Under relatively mild and natural conditions, we establish integral period relations for the (real or imaginary) quadratic base change of an elliptic cusp form. This answers a conjecture of Hida regarding the congruence ideal controlling the congruences between this base change and other eigenforms which are not base change. As a corollary, we establish the Bloch–Kato conjecture for adjoint modular Galois representations twisted by an even quadratic character. In the odd case, we formulate a conjecture linking the degree two topological period attached to the base change Bianchi modular form, the cotangent complex of the corresponding Hecke algebra and the archimedean regulator attached to some Beilinson–Flach element.

在相对温和和自然的条件下，我们建立了椭圆尖顶形式的（实或虚）二次基变化的积分周期关系。这回答了飞驒关于控制该碱基变化与其他非碱基变化的本征形式之间的一致性的全同理想的猜想。作为推论，我们为由偶二次字符扭曲的伴随模伽罗瓦表示建立了 Bloch-Kato 猜想。在奇怪的情况下，我们提出了一个猜想，将附加于基变化 Bianchi 模形式的二次拓扑周期、相应的 Hecke 代数的余切复数和附加于某个 Beilinson-Flach 元素的阿基米德调节器联系起来。