Exact critical values of the symmetric fourth $L$ function and vector valued Siegel modular forms

Abstract

Exact critical values of symmetric fourth $L$ function of the Ramanujan Delta function $\Delta$ were conjectured by Don Zagier in 1977. They are given as products of explicit rational numbers, powers of $\pi$, and the cube of the inner product of $\Delta$. In this paper, we prove that the ratio of these critical values are as conjectured by showing that the critical values are products of the same explicit rational numbers, powers of $\pi$, and the inner product of some vector valued Siegel modular form of degree two. Our method is based on the Kim-Ramakrishnan-Shahidi lifting, the pullback formulas, and differential operators which preserve automorphy under restriction of domains. We also show a congruence between a lift and a non-lift. Furthermore, we show the algebraicity of the critical values of the symmetric fourth $L$ function of any elliptic modular form and give some conjectures in general case.