= Solution
Take the constant-parameter subgroup of the local transformation: $\delta\phi=-iq\epsilon\phi$, $\delta\phi^*=iq\epsilon\phi^*$, $\delta A_\mu=0$. The momenta conjugate to the <complex scalar field> <gradients> are $\Pi_\phi^\mu=(D^\mu\phi)^*$ and $\Pi_{\phi^*}^\mu=D^\mu\phi$. <Noether theorem> gives
$$
j^\mu=\Pi_\phi^\mu\frac{\delta\phi}{\epsilon}+\Pi_{\phi^*}^\mu\frac{\delta\phi^*}{\epsilon}=iq[\phi^*D^\mu\phi-\phi(D^\mu\phi)^*]=J^\mu.
$$
The <scalar field> equations are $D_\mu D^\mu\phi+m^2\phi=0$ and their <complex conjugate>. Applying the <product rule>, the gauge-potential terms cancel to give
$$
\partial_\mu J^\mu=iq\left[\phi^*D_\mu D^\mu\phi-\phi(D_\mu D^\mu\phi)^*\right]=iq[-m^2|\phi|^2+m^2|\phi|^2]=\boxed{0}.
$$
This establishes conservation of the <Noether current of scalar electrodynamics> <on shell>, and conservation of its integrated <electric charge> when there is no boundary flux. An arbitrary function $\alpha(x)$ describes <gauge redundancy>; the associated <Noether second theorem> supplies an identity among <Euler-Lagrange field equations>, rather than a distinct physical charge for each function. Indeed its on-shell current can be written $j^\mu[\alpha]=\alpha J^\mu-F^{\mu\nu}\partial_\nu\alpha=-\partial_\nu(\alpha F^{\mu\nu})$. The <antisymmetric tensor> makes its <divergence> vanish. Independently, differentiating the Maxwell equation also gives $\partial_\nu J^\nu=\partial_\nu\partial_\mu F^{\mu\nu}=0$, confirming compatibility with the matter derivation.<Quantum Field Theory by David Tong>[§1.3.4, internal phase symmetry; §6.1, gauge symmetry and its constant subgroup]{ref}
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