A closure operator respecting the modular $j$-function

TL;DR

This paper proves certain cases of the Existential Closedness problem for the modular j-function by establishing a natural closure operator in fields with functions mimicking its algebraic properties.

Contribution

It introduces a new closure operator for fields with functions similar to the modular j-function, advancing understanding of its algebraic and model-theoretic properties.

Findings

Unconditional cases of the Existential Closedness problem are proven.

A natural closure operator is defined in fields with modular j-function-like functions.

A method to find convenient generators in finitely generated fields is developed.

Abstract

We prove some unconditional cases of the Existential Closedness problem for the modular $j$-function. For this, we show that for any finitely generated field we can find a "convenient" set of generators. This is done by showing that in any field equipped with functions replicating the algebraic behaviour of the modular $j$-function and its derivatives, one can define a natural closure operator in three equivalent different ways.