A closure operator respecting the modular $j$-function

Abstract

We give three equivalent characterisations of the natural closure operator on a field equipped with functions replicating the algebraic behaviour of the modular j-function and its derivatives, following similar work on exponential fields. As an application of these results, we give unconditional cases of a theorem of Eterović-Herrero on the Existential Closedness problem for the complex j-function which previously assumed the modular version of Schanuel's conjecture.

A differential approach to the Ax-Schanuel, I

“…Given a differential field subfield K of U and υ a tuple of elements from U, the complete type of υ over K, denoted tp(υ/K), is the set of all L m -formulas with parameters from K that υ satisfies. It is not hard to see that the set 4 The description given here is not a first order axiomatization. We refer the reader to [24] for the basic model theory of m-DCF0.…”

“…Over the past several years, in a series of works Aslanyan, and later Aslanyan, Kirby and Eterović [1,2,5,6,4] develop the connection between Ax-Schanuel type transcendence statements and the existential closedness of certain reducts of differentially closed fields related to equations satisfied by the j-function. This series of work builds on the earlier program of Kirby, Zilber and others mainly around the exponential function, see e.g.…”

“…Intermediate differential algebraic results from Ax's work are utilized in the program studying existential closedness results around the exponential function. A different approach was required in [1,2,5,6,4] for studying the j-function, in part since the intermediate results of the functional transcendence results of [31] take place in the o-minimal rather than abstract differential setting. For instance, the motivation for a differential algebraic proof of Ax-Schanuel results and this issue is pointed out specifically following Theorem 1.3 of [5] and in Section 4.4 of [4].…”

“…A different approach was required in [1,2,5,6,4] for studying the j-function, in part since the intermediate results of the functional transcendence results of [31] take place in the o-minimal rather than abstract differential setting. For instance, the motivation for a differential algebraic proof of Ax-Schanuel results and this issue is pointed out specifically following Theorem 1.3 of [5] and in Section 4.4 of [4]. Our results open up the possibility of adapting a similar approach to existential closedness around more general automorphic functions since the general technique of our proof is differential algebraic along lines similar to Ax's work.…”