Number Theory: Arithmetic in Shangri-La

The following sections are included:

• PREFACE

• CONTENTS

The following sections are included:

• Definitions and basic properties
  • Elliptic modular forms
  • Siegel modular forms of degree 2
  • Jacobi forms
  • Jacobi forms with levels
  • Theta decomposition
  • Structure theorem for weak Jacobi forms

• Applications
  • Structure of Siegel modular forms
  • Formal series of Jacobi forms
  • Index-Level change of Jacobi forms

• References

The following sections are included:

• Theme and results
  • Diophantine inequalities
  • Additive cubic forms
  • Linear forms in primes
  • Further applications
  • A related diophantine inequality

• The Fourier transform method
  • Some classical integrals
  • Counting solutions of diophantine inequalities
  • Weighted counting
  • The central interval
  • The interference principle

• Classical mean square methods
  • Plancherel's identity
  • Some mean values
  • The amplification technique
  • Linear forms in primes
  • Bessel's inequality

• Semi-classical averaging
  • Another mean square approach
  • Exponential sums over test sequences
  • Potential applications

• Fourier analysis of exceptional sets
  • An illustrative example
  • A quadratic average
  • Some brief heckling
  • An inequality involving quadratic polynomials
  • An application of Vinogradov's method
  • Linear forms in primes, yet again

• Outstanding arts
  • Smooth cubic Weyl sums
  • Senary cubic forms
  • Two technical estimates
  • The lower bound variant
  • An auxiliary inequality
  • Additive forms of large degree
  • Proof of Theorem 1.8
  • Proof of Theorem 1.9

• An appendix on inhomogeneous polynomials
  • The counting integral
  • The central interval
  • The complementary compositum

• References

The following sections are included:

• Downloading updates

• References

The following sections are included:

• Introduction

• Notations and Some Lemmas

• Proofs

• Further Remark

• References

The following sections are included:

• The matrix of a finite abelian group
  • Matrix of a finite group
  • Matrix of a finite abelian group
  • Matrix of a cyclic group
  • The group ring of a cyclic group and the algebra of circulants
  • The group ring F[G] of a finite abelian group G

• Finite Fourier Transform associated with a finite abelian group
  • Generalized Finite Fourier Transform
  • Case of a cyclic group: Finite Fourier Transform

• Hamming weight and Generalized Finite Fourier Transform

• The matrix of a finite group
  • An example: 𝔖₃
  • The general case
  • Frobenius

• References

The following sections are included:

• Introduction

• Quadratic forms

• Even positive definite quaternary quadratic form with prime discriminant

• A modification of Eichler's sum

• References

The following sections are included:

• Introduction

• Auxiliary results

• Multilinear Weyl sums

• Proof of Theorem 1.2

• References

The following sections are included:

• Introductions

• Theorems

• References

The following sections are included:

• Introduction

• The least quadratic non-residue
  • Estimates for character sums and individual bounds for n_χ
  • Linnik's large sieve and almost bound for n_χ

• Automorphic L-functions of GL_m(𝔸_𝕂)
  • The analytic strong multiplicity one theorem for GL_m(𝔸_𝕂)
  • Sign changes of the coefficients of L-functions for GL_m(𝔸_ℚ)

• Classic modular forms
  • The first negative Hecke eigenvalue
  • Statistic study of the first sign-change
  • Recognition of newforms by signs of Hecke eigenvalues
  • Recognition of newforms by values of Hecke eigenvalues
  • Matching signs of Hecke eigenvalues of two newforms
  • The number of Hecke eigenvalues of same signs

• Symmetric square L-functions
  • The first negative coefficient of symmetric square L-functions
  • Positive and negative values over a positive density of primes
  • Ω_±-results over integers
  • The number of coefficients of L(s, sym²f) of same signs

• Half-integral weight case
  • Theta functions
  • Half-integral weight forms and Shimura's theory
  • Sign-change problem of half-integral weight cusp forms

• References

The following sections are included:

• Integral power sums of cusp forms

• Symmetric power L-functions and their Rankin-Selberg L-functions

• Main ingredients in the proof

• Further refinement

• Further Discussion

• References

The following sections are included:

• The twists of Hessian elliptic curves

• The proof of Theorem 1.2

• The twists over cyclic cubic fields

• The elliptic 3-folds
  • The general case
  • The case of the twists over relative cyclic cubic extensions

• References

The following sections are included:

• Introduction

• Proof of Theorem 1

• Proof of Theorem 2

• Applications and duality

• References

The following sections are included:

• Congruences

• Identities

• Diophantine equations

• Algebraic K-theory

• References

The following sections are included:

• Introduction

• Conjectures on number-theoretic sequences
  • Conjectures on sequences involving primes
  • Conjectures on other number-theoretic sequences

• Conjectures on combinatorial sequences

• References

The following sections are included:

• INDEX