Exact critical values of the symmetric fourth L function and vector valued Siegel modular forms
Exact critical values of the symmetric fourth L function and vector valued Siegel modular forms
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对称第四 L 函数和向量值 Siegel 模形式的精确临界值
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
H. Katsurada
中科院分区：
文献类型：
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作者：
T. Ibukiyama;H. Katsurada
. 1977年Don Zagier证明了Ramanujan Delta函数的对称第四L函数的精确临界值。它们是显式有理数、π的幂和π的内积的立方的乘积。本文证明了这些临界值之比是由相同的显式有理数、π的幂和某个二次向量值Siegel模形式的内积的乘积所确定的.我们的方法是基于Kim-Ramakrishnan-Shahidi提升，拉回公式，并保持自同构的域的限制下的微分算子。我们还显示了电梯和非电梯之间的一致性。进一步证明了任意椭圆模形式的对称第四L函数临界值的代数性，并给出了一般情况下的一些证明。
. Exact critical values of symmetric fourth L function of the Ramanujan Delta function ∆ 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 ∆. 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 L function of any elliptic modular form and give some conjectures in general case.