Poor man's transcendence for Frobenius traces of elliptic curves

TL;DR

This paper proves that the sequence of Frobenius traces of a non-CM elliptic curve over  is transcendental over , using a novel approach involving the 'poor man's ade8le ring' to analyze its properties.

Contribution

It introduces a new method to establish the transcendence of Frobenius trace sequences using the 'poor man's ade8le ring', a novel conceptual framework.

Findings

Frobenius trace sequence is transcendental over .

New approach via the 'poor man's ade8le ring' to study elliptic curve sequences.

Provides insights into the arithmetic nature of Frobenius traces.

Abstract

Let $E$ be an elliptic curve without complex multiplication defined over $\mathbb Q$. Viewing the sequence of its Frobenius traces $(a_p(E))_p$ indexed by primes $p$ as an element in the "poor man's ad\`ele ring", we prove its transcendence over $\mathbb Q$.