This book collects the papers presented at the Conference on Number Theory, held at the Kerala School of Mathematics, Kozhikode, Kerala, India, from December 10–14, 2018. The conference aimed at bringing the active number theorists and researchers in automorphic forms and allied areas to demonstrate their current research works. This book benefits young research scholars, postdoctoral fellows, and young faculty members working in these areas of research.
Author(s): B. Ramakrishnan, Bernhard Heim, Brundaban Sahu
Series: Springer Proceedings in Mathematics & Statistics, 340
Publisher: Springer Singapore
Year: 2021
Language: English
Pages: 233
City: Singapore
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
