Modular Forms of Half Integer Weight

1993 ◽  
pp. 176-222
Author(s):  
Neal Koblitz
Keyword(s):  
Modular Forms ◽  
Half Integer ◽  
Integer Weight

scholarly journals CONGRUENCES FOR ANDREWS' SMALLEST PARTS PARTITION FUNCTION AND NEW CONGRUENCES FOR DYSON'S RANK

2010 ◽  
Vol 06 (02) ◽  
pp. 281-309 ◽  
Author(s):  
F. G. GARVAN
Keyword(s):  
Partition Function ◽  
Rank Function ◽  
Maass Forms ◽  
Hecke Eigenforms ◽  
Half Integer ◽  
Integer Weight

Let spt (n) denote the total number of appearances of smallest parts in the partitions of n. Recently, Andrews showed how spt (n) is related to the second rank moment, and proved some surprising Ramanujan-type congruences mod 5, 7 and 13. We prove a generalization of these congruences using known relations between rank and crank moments. We obtain explicit Ramanujan-type congruences for spt (n) mod ℓ for ℓ = 11, 17, 19, 29, 31 and 37. Recently, Bringmann and Ono proved that Dyson's rank function has infinitely many Ramanujan-type congruences. Their proof is non-constructive and utilizes the theory of weak Maass forms. We construct two explicit nontrivial examples mod 11 using elementary congruences between rank moments and half-integer weight Hecke eigenforms.

On the Sizes of Gaps in the Fourier Expansion of Modular Forms

2005 ◽  
Vol 57 (3) ◽  
pp. 449-470 ◽  
Author(s):  
Emre Alkan
Keyword(s):  
Cusp Form ◽  
Modular Forms ◽  
Fourier Expansion ◽  
Complex Multiplication ◽  
The Riemann Zeta Function ◽  
Classical Paper ◽  
Riemann Zeta ◽  
Image Position ◽  
Almost All ◽  
Integer Weight

AbstractLet be a cusp form with integer weight k ≥ 2 that is not a linear combination of forms with complex multiplication. For n ≥ 1, letConcerning bounded values of i f (n) we prove that for ∊ > 0 there exists M = M(∊, f ) such that Using results of Wu, we show that if f is a weight 2 cusp form for an elliptic curve without complex multiplication, then . Using a result of David and Pappalardi, we improve the exponent to for almost all newforms associated to elliptic curves without complex multiplication. Inspired by a classical paper of Selberg, we also investigate i f (n) on the average using well known bounds on the Riemann Zeta function.

SYMMETRY INSIGHTS FOR DESIGN OF SUPERCOMPUTER NETWORK TOPOLOGIES: ROOTS AND WEIGHTS LATTICES

2012 ◽  
Vol 26 (31) ◽  
pp. 1250169 ◽  
Author(s):  
YUEFAN DENG ◽  
ALEXANDRE F. RAMOS ◽  
JOSÉ EDUARDO M. HORNOS
Keyword(s):  
Weyl Group ◽  
Lie Algebra ◽  
Root Systems ◽  
Hardware Architecture ◽  
Classification Theorem ◽  
Simple Complex ◽  
Half Integer ◽  
Network Topologies ◽  
Integer Weight

We present a family of networks whose local interconnection topologies are generated by the root vectors of a semi-simple complex Lie algebra. Cartan classification theorem of those algebras ensures those families of interconnection topologies to be exhaustive. The global arrangement of the network is defined in terms of integer or half-integer weight lattices. The mesh or torus topologies that network millions of processing cores, such as those in the IBM BlueGene series, are the simplest member of that category. The symmetries of the root systems of an algebra, manifested by their Weyl group, lends great convenience for the design and analysis of hardware architecture, algorithms and programs.

scholarly journals ARITHMETIC PROPERTIES OF CERTAIN LEVEL ONE MOCK MODULAR FORMS

2010 ◽  
Vol 06 (01) ◽  
pp. 185-202 ◽  
Author(s):  
MATTHEW BOYLAN
Keyword(s):  
Theta Function ◽  
Modular Forms ◽  
Hecke Operators ◽  
Mock Theta Function ◽  
Maass Forms ◽  
Maass Form ◽  
Arithmetic Properties ◽  
Holomorphic Projection ◽  
Harmonic Weak Maass Form ◽  
Integer Weight

In a recent work, Bringmann and Ono [4] show that Ramanujan's f(q) mock theta function is the holomorphic projection of a harmonic weak Maass form of weight 1/2. In this paper, we extend the work of Ono in [13]. In particular, we study holomorphic projections of certain integer weight harmonic weak Maass forms on SL 2(ℤ) using Hecke operators and the differential theta-operator.

POWERS OF THE ETA-FUNCTION AND HECKE OPERATORS

2012 ◽  
Vol 08 (03) ◽  
pp. 599-611 ◽  
Author(s):  
ALEXANDER CARNEY ◽  
ANASTASSIA ETROPOLSKI ◽  
SARAH PITMAN
Keyword(s):  
Partition Function ◽  
Large Class ◽  
Delta Function ◽  
Hecke Operators ◽  
Faber Polynomials ◽  
Partition Functions ◽  
Fast Calculation ◽  
Half Integer ◽  
Integer Weight

Half-integer weight Hecke operators and their distinct properties play a major role in the theory surrounding partition numbers and Dedekind's eta-function. Generalizing the work of Ono in [K. Ono, The partition function and Hecke operators, Adv. Math.228 (2011) 527–534], here we obtain closed formulas for the Hecke images of all negative powers of the eta-function. These formulas are generated through the use of Faber polynomials. In addition, congruences for a large class of powers of Ramanujan's Delta-function are obtained in a corollary. We further exhibit a fast calculation for many large values of vector partition functions.

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