$2$-adic properties of Hecke traces of singular moduli

Boylan, Matthew

doi:10.4310/mrl.2005.v12.n4.a12

Cited by 9 publications

(11 citation statements)

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“…For small primes one expects that more will be true. For example, Boylan [4] has given an elegant and precise description of the 2-adic properties of the numbers tr m (d). Finally, we mention that methods similar to those below have recently been used in joint work with Bringmann and Lovejoy [1] to study congruences for mock modular forms of weight 3/2 associated to various partition generating functions.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Hecke relations for traces of singular moduli

Ahlgren

2011

Bulletin of the London Mathematical Society

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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).Each J m can be expressed as a polynomial in j; for example, we have J 0 (τ ) = 1, (τ ) + 159 768 = q −2 + 42 987 520q + . . . , J 3 (τ ) = j(τ ) 3 − 2232j(τ ) 2 + 1 069 956j(τ ) − 36 866 976 = q −3 + 2 592 899 910q + 12 756 069 900 288q 2 + 9 529 320 689 550 144q 3 + . . . . Alternatively, we may define J m := J|T 0 (m), where T 0 (m) is the Hecke operator of index m and weight 0 (normalized so that its action preserves integrality). For each m 0 we define the mth Hecke trace of the singular moduli of discriminant −d as tr m (d) := Q∈Q d /Γ

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Hecke relations for traces of singular moduli

Ahlgren

2011

Bulletin of the London Mathematical Society

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“…Furthermore, when λ = 1 and 1 − λ = 0, Theorem 2.1 (3) and (4) shows that the class number summands obey the alleged duality. To complete the proof one simply observes that we may transform the formula for b λ (−m; n) into the formula for −b 1−λ (−n; m).…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

Arithmetic properties of coefficients of half-integral weight Maass–Poincaré series

Bringmann

Ono

2006

Math. Ann.

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Zagier [23] proved that the generating functions for the traces of level 1 singular moduli are weight 3/2 modular forms. He also obtained generalizations for "twisted traces", and for traces of special non-holomorphic modular functions. Using properties of Kloosterman-Salié sums, and a well known reformulation of Salié sums in terms of orbits of CM points, we systematically show that such results hold for arbitrary weakly holomorphic and cuspidal half-integral weight Poincaré series in Kohnen's 0 (4) plus-space. These results imply the aforementioned results of Zagier, and they provide exact formulas for such traces.

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“…An enormous amount of research has followed; while it is impossible to provide a complete list, we mention the papers [3,5,11,12,15,16,[24][25][26]29] as well as the references therein.…”

Section: Introductionmentioning

confidence: 99%

Mock modular grids and Hecke relations for mock modular forms

Ahlgren

Kim

2012

Forum Mathematicum

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We construct and study several infinite grids of mock modular forms. The "rows" of each grid encode an infinite family of forms, while the "columns" encode a second infinite family. Each of these grids contains a generating function of interest as its first entry (for example, the "smallest parts" function of Andrews, or a mock theta function of Ramanujan). We show that the Hecke operators transform these grids in systematic ways, and from this we deduce many corollaries for the generating functions under consideration. For example, we recover the systematic families of congruences for the smallest parts function which have recently been discovered by various authors, and we obtain families of Hecke relations in the grid associated to the mock theta function f .q/.

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$2$-adic properties of Hecke traces of singular moduli

Cited by 9 publications

References 1 publication

Hecke relations for traces of singular moduli

Hecke relations for traces of singular moduli

Arithmetic properties of coefficients of half-integral weight Maass–Poincaré series

Mock modular grids and Hecke relations for mock modular forms

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