# On a non-critical symmetric square $L$-value of the congruent number elliptic curves

## TL;DR

This paper provides a straightforward proof of a formula relating the symmetric square L-value at 3 for congruent number elliptic curves to elliptic trilogarithms evaluated at torsion points, revealing new connections in number theory.

## Contribution

It introduces a simple proof of a formula linking the symmetric square L-value to elliptic trilogarithms on congruent number elliptic curves.

## Key findings

- Explicit formula for L(Sym^2(E_d),3) in terms of elliptic trilogarithms
- Connection between L-values and torsion points on elliptic curves
- Simplification of previous complex proofs in the area

## Abstract

The congruent number elliptic curves are defined by $E_d: y^2=x^3-d^2x$, where $d\in \mathbb{N}.$ We give a simple proof of a formula for $L(\mathrm{Sym}^2(E_d),3)$ in terms of the determinant of the elliptic trilogarithm evaluated at some degree zero divisors supported on the torsion points on $E_d(\overline{\mathbb{Q}})$.