It has been well established that congruences between automorphic forms have far-reaching applications in arithmetic. In this paper, we construct congruences for Siegel–Hilbert modular forms defined over a totally real field of class number 1. As an application of this general congruence, we produce congruences between paramodular Saito–Kurokawa lifts and non-lifted Siegel modular forms. These congruences are used to produce evidence for the Bloch–Kato conjecture for elliptic newforms of square-free level and odd functional equation.

中文翻译：

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Siegel 模数形式的同余质数及其在 Bloch-kato 猜想中的应用

众所周知，自守形式之间的同余在算术中具有深远的应用。在本文中，我们为定义在第 1 类的完全实数域上的 Siegel-Hilbert 模形式构造同余。作为这种一般同余的应用，我们在准模 Saito-Kurokawa 升降机和非升降 Siegel 模形式之间产生同余。这些同余用于为无平方能级和奇函数方程的椭圆新形式的 Bloch-Kato 猜想提供证据。