Determining Modular Forms of Half-Integral Weight by Central Values of Convolution L-Functions

2020 ◽  
pp. 157-172
Author(s):  
Manish Kumar Pandey ◽  
B. Ramakrishnan
Keyword(s):  
Modular Forms ◽  
Central Values ◽  
Half Integral Weight ◽  
Integral Weight

On p-Adic Properties of Central L-Values of Quadratic Twists of an Elliptic Curve

2010 ◽  
Vol 62 (2) ◽  
pp. 400-414 ◽  
Author(s):  
Kartik Prasanna
Keyword(s):  
Elliptic Curve ◽  
Modular Forms ◽  
Central Values ◽  
Quadratic Twists ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Half Integral Weight Forms ◽  
Refined Version

AbstractWe study p-indivisibility of the central values L(1, Ed) of quadratic twists Ed of a semi-stable elliptic curve E of conductor N. A consideration of the conjecture of Birch and Swinnerton-Dyer shows that the set of quadratic discriminants d splits naturally into several families ℱS, indexed by subsets S of the primes dividing N. Let δS = gcdd∈ℱSL(1, Ed)alg, where L(1, Ed)alg denotes the algebraic part of the central L-value, L(1, Ed). Our main theorem relates the p-adic valuations of δS as S varies. As a consequence we present an application to a refined version of a question of Kolyvagin. Finally we explain an intriguing (albeit speculative) relation betweenWaldspurger packets on and congruences of modular forms of integral and half-integral weight. In this context, we formulate a conjecture on congruences of half-integral weight forms and explain its relevance to the problem of p-indivisibility of L-values of quadratic twists.

ARITHMETIC TRACES OF NON-HOLOMORPHIC MODULAR INVARIANTS

2010 ◽  
Vol 06 (01) ◽  
pp. 69-87 ◽  
Author(s):  
ALISON MILLER ◽  
AARON PIXTON
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Algebraic Numbers ◽  
Heegner Points ◽  
Positive Weight ◽  
Poincare Series ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Modular Invariants

We extend results of Bringmann and Ono that relate certain generalized traces of Maass–Poincaré series to Fourier coefficients of modular forms of half-integral weight. By specializing to cases in which these traces are usual traces of algebraic numbers, we generalize results of Zagier describing arithmetic traces associated to modular forms. We define correspondences [Formula: see text] and [Formula: see text]. We show that if f is a modular form of non-positive weight 2 - 2 λ and odd level N, holomorphic away from the cusp at infinity, then the traces of values at Heegner points of a certain iterated non-holomorphic derivative of f are equal to Fourier coefficients of the half-integral weight modular forms [Formula: see text].

scholarly journals On the Fourier expansions of Jacobi forms of half-integral weight

2006 ◽  
Vol 2006 ◽  
pp. 1-11
Author(s):  
B. Ramakrishnan ◽  
Brundaban Sahu
Keyword(s):  
Modular Forms ◽  
Jacobi Form ◽  
Jacobi Forms ◽  
Number Of Components ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Fourier Expansions ◽  
The Given ◽  
The Relationship ◽  
Vector Valued

Using the relationship between Jacobi forms of half-integral weight and vector valued modular forms, we obtain the number of components which determine the given Jacobi form of indexp,p2orpq, wherepandqare odd primes.

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