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Let f be a Hecke eigenform of weight k, level 1, genus 1. Let \(E^k_{2,1}(f)\) be its genus-2 Klingen–Eisenstein series. Let F be a genus-2 cusp form whose Hecke eigenvalues are congruent modulo \({\mathfrak {q}}\) to those of \(E^k_{2,1}(f)\), where \({\mathfrak {q}}\) 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 \({\mathfrak {q}}\) should also appear in the denominator of the “algebraic part” of the rightmost critical value of the tensor product L-function \(L(s,f\otimes F)\), 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.

Kurokawa–Mizumoto congruences and degree-8 L-values