Improved Beilinson–Flach class conjecture

Let $\mathcal{C}$ be the family locus under consideration, let $\kappa_{{\mathbf{g}},{\mathbf h}}\vert_{\mathcal{C}}$ be the restriction of the Beilinson–Flach cohomology class, and let $H^1_{\operatorname{bal}}(\mathbb{Q},\mathbb{V}_{{\mathbf{g}}{\mathbf h}}^\dagger\vert_{\mathcal{C}})$ be the balanced Selmer group. Improved Beilinson–Flach class conjecture. There exists a cohomology class $\widehat{\kappa}_{{\mathbf{g}},{\mathbf h}}\in H^1_{\operatorname{bal}}(\mathbb{Q},\mathbb{V}_{{\mathbf{g}}{\mathbf h}}^\dagger\vert_{\mathcal{C}})$ such that

$$\kappa_{{\mathbf{g}},{\mathbf h}}\vert_{\mathcal{C}} = \left( 1 - \chi_g(p)\frac{a_p({\mathbf h})}{a_p({\mathbf{g}})} \right) \widehat{\kappa}_{{\mathbf{g}},{\mathbf h}}.$$

This is the expected improved cohomology class corresponding to the exceptional-zero factor in the interpolation property of the Beilinson–Flach element; its construction is left for future work.