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

We consider smooth projective curves C/𝔽 over a finite field and their symmetric squares C (2) . For a global function field K/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 fit there. Our methods also lead to an explicit bound on the number of K-rational points of C (2) satisfying an additional condition. Some of our results apply to arbitrary constant subvarieties of abelian varieties, however we produce examples which show that not all of our stronger conclusions extend.