Exact critical values of the symmetric fourth <i>L</i> function and vector valued Siegel modular forms
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Exact critical values of symmetric fourth <i>L</i> function of the Ramanujan Delta function Δ were conjectured by Don Zagier in 1977. They are given as products of explicit rational numbers, powers of π, and the cube of the inner product of Δ. 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 π, 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 <i>L</i> function of any elliptic modular form and give some conjectures in general case.
- Journal of the Mathematical Society of Japan
Journal of the Mathematical Society of Japan 66(1), 139-160, 2014
The Mathematical Society of Japan