Modular Forms and Related Topics in Number Theory: Kozhikode, India, December 10–14, 2018

Preface
Conference on Number Theory
	List of Speakers
	List of Participants
Contents
On Vanishing of Hecke Operators
	1 Statement of Results
	2 Proof of Theorem 3
	3 Proof of Theorem 2
	4 Further Remarks
	References
On a Polynomial Method of Rónyai in the Study of Zero-Sum Theorems
	1 Introduction
	2 The Constant s (mathbbZnd)
	3 Weighted Zero-Sum Problems and Rónyai\'s Method
	References
p-adic Asai L-functions Attached to Bianchi Cusp Forms
	1 Preliminaries
	2 Complex Valued Distributions
	3 Rationality Result for Twisted Asai L-values
	4 Rationality of Eisenstein Cohomology Classes
	5 Towards Integrality
	6 Constructing Bounded Distributions
	References
Arithmetic Properties of Vector-Valued Siegel Modular Forms
	1 Introduction
	2 Generalities
	3 Construction of Integral Modular Forms of Degree n from Eisenstein Series of Degree 2n
	4 Linearized Pullback Formula
		4.1 On Vector-Valued Klingen–Eisenstein Series
		4.2 Pullback Formula
	5 An Application to Congruences for Vector-Valued Modular Forms
	References
Omega Results for Fourier Coefficients of Half-Integral Weight and Siegel Modular Forms
	1 Introduction
	2 Notation and Preliminaries
	3 Proof of Theorem 1.1
	4 Application to Siegel Modular Forms
		4.1 Proof of Theorem 1.2
		4.2 Proof of Theorem 1.3
	References
On Hecke Theory for Hermitian Modular Forms
	1 Introduction
	2 The Hecke Algebra for the Hermitian Modular Group
	3 The Inert Part of the Hecke Algebra
	4 Hermitian Modular Forms
	5 Hecke Operators
	6 The Siegel-Eisenstein Series
	References
Sign Changes of the Ramanujan τ-Function
	1 Introduction
	2 Root Distribution
		2.1 Kostant\'s Theorem
		2.2 The Polynomials Pn(x)
	3 Modular Forms with Real Weight
	4 Patterns of Sign Distribution
	5 Neighbours of Δ
	6 Open Questions
	References
The Central Limit Theorem in Algebra and Number Theory
	1 Introduction
	2 Review of Some Probabilistic Concepts
	3 The Evolution of the Central Limit Theorem
	4 The Evolution of Probabilistic Number Theory
	5 Goncharov\'s Theorem
	6 The Connection to Stirling Numbers of the First Kind
	7 Normal Number of Prime Factors of Fourier Coefficients of Modular Forms
	8 Probabilistic Connections to the Riemann Hypothesis
	9 Concluding Remarks
	References
Rankin–Cohen Brackets and Identities Among Eigenforms II
	1 Introduction and Statement of the Main Result
	2 Preliminaries
		2.1 Eigenforms
		2.2 The Isomorphism
		2.3 L-Functions Attached to Modular Forms
	3 Eisenstein Series
		3.1 Eisenstein Series of Weight 2
	4 Auxiliary Results
	5 Proof of Theorem 2
		5.1 First Case
		5.2 Second Case
		5.3 Third Case
	6 Further Remarks
	References
Rankin–Cohen Type Operators  for Hilbert–Jacobi Forms
	1 Introduction
	2 Preliminaries and Statement of Result
	3 Intermediate Results
	4 Proof of Theorem 1
	5 Concluding Remark
	References
Determining Modular Forms of Half-Integral Weight by Central Values of Convolution L-Functions
	1 Introduction and Statement of Results
	2 Notations and Preliminaries
	3 Rankin–Selberg L-Functions
	4 Approximate Functional Equation
	5 An Auxiliary Theorem
	6 Estimation of the Main Term M|D|(k,ell)
	7 Estimation of the Error Term Eg,|D|(k, ell)
		7.1 Preliminary Results
		7.2 Error Term Estimation
	8 Proof of the Main Theorem
	References
On the Number of Representations of a Natural Number by Certain Quaternary Quadratic Forms
	1 Introduction
	2 Preliminaries and Statement of Results
		2.1 Simplification of Some of the Formulas and Determining Universal Property
		2.2 Remarks on Equivalence of Formulas
	3 Proofs
		3.1 A Basis for M2(48, χ0)
		3.2 A Basis for M2(48, χ8)
		3.3 A Basis for M2(48, χ12)
		3.4 A Basis for M2(48, χ24)
		3.5 Proofs of Theorms
	4 Tables for Theorems 2.1 and 2.3
	References
Identities from Partition of the Symmetric Group Sn
	1 Introduction
	2 Proof of Theorem 1
	3 Proofs of Corollaries 1, 2 and 3
		3.1 Proof of Corollary 1
		3.2 Proof of Corollary 2
		3.3 Proof of Corollary 3
	4 Proofs of Theorem 2 and Corollary 4
		4.1 Proof of Theorem 2
		4.2 Proof of Corollary 4
	5 Proofs of Lemma 1 and Theorems 3, 4, 5
		5.1 Proof of Lemma 1
		5.2 Proof of Theorem 3
		5.3 Proof of Theorem 4
		5.4 Proof of Theorem 5
	References
A Certain Kernel Function for L-Values of Half-Integral Weight Hecke Eigenforms
	1 Introduction
	2 Notations and Preliminaries
	3 Statement of Results
	4 Proof of the Theorem
	5 Applications
	6 Appendix
	References
On Admissible Set of Primes in Real Quadratic Fields
	1 Introduction
	2 Proof of the Theorem 2
	3 Proof of Theorem 3
	References