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arXiv 0709.1645math.NT
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关于Zeta函数和Siegel模形式族

On Zeta Functions and Families of Siegel Modular Forms

Alexei Panchishkin

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in English and in Russian, 2 figures
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摘要：

设$p$是素数，$Γ=\Sp_g(\Z)$是亏格为$g$的Siegel模群.我们研究$p$ -adic族的zeta函数和Siegel模形式.用附在$\Sp_g$上的动机$L$ -函数来描述Siegel模形式的$L$ -函数，并给出它们的解析性质.讨论了旋量$L$ -函数和$p$ -adic构造的临界值.建立了高等亏格的Rankin引理。给出了从$ GSp_{2m} \times GSp_{2m}$提升到$GSp_{4m}$（属$g=4m$）的一般猜想.利用Ikeda-Miyawaki构造给出了Siegel模形式的$p$ -adic族的构造.

英文摘要：

Let $p$ be a prime, and let $Γ=\Sp_g(\Z)$ be the Siegel modular group of genus $g$. We study $p$-adic families of zeta functions and Siegel modular forms. $L$-functions of Siegel modular forms are described in terms of motivic $L$-functions attached to $\Sp_g$, and their analytic properties are given. Critical values for the spinor $L$-functions and $p$-adic constructions are discussed. Rankin's lemma of higher genus is established. A general conjecture on a lifting from $ GSp_{2m} \times GSp_{2m}$ to $GSp_{4m}$ (of genus $g=4m$) is formulated. Constructions of $p$-adic families of Siegel modular forms are given using Ikeda-Miyawaki constructions.

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