Ap\'ery limits for elliptic $L$-values

TL;DR

This paper introduces a construction that realizes specific quotients of elliptic curve L-values as Apéry limits derived from recurrence solutions.

Contribution

It establishes a method to connect elliptic curve L-values with Apéry limits through recurrence equations.

Findings

Identifies conditions for Apéry limits to equal elliptic L-value quotients

Provides explicit constructions linking recurrences to elliptic L-values

Enhances understanding of the relationship between recurrence solutions and special values of L-functions

Abstract

For an (irreducible) recurrence equation with coefficients from $\mathbb Z[n]$ and its two linearly independent rational solutions $u_n,v_n$, the limit of $u_n/v_n$ as $n\to\infty$, when exists, is called the Ap\'ery limit. We give a construction that realises certain quotients of $L$-values of elliptic curves as Ap\'ery limits.