Kurokawa–Mizumoto congruences and degree-8 L-values

Abstract

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.