p-adic Gross-Zagier formula at critical slope and a conjecture of Perrin-Riou

2018/11/20 by Büyükboduk, Kazim, Pollack, Robert, Sasaki, Shu
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1811.08216

Abstract

Let p be an odd prime. Given an imaginary quadratic field K=ℚ(√(-D_K)) where p splits with D_K>3, and a p-ordinary newform f ∈ S_k(Γ₀(N)) such that N verifies the Heegner hypothesis relative to K, we prove a p-adic Gross-Zagier formula for the critical slope p-stabilization of f (assuming that it is non-θ-critical). In the particular case when f=f_A is the newform of weight 2 associated to an elliptic curve A that has good ordinary reduction at p, this allows us to verify a conjecture of Perrin-Riou. The p-adic Gross-Zagier formula we prove has applications also towards the Birch and Swinnerton-Dyer formula for elliptic curves of analytic rank one.

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