/j.aim.2013.05.028

• Partition function

We prove that the coefficients of certain weight -1/2 harmonic Maass forms are %26quot;traces%26quot; 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 -24n + 1 ring class field.

• 出版日期2013-10-20