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We consider smooth projective curves C/$\mathbb{F}$ over a finite field andtheir symmetric squares $C^{(2)}$. For a global function field $K/\mathbb{F}$,we study the $K$-rational points of $C^{(2)}$. We describe the adelic points of$C^{(2)}$ surviving Frobenius descent and how the $K$-rational points fitthere. Our methods also lead to an explicit bound on the number of $K$-rationalpoints of $C^{(2)}$ satisfying an additional condition. Some of our resultsapply to arbitrary constant subvarieties of abelian varieties, however weproduce examples which show that not all of our stronger conclusions extend.

Rational points on symmetric squares of constant algebraic curves over function fields