Journal of the Mathematical Society of Japan

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

Tomoyoshi IBUKIYAMA and Hidenori KATSURADA

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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

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

First available in Project Euclid: 24 January 2014

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Primary: 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
Secondary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11F11: Holomorphic modular forms of integral weight 11F60: Hecke-Petersson operators, differential operators (several variables) 11F33: Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50]

Siegel modular forms critical values $L$ functions differential operators


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.

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