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On the number of prime divisors and radicals of non-zero Fourier coefficients of Hilbert cusp forms

Sunil L Naik

Abstract

In this article, we derive lower bounds for the number of distinct prime divisors of families of non-zero Fourier coefficients of non-CM primitive cusp forms and more generally of non-CM primitive Hilbert cusp forms. In particular, for the Ramanujan $Δ$-function, we show that for any $ε> 0$, there exist infinitely many natural numbers $n$ such that $τ(p^n)$ has at least $$ 2^{(1-ε) \frac{\log n}{\log\log n}} $$ distinct prime factors for almost all primes $p$. This improves and refines the existing bounds. We also study lower bounds for absolute norms of radicals of non-zero Fourier coefficients of Modular forms alluded to above.

On the number of prime divisors and radicals of non-zero Fourier coefficients of Hilbert cusp forms

Abstract

In this article, we derive lower bounds for the number of distinct prime divisors of families of non-zero Fourier coefficients of non-CM primitive cusp forms and more generally of non-CM primitive Hilbert cusp forms. In particular, for the Ramanujan -function, we show that for any , there exist infinitely many natural numbers such that has at least distinct prime factors for almost all primes . This improves and refines the existing bounds. We also study lower bounds for absolute norms of radicals of non-zero Fourier coefficients of Modular forms alluded to above.
Paper Structure (26 sections, 17 theorems, 115 equations)

This paper contains 26 sections, 17 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Statements of Results for Hilbert Modular forms
  3. Notations and Preliminaries
  4. Prerequisites from elliptic modular forms
  5. Prerequisites from Hilbert cusp forms
  6. Height of an algebraic number
  7. Prerequisites from Lucas sequence
  8. Statement of the Frey's ABC-conjecture
  9. Number of prime factors of Fourier coefficients
  10. Proof of Theorem \ref{['tau']}
  11. Proof of Remark \ref{['rmk1.1']}
  12. Proof of Theorem \ref{['EHE']}
  13. Proof of Theorem \ref{['thm1']}
  14. Radical of Fourier coefficients
  15. Proof of Corollary \ref{['cor1']} and Corollary \ref{['cor2']}
  16. ...and 11 more sections

Key Result

Theorem 1

For any $\epsilon >0$, there exists infinitely many natural numbers $n$ such that for almost all primes $p$.

Theorems & Definitions (31)

  • Theorem 1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 2
  • Corollary 3
  • Remark 1.4
  • Theorem 4
  • Corollary 5
  • Theorem 6
  • ...and 21 more