## Journal of the Mathematical Society of Japan

### 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.

#### Article information

Source
J. Math. Soc. Japan, Volume 66, Number 1 (2014), 139-160.

Dates
First available in Project Euclid: 24 January 2014

https://projecteuclid.org/euclid.jmsj/1390600840

Digital Object Identifier
doi:10.2969/jmsj/06610139

Mathematical Reviews number (MathSciNet)
MR3161396

Zentralblatt MATH identifier
1291.11087

#### Citation

IBUKIYAMA, Tomoyoshi; KATSURADA, Hidenori. Exact critical values of the symmetric fourth $L$ function and vector valued Siegel modular forms. J. Math. Soc. Japan 66 (2014), no. 1, 139--160. doi:10.2969/jmsj/06610139. https://projecteuclid.org/euclid.jmsj/1390600840

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