The Sato$$--$$Tate distributions for genus $$2$$ curves (conjecturally) describe the statistics of numbers of rational points on the curves. In this talk, we explicitly compute the auto-correlation functions of Sato$$--$$Tate distributions for genus $$2$$ curves as sums of irreducible characters of symplectic groups. Our computations bring about families of identities involving irreducible characters of symplectic groups, which have interest in their own rights.

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