p-Adic limit of the Fourier coefficients of weakly holomorphic modular forms of half integral weight

Serre obtained the p-adic limit of the integral Fourier coefficients of modular forms on SL₂(ℤ) for p = 2,3,5,7. In this paper, we extend the result of Serre to weakly holomorphic modular forms of half integral weight on Γ₀(4N) for N = 1,2,4. The proof is based on linear relations among Fourier coefficients of modular forms of half integral weight. As applications to our main result, we obtain congruences on various modular objects, such as those for Borcherds exponents, for Fourier coefficients of quotients of Eisentein series and for Fourier coefficients of Siegel modular forms on the Maass Space.