Independence between coefficients of two modular forms

Dohoon Choi, Subong Lim

Research output: Contribution to journalArticle

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 Sep 1

Keywords

  • Fourier coefficient
  • Galois representation
  • Modular form

ASJC Scopus subject areas

  • Algebra and Number Theory

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