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Half-integral weight p-adic coupling of weakly holomorphic and holomorphic modular forms

Research in Number Theory20151:26

  • Received: 17 June 2015
  • Accepted: 5 October 2015
  • Published:


In this paper, we consider p-adic limits of \(\beta ^{-n}g|U_{p^{2}}^{n}\) for half-integral weight weakly holomorphic Hecke eigenforms g with eigenvalue λ p =β+β under \(T\phantom {\dot {i}\!}_{p^{2}}\) and prove that these equal classical Hecke eigenforms of the same weight. This result parallels the integral weight case, but requires a much more careful investigation due to a more complicated structure of half-integral weight weakly holomorphic Hecke eigenforms.


  • Weakly holomorphic modular forms
  • p-adic congruences
  • Half-integral weight
  • Hecke eigenforms
  • Shimura lift
  • Shintani lift