Let \(K\) be an imaginary quadratic field and let \(p\ge 5\) be a prime that splits as \(\mathfrak p \mathfrak p^c\) in \(K\). Let \(D_{\infty}\) be the anticyclotomic \(\mathbb Z_p\)-extension of \(K\). Let \(f\) be a normalized \(p\)-ordinary non-CM newform of weight \(k\) for \(\Gamma_0(N)\). Write \(N=p^a N^+ N^-\), where \(N^+\) is divisible only by primes other than \(p\) that split in \(K\) and where \(N^-\) is assumed to be squarefree and is divisible only by primes that are inert in \(K\). Let \(\alpha\) be a ring class character mod \(\mathfrak f \mathfrak p^{\infty}\) for which \(\alpha(\mathfrak p)\ne \alpha(\mathfrak p^c)\). Under certain assumptions on \(\mathfrak f\), the newform \(f\), and the Galois representation attached to \(f\), the authors prove the following: If \(N\) is divisible by an odd number of primes and \(p^2\nmid N\), then the characteristic ideal of a certain Greenberg Selmer group \(\mathcal{X}(f\otimes \alpha/D_{\infty})\) contains the projection \(\mathcal{L}_{f, 0}^{(\alpha)}\) of the Hida-Perrin-Riou \(p\)-adic \(L\)-function after inverting \(p\). If in addition \(p\nmid N\), then \(\mathcal{L}_{f, 0}^{(\alpha)}\) generates this characteristic ideal. If \(N\) is divisible by an even number of primes and \(p\nmid N\), then \(\mathcal{L}_{f, 0}^{(\alpha)}=0\) and \(\mathcal{X}(f\otimes \alpha/D_{\infty})\) has rank one. Moreover, the derivative of the Hida–Perrin-Riou \(p\)-adic \(L\)-function in the cyclotomic direction, restricted to the anticyclotomic line, is contained in the module obtained as a certain regulator times the characteristic ideal of \(\mathcal{X}(f\otimes \alpha/D_{\infty})_{\text{tor}}\) tensored with a finite extension of \(\mathbb Q_p\). Under additional hypotheses, it is shown that this derivative actually generates the module.
A new aspect of the present work is the use of Beilinson-Flach elements along the anticyclotomic tower, which allows both the definite and indefinite cases to be treated simultaneously.