Notes on Ramanujan’s singular moduli

1999 ◽  
pp. 7-16 ◽  
Author(s):  
Bruce Berndt ◽  
Heng Huat Chan
Keyword(s):  
Singular Moduli

scholarly journals LINEAR INDEPENDENCE OF POWERS OF SINGULAR MODULI OF DEGREE THREE

2018 ◽  
Vol 99 (1) ◽  
pp. 42-50
Author(s):  
FLORIAN LUCA ◽  
ANTONIN RIFFAUT
Keyword(s):  
Number Theory ◽  
Number Field ◽  
Linear Independence ◽  
Singular Moduli ◽  
Positive Integers

We show that two distinct singular moduli $j(\unicode[STIX]{x1D70F}),j(\unicode[STIX]{x1D70F}^{\prime })$, such that for some positive integers $m$ and $n$ the numbers $1,j(\unicode[STIX]{x1D70F})^{m}$ and $j(\unicode[STIX]{x1D70F}^{\prime })^{n}$ are linearly dependent over $\mathbb{Q}$, generate the same number field of degree at most two. This completes a result of Riffaut [‘Equations with powers of singular moduli’, Int. J. Number Theory, to appear], who proved the above theorem except for two explicit pairs of exceptions consisting of numbers of degree three. The purpose of this article is to treat these two remaining cases.

scholarly journals Equations with powers of singular moduli

2019 ◽  
Vol 15 (03) ◽  
pp. 445-468 ◽  
Author(s):  
Antonin Riffaut
Keyword(s):  
Rational Number ◽  
The Other ◽  
Other Hand ◽  
Singular Moduli ◽  
Cm Points ◽  
The One ◽  
Straight Lines ◽  
Positive Integers

We treat two different equations involving powers of singular moduli. On the one hand, we show that, with two possible (explicitly specified) exceptions, two distinct singular moduli [Formula: see text] such that the numbers [Formula: see text], [Formula: see text] and [Formula: see text] are linearly dependent over [Formula: see text] for some positive integers [Formula: see text], must be of degree at most [Formula: see text]. This partially generalizes a result of Allombert, Bilu and Pizarro-Madariaga, who studied CM-points belonging to straight lines in [Formula: see text] defined over [Formula: see text]. On the other hand, we show that, with obvious exceptions, the product of any two powers of singular moduli cannot be a non-zero rational number. This generalizes a result of Bilu, Luca and Pizarro-Madariaga, who studied CM-points belonging to a hyperbola [Formula: see text], where [Formula: see text].

scholarly journals Heegner divisors in generalized Jacobians and traces of singular moduli

2016 ◽  
Vol 10 (6) ◽  
pp. 1277-1300 ◽  
Author(s):  
Jan Hendrik Bruinier ◽  
Yingkun Li
Keyword(s):  
Generalized Jacobians ◽  
Singular Moduli

scholarly journals Completions and algebraic formulas for the coefficients of Ramanujan’s mock theta functions

2020 ◽  
Author(s):  
David Klein ◽  
Jennifer Kupka
Keyword(s):  
Theta Functions ◽  
Mock Theta Functions ◽  
Maass Forms ◽  
Theta Lift ◽  
Singular Moduli

Abstract We present completions of mock theta functions to harmonic weak Maass forms of weight $$\nicefrac {1}{2}$$ 1 2 and algebraic formulas for the coefficients of mock theta functions. We give several harmonic weak Maass forms of weight $$\nicefrac {1}{2}$$ 1 2 that have mock theta functions as their holomorphic part. Using these harmonic weak Maass forms and the Millson theta lift, we compute finite algebraic formulas for the coefficients of the appearing mock theta functions in terms of traces of singular moduli.

Singular Moduli (3)†

1936 ◽  
Vol s2-40 (1) ◽  
pp. 83-142 ◽  
Author(s):  
G. N. Watson
Keyword(s):  
Singular Moduli

scholarly journals On Singular Moduli for Arbitrary Discriminants

2014 ◽  
Vol 2015 (19) ◽  
pp. 9206-9250 ◽  
Author(s):  
Kristin Lauter ◽  
Bianca Viray
Keyword(s):  
Singular Moduli

scholarly journals ON THE p-ADIC GEOMETRY OF TRACES OF SINGULAR MODULI

2005 ◽  
Vol 01 (04) ◽  
pp. 495-497 ◽  
Author(s):  
BAS EDIXHOVEN
Keyword(s):  
Shimura Curves ◽  
Modular Curves ◽  
Genus Zero ◽  
Singular Moduli ◽  
Arbitrary Level ◽  
Local Nature ◽  
Do So

The aim of this article is to show that p-adic geometry of modular curves is useful in the study of p-adic properties of traces of singular moduli. In order to do so, we partly answer a question by Ono [7, Problem 7.30]. As our goal is just to illustrate how p-adic geometry can be used in this context, we focus on a relatively simple case, in the hope that others will try to obtain the strongest and most general results. For example, for p = 2, a result stronger than Theorem 2 is proved in [2], and a result on some modular curves of genus zero can be found in [8]. It should be easy to apply our method, because of its local nature, to modular curves of arbitrary level, as well as to Shimura curves.

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