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arXiv · 1811.08216

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

Kazim Büyükboduk·Robert Pollack·Shu Sasaki
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Abstract

Let $p$ be an odd prime. Given an imaginary quadratic field $K=\mathbb{Q}(\sqrt{-D_K})$ where $p$ splits with $D_K>3$, and a $p$-ordinary newform $f \in S_k(\Gamma_0(N))$ 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-$\theta$-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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Kazim Büyükboduk, Robert Pollack, Shu Sasaki. 2018-11-20. $p$-adic Gross-Zagier formula at critical slope and a conjecture of Perrin-Riou. https://arxiv.org/abs/1811.08216

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