Independence between coefficients of two modular forms

Dohoon Choi, Subong Lim

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)

Abstract

Let k be an even integer and Sk be the space of cusp forms of weight k on SL2(Z). Let S=⊕k∈2ZSk. For f,g∈S, we let R(f,g)be the set of ratios of the Fourier coefficients of f and g defined by R(f,g):={x∈P1(C)|x=[af(p):ag(p)]for some primep}, where af(n)(resp. ag(n))denotes the nth Fourier coefficient of f (resp. g). In this paper, we prove that if f and g are nonzero and R(f,g)is finite, then f=cg for some constant c. This result is extended to the space of weakly holomorphic modular forms on SL2(Z). We apply it to study the number of representations of a positive integer by a quadratic form.

Original languageEnglish
Pages (from-to)298-315
Number of pages18
JournalJournal of Number Theory
Volume202
DOIs
Publication statusPublished - 2019 Sept

Bibliographical note

Funding Information:
This paper was supported by Samsung Research Fund, Sungkyunkwan University, 2018. The authors are grateful to the referee for helpful comments and corrections. The authors also thank Jeremy Rouse for useful comments on the previous version of this paper.

Publisher Copyright:
© 2019 Elsevier Inc.

Keywords

  • Fourier coefficient
  • Galois representation
  • Modular form

ASJC Scopus subject areas

  • Algebra and Number Theory

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