We consider the three-dimensional $pm J$ model defined on a simple cubic lattice and study its behavior close to the multicritical Nishimori point where the paramagnetic-ferromagnetic, the paramagnetic-glassy, and the ferromagnetic-glassy transition lines meet in the T-p phase diagram (p characterizes the disorder distribution and gives the fraction of ferromagnetic bonds). For this purpose we perform Monte Carlo simulations on cubic lattices of size $Lle 32$ and a finite-size scaling analysis of the numerical results. The magnetic-glassy multicritical point is found at $p^*=0.76820(4)$, along the Nishimori line given by $2p-1={rm Tanh}(J/T)$. We determine the renormalization-group dimensions of the operators that control the renormalization-group flow close to the multicritical point, $y_1 = 1.02(5)$, $y_2 = 0.61(2)$, and the susceptibility exponent $eta = -0.114(3)$. The temperature and crossover exponents are $ u=1/y_2=1.64(5)$ and $phi=y_1/y_2 = 1.67(10)$, respectively. We also investigate the model-A dynamics, obtaining the dynamic critical exponent $z = 5.0(5)$.
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