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Letf  ∈    S    k  0   +  2    ⁢   (    Γ  0   ⁢   (   N  ⁢  p   )    )${f\in S_{k_{0}+2}(\Gamma_{0}(Np))}$    be a normalized  N -new eigenform withp   ∤  N    ${p\,{\nmid}\,N}$    and such thata  p  2   ≠   p    k  0   +  1${a_{p}^{2}\neq p^{k_{0}+1}}$    andord  p   ⁡   (   a  p   )    &lt;    k  0   +  1${\operatorname{ord}_{p}(a_{p})&lt;k_{0}+1}$   . By Coleman’s theory, there is a  p -adic family   ${\mathbf{F}}$    of eigenforms whose weightk  0   +  2    ${k_{0}+2}$    specialization is  f . Let  K  be a real quadratic field and let ψ be an unramified character ofGal  ⁡   (    K  ¯   /  K   )${\operatorname{Gal}(\overline{K}/K)}$   . Under mild hypotheses on the discriminant of  K  and the factorization of  N , we construct a  p -adic  L -functionℒ/  K   ,  ψ${\mathcal{L}_{\mathbf{F}/K,\psi}}$    interpolating the central critical values of the Rankin  L -functions associated to the base change to  K  of the specializations of   ${\mathbf{F}}$    in classical weight, twisted by ψ. When the character ψ is quadratic,ℒ/  K   ,  ψ${\mathcal{L}_{\mathbf{F}/K,\psi}}$    factors into a product of two Mazur-Kitagawa  p -adic  L -functions. If, in addition,   ${\mathbf{F}}$    has  p -new specialization in weightk  0   +  2    ${k_{0}+2}$   , then under natural parity hypotheses we may relate derivatives of each of the Mazur-Kitagawa factors ofℒ/  K   ,  ψ${\mathcal{L}_{\mathbf{F}/K,\psi}}$    atk  0    ${k_{0}}$    to Bloch–Kato logarithms of Heegner cycles. On the other hand the derivatives of our  p -adic  L -functions encodes the position of the so called Darmon cycles. As an application we prove rationality results about them, generalizing theorems of Bertolini–Darmon, Seveso, and Shahabi.

Modular p -adic L -functions attached to real quadratic fields and arithmetic applications