Integral Prasanna–Venkatesh conjecture for the adjoint regulator

Let $f$ be the modular form, $F$ the field, $p$ and $k$ the prime and weight from Theorem 5.11, and let $u_2(f_F)$, $\Omega_f^+$, $\Omega_f^-$, and $R_{f,\alpha}$ be respectively the congruence invariant, the two periods, and the archimedean regulator of the Beilinson–Flach element attached to $f$ and $\alpha$. Let $T_F$ be the Hecke algebra and $\lambda_{f_F}\colon T_F\to\mathcal O$ the eigensystem associated to $f_F$; write $L_{T_F/\mathcal O}$ for the cotangent complex and $\operatorname{char}$ for the characteristic ideal. The notation $a\sim b$ means equality up to multiplication by an element of $\mathcal O^\times$. Integral Prasanna–Venkatesh conjecture. If the hypotheses of Theorem 5.11 hold and $p>2k-1$, then

$$u_2(f_F)\sim \Omega_f^+\Omega_f^-\cdot R_{f,\alpha}\cdot \operatorname{char}\mathrm{H}_1\left(L_{T_F/\mathcal O}\otimes_{\lambda_{f_F}}\mathcal O\right).$$

This is presented as an integral version of a conjecture of Prasanna and Venkatesh, arising from a formulation of the Bloch–Kato conjecture. The stronger Fontaine–Laffaille bound $p>2k-1$ is required for the relation between the regulator and the special twisted adjoint $L$-value; the conjecture relates the congruence invariant to periods, the Beilinson–Flach regulator, and the defect of the Hecke algebra from being a complete intersection.