### Fourier coefficients of Half-integral weight modular forms modulo ell, by Ken Ono and Christopher Skinner

For each prime $\ell$, let $|\cdot|_\ell$ be an extension to $\overline \Q$
of the usual $\ell$-adic absolute value on $\Q$. Suppose $g(z) =
\sum_{n=0}^\infty c(n)q^n \in M_{k+\half}(N)$ is an eigenform whose
Fourier coefficients are algebraic integers.
Under a mild condition,
for all but finitely many primes $\ell$ there are infinitely many
square-free integers $m$ for which $|c(m)|_\ell = 1$. 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 $\Delta(z)$ is Ramanujan's cusp form and
$g(z)=\sum_{n=1}^{\infty}c(n)q^n$ is the cusp form
for which
$$L(\Delta_D,6)=\fracwithdelims(){\pi}{D}^6\frac{\sqrt{D}}{5!}\frac{\langle \Delta(z),\Delta(z)\rangle}
{\langle g(z),g(z)\rangle}\cdot c(D)^2,
$$
for fundamental discriminants $D>0,$ then for
$N\geq 1$
$$
\sum_{k=-\infty}^\infty c(N-k^2) \equiv \half \sum_{d|N}(\chi_{-1}(d)+\chi_{-1}(N/d))d^6
\pmod {61}. \tag{0}
$$

Ken Ono and Christopher Skinner <ono@math.ias.edu,ono@math.psu.edu,cmcls@math.princeton.edu>