## Abstract

Special values of the modular j function at imaginary quadratic points in the upper half-plane are known as singular moduli; these are algebraic integers that play many roles in number theory. Zagier proved that the traces (and more generally, the Hecke traces) of singular moduli are described by a multiply infinite family of weight 3/2 weakly holomorphic modular forms of level 4 (or, through what is sometimes called 'duality', by a multiply infinite family of weight 1/2 weakly holomorphic modular forms of level 4). Several authors have used this description to obtain relations and congruences for these traces modulo prime powers p ^{n} in various situations. We prove that the modular forms in question satisfy a simple relationship involving the Hecke operators T(p ^{2n}) for n≥1. As a corollary we obtain uniform relations for the traces (some of which were known in particular cases).

Original language | English (US) |
---|---|

Pages (from-to) | 99-105 |

Number of pages | 7 |

Journal | Bulletin of the London Mathematical Society |

Volume | 44 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2012 |

## ASJC Scopus subject areas

- Mathematics(all)