Approximations of the balanced triple product $p$-adic $L$-function

TL;DR

This paper develops an algorithm to approximate the balanced p-adic L-function at a specific point outside its usual domain, using geodesic methods on the Bruhat--Tits tree, with applications to elliptic curves.

Contribution

It introduces a novel algorithm for approximating the balanced p-adic L-function at non-interpolation points, extending previous methods to new cases involving elliptic curves.

Findings

Algorithm successfully approximates the p-adic L-function at (2,1,1).

Method leverages geodesics on the Bruhat--Tits tree for computation.

Applicable to cases with elliptic curves and specific functional equation signs.

Abstract

The main purpose of this note is to provide an algorithm for approximating the value of the balanced $p$-adic $L$-function, as constructed in [Hsi21], at the point $(2,1,1)$, which is lying outside of the interpolation region. The algorithmic procedure is obtained building on the work of [FM14] and considering finite length geodesics on the Bruhat--Tits tree for $GL_2(\mathbb{Q}_p)$. We are interested in the case where at least one of the Hida families is associated with an elliptic curve over the rationals and we further restrict ourselves to the case where only one finite local sign of the functional equation is $-1$.