  We study dynamic correlations for current and mass, as well as the associated
power spectra, in the one-dimensional conserved Manna sandpile. We show that,
in the thermodynamic limit, the variance of cumulative bond current up to time
$T$ grows subdiffusively as $T^{1/2-\mu}$ with the exponent $\mu \ge 0$
depending on the density regimes considered and, likewise, the power spectra of
current and mass at low frequency $f$ varies as $f^{1/2+\mu}$ and
$f^{-3/2+\mu}$, respectively; our theory predicts that, far from criticality,
$\mu = 0$ and, near criticality, $\mu = (\beta+1)/2 \nu_{\perp} z > 0$ with
$\beta$, $\nu_{\perp}$ and $z$ being the order-parameter, correlation-length
and dynamic exponents, respectively. The anomalous suppression of fluctuations
near criticality signifies a "dynamic hyperuniformity", characterized by a set
of fluctuation relations, in which current, mass and tagged-particle
displacement fluctuations are shown to have a precise quantitative relationship
with the density-dependent activity (or, it's derivative). In particular, the
relation, ${\mathcal{D}}_s(\bar{\rho}) = a(\bar{\rho}) / \bar{\rho}$, between
the self-diffusion coefficient ${\mathcal{D}}_s(\bar{\rho})$, activity
$a(\bar{\rho})$ and density $\bar{\rho}$ explains a previous simulation
observation [Eur. Phys. J. B \textbf{72}, 441 (2009)] that, near criticality,
the self-diffusion coefficient in the Manna sandpile has the same scaling
behavior as the activity.
