Fourier Coefficients of Half-Integral Weight Modular Forms Modulo &#8467

For each prime, let ||_ be an extension to of the usual -adic absolute value on. Suppose g (z) = ₍=₀^ c (n) qⁿ M₊+ (N) is an eigenform whose Fourier coefficients are algebraic integers. Under a mild condition, for all but finitely many primes there are infinitely many square-free integers m for which |c (m) |_ = 1. Consequently we obtain indivisibility results for ``algebraic parts'' of central critical values of modular L-functions and class numbers of imaginary quadratic fields. These results partially answer a conjecture of Kolyvagin regarding Tate-Shafarevich groups of modular elliptic curves. Similar results were obtained earlier by Jochnowitz by a completely different method. Our method uses standard facts about Galois representations attached to modular forms, and pleasantly uncovers surprising Kronecker-style congruences for L-function values. For example if (z) is Ramanujan's cusp form and g (z) =₍=₁^c (n) qⁿ is the cusp form for which L (D, 6) = () D⁶D5! c (D) ², for fundamental discriminants D>0, then for N 1 ₊=-^ c (N-k²) ₃|₍ (-₁ (d) +-₁ (N/d) ) d⁶ 61. 0