Kurokawa–Mizumoto congruences and degree-8 L-values

Let f be a Hecke eigenform of weight k, level 1, genus 1. Let E2,1k(f) be its genus-2 Klingen–Eisenstein series. Let F be a genus-2 cusp form whose Hecke eigenvalues are congruent modulo q to those of E2,1k(f), where q is a “large” prime divisor of the algebraic part of the rightmost critical value of the symmetric square L-function of f. We explain how the Bloch–Kato conjecture leads one to believe that q should also appear in the denominator of the “algebraic part” of the rightmost critical value of the tensor product L-function L(s, f⊗ F) , i.e. in an algebraic ratio obtained from the quotient of this with another critical value. Using pullback of a genus-5 Siegel–Eisenstein series, we prove this, under weak conditions.