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Let $\mathscr{P}$ be a symplectic polar space over a finite field$\mathbb{F}_q$, and $\mathscr{P}_m$ denote the collection of all$k$-dimensional totally isotropic subspace in $\mathscr{P}$. Let$\mathscr{F}_1\subset\mathscr{P}_{m_1}$ and$\mathscr{F}_2\subset\mathscr{P}_{m_2}$ satisfy $\dim(F_1\cap F_2)\ge t$ forany $F_1\in\mathscr{F}_1$ and $F_2\in\mathscr{F}_2$. We say they are cross$t$-intersecting families. Moreover, we say they are trivial if each member ofthem contains a fixed $t$-dimensional totally isotropic subspace. In thispaper, we show that cross $t$-intersecting families with maximum product ofsizes are trivial. We also describe the structure of non-trivial$t$-intersecting families with maximum product of sizes.

Cross $t$-intersecting families for symplectic polar spaces