CM congruence and trivial zeros of the Katz $p$-adic $L$-functions for CM fields
CM congruence and trivial zeros of the Katz $p$-adic $L$-functions for CM fields
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CM 域的 Katz $p$-adic $L$-函数的 CM 同余和平凡零点
DOI：
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发表时间：
2022
期刊：
影响因子：
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通讯作者：
Ming
中科院分区：
文献类型：
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作者：
Adel Betina;Ming
利用CM同余研究Katz p-adic L-函数的平凡零点.证明了一般CM域上Katz p-adic L-函数平凡零点的存在性，并在Leopoldt假设下建立了分圆p-adic L-函数平凡零点处的一阶导数公式.证明中的关键要素是CM域的p-adic Kronecker极限公式的一个特例和反分圆p-adic L-函数在平凡零点处的首项公式，通过CM和非CM的Hilbert尖点形式的希达族之间的显式同余关系.
The aim of this paper is to investigate the trivial zeros of the Katz p-adic L-functions by the CM congruence. We prove the existence of trivial zeros of the Katz p-adic L-functions for general CM fields and establish a first derivative formula of the cyclotomic p-adic L-functions at trivial zeros under some Leopoldt hypothesis. The crucial ingredients in our proof are a special case of p-adic Kronecker limit formula for CM fields and a leading term formula of anticyclotomic p-adic L-functions at trivial zeros via the explicit congruences between CM and non-CM Hida families of Hilbert cusp forms.