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

Abstract

Abstract Let f ∈ S k 0 + 2 ⁢ ( Γ 0 ⁢ ( N ⁢ p ) ) ${f\in S_{k_{0}+2}(\Gamma_{0}(Np))}$ be a normalized N-new eigenform with p ∤ N ${p\,{\nmid}\,N}$ and such that a p 2 ≠ p k 0 + 1 ${a_{p}^{2}\neq p^{k_{0}+1}}$ and ord p ⁡ ( a p ) < k 0 + 1 ${\operatorname{ord}_{p}(a_{p})<k_{0}+1}$ . By Coleman’s theory, there is a p-adic family 𝐅 ${\mathbf{F}}$ of eigenforms whose weight k 0 + 2 ${k_{0}+2}$ specialization is f. Let K be a real quadratic field and let ψ be an unramified character of Gal ⁡ ( 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-function ℒ 𝐅 / 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 𝐅 ${\mathbf{F}}$ in classical weight, twisted by ψ. When the character ψ is quadratic, ℒ 𝐅 / K , ψ ${\mathcal{L}_{\mathbf{F}/K,\psi}}$ factors into a product of two Mazur-Kitagawa p-adic L-functions. If, in addition, 𝐅 ${\mathbf{F}}$ has p-new specialization in weight k 0 + 2 ${k_{0}+2}$ , then under natural parity hypotheses we may relate derivatives of each of the Mazur-Kitagawa factors of ℒ 𝐅 / K , ψ ${\mathcal{L}_{\mathbf{F}/K,\psi}}$ at k 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.