Plectic points and Hida-Rankin 𝑝-adic 𝐿-functions

Hernández Barrios, Víctor; Molina Blanco, Santiago

doi:10.1090/tran/9432

Plectic points and Hida-Rankin $p$-adic $L$-functions
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by Víctor Hernández Barrios and Santiago Molina Blanco;

Trans. Amer. Math. Soc.

DOI: https://doi.org/10.1090/tran/9432

Published electronically: May 1, 2025

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Abstract:

Plectic points were introduced by Fornea and Gehrmann [Adv. Math. 414 (2023)] as certain tensor products of local points on elliptic curves over arbitrary number fields $F$. They are constructed in rank $r\leq [F:\mathbb {Q}]$-situations, and they conjecturally come from $p$-adic regulators of basis of the Mordell-Weil group defined over dihedral extensions of $F$.

In this article we define anticyclotomic $p$-adic L-functions with variables of type weight and level attached to a family of overconvergent modular symbols defined over totally real fields $F$ and a quadratic extension $K/F$. Their restriction to the weight space provides Hida-Rankin $p$-adic L-functions.

If such a family passes through an overconvergent modular symbol attached to a modular elliptic curve $E$, we obtain a $p$-adic Gross-Zagier formula that computes higher derivatives of such Hida-Rankin $p$-adic L-functions in terms of plectic points. This result generalizes that of Bertolini and Darmon [Ann. of Math. (2) 170 (2009), pp. 343–370], which has been key to the recent approach towards the algebraicity of Darmon points.

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Plectic points and Hida-Rankin $p$-adic $L$-functions
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Abstract: