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Hecke relations for traces of singular moduli
Author(s) -
Ahlgren Scott
Publication year - 2012
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdr072
Subject(s) - mathematics , modular form , holomorphic function , moduli , pure mathematics , hecke operator , congruence relation , eisenstein series , duality (order theory) , modular equation , moduli space , algebraic number , simple (philosophy) , algebra over a field , mathematical analysis , moduli of algebraic curves , philosophy , epistemology , physics , quantum mechanics
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 2 n ) for n ⩾1. As a corollary we obtain uniform relations for the traces (some of which were known in particular cases).